# ParMooN -- A modernized program package based on mapped finite elements

*Authors*

- Wilbrandt, Ulrich
- Bartsch, Clemens
- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - Alia, Najib
- Anker, Felix
- Blank, Laura
- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Ganesan, Sashikumaar
- Giere, Swetlana
- Matthies, Gunar
- Meesala, Raviteja
- Shamim, Abdus
- Venkatensan, Jagannath
- John, Volker

ORCID: 0000-0002-2711-4409

*2010 Mathematics Subject Classification*

- 65Y05

*Keywords*

- Mapped finite elements, Geometric multigrid method, Parallelization

*DOI*

*Appeared in*

- Comput. Math. Appl., 74 (2017), pp. 74--88, DOI 10.1016/j.camwa.2016.12.020 .

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# Existence of weak solutions for improved Nernst--Planck--Poisson models of compressible reacting electrolytes

*Authors*

- Dreyer, Wolfgang
- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500 - Gajewski, Paul
- Guhlke, Clemens

*2010 Mathematics Subject Classification*

- 35Q35 76T30 78A57 35Q30 76N10 35M33 35D30 35B45

*2008 Physics and Astronomy Classification Scheme*

- 82.45.Gj 82.45.Mp 82.60.Lf

*Keywords*

- Electrolyte, electrochemical interface, chemical reaction, compressible fluid, Navier-Stokes equations, advection-diffusion-reaction equations, PDE system of mixed-type, a-priori estimates, weak solution

*Abstract*

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# Global existence results for viscoplasticity at finite strain

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Rossi, Riccarda
- Savaré, Giuseppe

*2010 Mathematics Subject Classification*

- 74C20 74H20 35Q74 49S05

*Keywords*

- Viscoplasticity, gradient plasticity with hardening, multiplicative decomposition, energy-dissipation principle for, generalized metric gradient systems, left-invariant dissipation potential, non-convex energy functional

*Abstract*

*generalized gradient system*, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite-strain elasticity as well as the multiplicative decomposition of finite-strain plasticity. Moreover, the dissipation potential depends on the left-invariant plastic rate and thus, depends on the plastic state variable.

The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energy-dissipation-balance (EDB) and energy-dissipation-inequality (EDI) solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory.

*Appeared in*

- Arch. Ration. Mech. Anal., 227 (2018), pp. 423--475 (published online on 20.09.2017), DOI 10.1007/s00205-017-1164-6 .

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# Solvability of an unsaturated porous media flow problem with thermomechanical interaction

*Authors*

- Albers, Bettina

ORCID: 0000-0003-4460-9152 - Krejčí, Pavel
- Rocca, Elisabetta

*2010 Mathematics Subject Classification*

- 76S05 74H20 74N30

*Keywords*

- Porous Media, Existence of solutions, Hysteresis

*Abstract*

A PDE system consisting of the momentum balance, mass balance, and energy balance equations for displacement, capillary pressure, and temperature as a model for unsaturated fluid flow in a porous viscoelastoplastic solid is shown to admit a solution under appropriate assumptions on the constitutive behavior. The problem involves two hysteresis operators accounting for plastic and capillary hysteresis.

*Appeared in*

- SIAM J. Math. Anal., 48:6 (2016) pp. 4175--4201, changed name of first author to B. Detmann.

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# On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Rautenberg, Carlos N.

ORCID: 0000-0001-9497-9296

*2010 Mathematics Subject Classification*

- 47J20 49J40 49M15 65J15 65K10

*Keywords*

- Quasi-variational inequality, gradient constraints, obstacle problem, semismooth Newton method

*Abstract*

A class of abstract nonlinear evolution quasi-variational inequality (QVI) problems in function space is considered. The abstract framework developed in this paper includes constraint sets of obstacle and gradient type. The paper address the existence, uniqueness and approximation of solutions when the constraint set mapping is of a special form. Uniqueness is addressed through contractive behavior of a nonlinear mapping whose fixed points are solutions to the QVI. An axiomatic semi-discrete approximation scheme is developed, which is proven to be convergent and which is numerically implemented. The paper ends by a report on numerical tests for several nonlinear constraints of gradient-type.

*Appeared in*

- Port. Math., 74 (2017), pp. 1--35.

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# A class of stochastic algorithms for the Wigner equation

*Authors*

- Wagner, Wolfgang
- Muscato, Orazio

*2010 Mathematics Subject Classification*

- 65C05 60J25 81Q05

*Keywords*

- Wigner equation, stochastic algorithms, numerical experiments

*DOI*

*Abstract*

A class of stochastic algorithms for the numerical treatment of the Wigner equation is introduced. The algorithms are derived using the theory of pure jump processes with a general state space. The class contains several new algorithms as well as some of the algorithms previously considered in the literature. The approximation error and the efficiency of the algorithms are analyzed. Numerical experiments are performed in a benchmark test case, where certain advantages of the new class of algorithms are demonstrated.

*Appeared in*

- SIAM J. Sci. Comput., 38 (2016) pp. A1483--A1507.

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# Dynamical large deviations of countable reaction networks under a weak reversibility condition

*Authors*

- Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857 - Renger, D. R. Michiel

ORCID: 0000-0003-3557-3485

*2010 Mathematics Subject Classification*

- 60F10 60J25 80A30 82C22

*Keywords*

- chemical reaction networks, Markov processes, large deviations

*DOI*

*Abstract*

A dynamic large deviations principle for a countable reaction network including coagulation--fragmentation models is proved. The rate function is represented as the infimal cost of the reaction fluxes and a minimiser for this variational problem is shown to exist. A weak reversibility condition is used to control the boundary behaviour and to guarantee a representation for the optimal fluxes via a Lagrange multiplier that can be used to construct the changes of measure used in standard tilting arguments. Reflecting the pure jump nature of the approximating processes, their paths are treated as elements of a BV function space.

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# Functional a posteriori error estimation for stationary reaction-convection-diffusion problems

*Authors*

- Eigel, Martin
- Samrowski, Tatiana

*2010 Mathematics Subject Classification*

- 65N30 65N15 65J15 65N22 65J10

*Abstract*

A functional type a posteriori error estimator for the finite element discretisation of the stationary reaction-convection-diffusion equation is derived. In case of dominant convection, the solution for this class of problems typically exhibits boundary layers and shock-front like areas with steep gradients. This renders the accurate numerical solution very demanding and appropriate techniques for the adaptive resolution of regions with large approximation errors are crucial. Functional error estimators as derived here contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. To evaluate the error estimator, a minimisation problem is solved which does not require any Galerkin orthogonality or any specific properties of the employed approximation space. Based on a set of numerical examples, we assess the performance of the new estimator and compare it with some classic a posteriori error estimators often used in practice. It is observed that the new estimator exhibits a good efficiency also with convection-dominated problem settings.

*Appeared in*

- Comput. Methods Appl. Math., 14 (2014) pp. 135--150.

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# Optimal selection of the regularization function in a generalized total variation model. Part I: Modelling and theory

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Rautenberg, Carlos N.

ORCID: 0000-0001-9497-9296

*2010 Mathematics Subject Classification*

- 94A08 68U10 49K20 49K30 49K40 49M37 65K15

*Keywords*

- Image restoration, generalized total variation regularization, spatially distributed regularization weight, Fenchel predual, bilevel optimization, variance corridor

*Abstract*

A generalized total variation model with a spatially varying regularization weight is considered. Existence of a solution is shown, and the associated Fenchel-predual problem is derived. For automatically selecting the regularization function, a bilevel optimization framework is proposed. In this context, the lower-level problem, which is parameterized by the regularization weight, is the Fenchel predual of the generalized total variation model and the upper-level objective penalizes violations of a variance corridor. The latter object relies on a localization of the image residual as well as on lower and upper bounds inspired by the statistics of the extremes.

*Appeared in*

- J. Math. Imaging Vision, 59 (2017), pp. 498--514.

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# Reliable averaging for the primal variable in the Courant FEM and hierarchical error estimators on red-refined meshes

*Authors*

- Carstensen, Carsten
- Eigel, Martin

*2010 Mathematics Subject Classification*

- 35J15 46E22 49Q10 49K20 49K40

*Keywords*

- a posteriori, error analysis, finite element method, averaging, smoothing, hierarchical estimator, adaptivity, mesh refinement, convergence

*Abstract*

A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a red-refined triangular mesh is presented for the 2D Poisson model problem. Reliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to pi/2. The error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes. The evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term. The results also allows the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation. Several numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks.

*Appeared in*

- Comput. Methods Appl. Math., 16 (2016) pp. 213--230.

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# Traveling wave modeling of nonlinear dynamics in multisection semiconductor lasers

*Authors*

- Radziunas, Mindaugas

*2010 Mathematics Subject Classification*

- 78A60 35Q60 78M34 37N20 78-04 35-04

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.-v 02.30.Oz 02.60.Cb.

*Keywords*

- traveling wave model, semiconductor laser, narrow waveguide, edge emitting laser, multisection laser, ring laser, simulations, analysis

*DOI*

*Abstract*

A hierarchy of 1 (time) + 1 (space) dimensional first-order partial differential equation (traveling wave) models is used for a description of dynamics in individual semiconductor lasers, various multisection semiconductor lasers, and coupled laser systems. Consequent modifications of the basic traveling wave model allow for taking into account different physical effects such as the gain dispersion, the thermal detuning, the spatial hole burning of carriers, the nonlinear gain saturation, or various carrier exchange processes in quantum dot lasers. For illustration, the model was applied for simulations of dynamics in complex ring laser with four branches of filtered feedback. Finally, several advanced techniques for model analysis such as calculation of instantaneous optical modes, finding of steady states, and numerical continuation and bifurcation analysis of the model equations were discussed and illustrated by example simulations.

*Appeared in*

- J. Piprek, ed., vol. 2 of Handbook of Optoelectronic Device Modeling and Simulation: Lasers, Modulators, Photodetectors, Solar Cells, and Numerical Methods, CRC Press, Boca Raton, 2017, pp. 153--182.

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# Stochastic topology optimisation with hierarchical tensor reconstruction

*Authors*

- Eigel, Martin
- Neumann, Johannes
- Schneider, Reinhold
- Wolf, Sebastian

*2010 Mathematics Subject Classification*

- 35R60 47B80 60H35 65C20 65N12 65N22 65J10

*Keywords*

- partial differential equations with random coefficients, tensor representation, tensor train, uncertainty quantification, topology optimization, phase field, adaptive methods, low-rank, tensor reconstruction, risk measures

*DOI*

*Abstract*

A novel approach for risk-averse structural topology optimization under uncertainties is presented which takes into account random material properties and random forces. For the distribution of material, a phase field approach is employed which allows for arbitrary topological changes during optimization. The state equation is assumed to be a high-dimensional PDE parametrized in a (finite) set of random variables. For the examined case, linearized elasticity with a parametric elasticity tensor is used. Instead of an optimization with respect to the expectation of the involved random fields, for practical purposes it is important to design structures which are also robust in case of events that are not the most frequent. As a common risk-aware measure, the Conditional Value at Risk (CVaR) is used in the cost functional during the minimization procedure. Since the treatment of such high-dimensional problems is a numerically challenging task, a representation in the modern hierarchical tensor train format is proposed. In order to obtain this highly efficient representation of the solution of the random state equation, a tensor completion algorithm is employed which only required the pointwise evaluation of solution realizations. The new method is illustrated with numerical examples and compared with a classical Monte Carlo sampling approach.

*Appeared in*

- Computer Methods Appl. Math. Engrg., 334 (2018), pp. 470--482, DOI 10.1016/j.cma.2018.02.003; changed title: Risk averse stochastic structural topology optimization.

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# Effect of chromatic dispersion on multimode laser dynamics: Delay differential model

*Authors*

- Vladimirov, Andrei G.
- Huyet, Guillaume
- Pimenov, Alexander

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Ah, 42.60.Mi, 42.60.Pk, 42.65.Sf

*Keywords*

- Laser dynamics, delay differential model, chromatic dispersion

*Abstract*

A set of differential equations with distributed delay is derived for modeling of multimode ring lasers with intracavity chromatic dispersion. Analytical stability analysis of continuous wave regimes is performed and it is demonstrated that sufficiently strong anomalous dispersion can destabilize these regimes.

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# A semismooth Newton method with analytical path-following for the $H^1$-projection onto the Gibbs simplex

*Authors*

- Adam, Lukáš

ORCID: 0000-0001-8748-4308 - Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Surowiec, Thomas M.

*2010 Mathematics Subject Classification*

- 49M15 90C20

*Keywords*

- Gibbs simplex, metric projection, semismooth Newton, path-following, Ginzburg-Landau energy, multiphase field models, inpainting, data classification

*DOI*

*Abstract*

An efficient, function-space-based second-order method for the $H^1$-projection onto the Gibbs-simplex is presented. The method makes use of the theory of semismooth Newton methods in function spaces as well as Moreau-Yosida regularization and techniques from parametric optimization. A path-following technique is considered for the regularization parameter updates. A rigorous first and second-order sensitivity analysis of the value function for the regularized problem is provided to justify the update scheme. The viability of the algorithm is then demonstrated for two applications found in the literature: binary image inpainting and labeled data classification. In both cases, the algorithm exhibits mesh-independent behavior.

*Appeared in*

- IMA J. Numer. Anal., (2018), published online on 07.06.2018, DOI 10.1093/imanum/dry034 .

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# Direct and inverse elastic scattering from anisotropic media

*Authors*

- Bao, Gang
- Hu, Guanghui
- Yin, Tao
- Sun, Jiguang

*2010 Mathematics Subject Classification*

- 35R30 74B05 78A46

*Keywords*

- linear elasticity, Lamé system, variational approach, Fréchet derivative, Dirichlet-to-Neumann map, inverse scattering

*DOI*

*Abstract*

Assume a time-harmonic elastic wave is incident onto a penetrable anisotropic body embedded into a homogeneous isotropic background medium. We propose an equivalent variational formulation in a truncated bounded domain and show uniqueness and existence of weak solutions by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fréchet derivative of the near-field solution operator with respect to the scattering interface is derived. As an application, we design a descent algorithm for recovering the interface from the near-field data of one or several incident directions and frequencies. Numerical examples in 2D are demonstrated to show the validity and accuracy of our methods.

*Appeared in*

- J. Math. Pures Appl., 117 (2018), pp. 263--301.

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# Numerical simulation of carrier transport in semiconductor devices at cryogenic temperatures

*Authors*

- Kantner, Markus
- Koprucki, Thomas

ORCID: 0000-0001-6235-9412

*2008 Physics and Astronomy Classification Scheme*

- 02.60.Cb, 47.11.Df, 72.20.-i

*Keywords*

- cryogenic temperatures, drift-diffusion, transport, device simulation

*DOI*

*Abstract*

At cryogenic temperatures the electron-hole plasma in semiconductor materials becomes strongly degenerate, leading to very sharp internal layers, extreme depletion in intrinsic domains and strong nonlinear diffusion. As a result, the numerical simulation of the drift-diffusion system suffers from serious convergence issues using standard methods. We consider a one-dimensional p-i-n diode to illustrate these problems and present a simple temperature-embedding scheme to enable the numerical simulation at cryogenic temperatures. The method is suitable for forward-biased devices as they appear e.g. in optoelectronic applications.

*Appeared in*

- Opt. Quantum Electron., 48 (2016), pp. 543/1--543/7, DOI 10.1007/s11082-016-0817-2 .

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# Another phase transition in the Axelrod model

*Authors*

- Stivala, Alex
- Keeler, Paul

ORCID: 0000-0002-2063-1075

*2010 Mathematics Subject Classification*

- 91D10 82C80 82C44

*Keywords*

- Culture dissemination, lattice model, mean-field analysis

*DOI*

*Abstract*

Axelrod's model of cultural dissemination, despite its apparent simplicity, demonstrates complex behavior that has been of much interest in statistical physics. Despite the many variations and extensions of the model that have been investigated, a systematic investigation of the effects of changing the size of the neighborhood on the lattice in which interactions can occur has not been made. Here we investigate the effect of varying the radius R of the von Neumann neighborhood in which agents can interact. We show, in addition to the well-known phase transition at the critical value of q, the number of traits, another phase transition at a critical value of R, and draw a q - R phase diagram for the Axelrod model on a square lattice. In addition, we present a mean-field approximation of the model in which behavior on an infinite lattice can be analyzed.

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# Corrector estimates for a class of imperfect transmission problems

*Authors*

- Reichelt, Sina

*2010 Mathematics Subject Classification*

- 35B10 35B27 35B40 35D30 35J25

*Keywords*

- Homogenization, corrector results, error estimates, periodic unfolding, two-component domain, periodic interface, imperfect transmission

*DOI*

*Abstract*

Based on previous homogenization results for imperfect transmission problems in two-component domains with periodic microstructure, we derive quantitative estimates for the difference between the microscopic and macroscopic solution. This difference is of order ερ, where ε > 0 describes the periodicity of the microstructure and ρ ∈ (0 , ½] depends on the transmission condition at the interface between the two components. The corrector estimates are proved without assuming additional regularity for the local correctors using the periodic unfolding method.

*Appeared in*

- Asymptot. Anal., 105 (2017), pp. 3--26, DOI 10.3233/ASY-171432 .

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# Optimal selection of the regularization function in a generalized total variation model. Part II: Algorithm, its analysis and numerical tests

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Rautenberg, Carlos N.

ORCID: 0000-0001-9497-9296 - Wu, Tao
- Langer, Andreas

*2010 Mathematics Subject Classification*

- 94A08 68U10 49K20 49K30 49K40 49M37 65K15

*Keywords*

- Image restoration, generalized total variation regularization, spatially distributed regularization weight, Fenchel predual, bilevel optimization, variance corridor, pprojected gradient method, convergence analysis

*Abstract*

Based on the generalized total variation model and its analysis pursued in part I (WIAS Preprint no. 2235), in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distributed regularization weight. Further, its numerical realization is discussed and results obtained for image denoising and deblurring as well as Fourier and wavelet inpainting are reported on.

*Appeared in*

- J. Math. Imaging Vision, 59 (2017), pp. 515--533.

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# Chance constraints in PDE constrained optimization

*Authors*

- Farshbaf Shaker, Mohammad Hassan
- Henrion, René
- Hömberg, Dietmar

*2010 Mathematics Subject Classification*

- 90C15 49J20

*Keywords*

- chance constraints, PDE constrained optimization

*DOI*

*Abstract*

Chance constraints represent a popular tool for finding decisions that enforce a robust satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in a finite-dimensional setting. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties to a setting of control problems subject to (uniform) state chance constraints.

*Appeared in*

- Set-Valued Var. Anal., (2017), published online on 11.10.2017, DOI 10.1007/s11228-017-0452-5 .

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# Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements

*Authors*

- Lederer, Philip L.
- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Merdon, Christian
- Schöberl, Joachim

*2010 Mathematics Subject Classification*

- 65N12 65N12 76D07 76D05 76M10

*Keywords*

- incompressible Navier--Stokes equations, mixed finite elements, pressure robustness, exact divergence-free velocity reconstruction, flux equilibration

*Abstract*

Classical inf-sup stable mixed finite elements for the incompressible (Navier--)Stokes equations are not pressure-robust, i.e., their velocity errors depend on the continuous pressure. How-ever, a modification only in the right hand side of a Stokes discretization is able to reestablish pressure-robustness, as shown recently for several inf-sup stable Stokes elements with discontinuous discrete pressures. In this contribution, this idea is extended to low and high order Taylor--Hood and mini elements, which have continuous discrete pressures. For the modification of the right hand side a velocity reconstruction operator is constructed that maps discretely divergence-free test functions to exactly divergence-free ones. The reconstruction is based on local H(div)-conforming flux equilibration on vertex patches, and fulfills certain orthogonality properties to provide consistency and optimal a-priori error estimates. Numerical examples for the incompressible Stokes and Navier--Stokes equations confirm that the new pressure-robust Taylor--Hood and mini elements converge with optimal order and outperform signi--cantly the classical versions of those elements when the continuous pressure is comparably large.

*Appeared in*

- SIAM J. Numer. Anal., 55 (2017) pp. 1291--1314.

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# Construction of generalized pendulum equations with prescribed maximum number of limit cycles of the second kind

*Authors*

- Schneider, Klaus R.
- Grin, Alexander

*2010 Mathematics Subject Classification*

- 34C05 34C23

*Keywords*

- generalized pendulum equation, limit cycles of the second kind, limit cycle of multiplicity three, bifurcation behavior, Dulac--Cherkas function

*DOI*

*Abstract*

Consider a class of planar autonomous differential systems with cylindric phase space which represent generalized pendulum equations. We describe a method to construct such systems with prescribed maximum number of limit cycles which are not contractible to a point (limit cycles of the second kind). The underlying idea consists in employing Dulac-Cherkas functions. We also show how this approach can be used to control the bifurcation of multiple limit cycles.

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# Near-field imaging of obstacles with the factorization method: Fluid-solid interaction

*Authors*

- Yin, Tao
- Hu, Guanghui
- Xu, Liwei
- Zhang, Bo

*2010 Mathematics Subject Classification*

- 35R30 74B05 78A46

*Keywords*

- inverse scattering, fluid-solid interaction problem, factorization method, Helmholtz equation, Navier equation, near-field imaging, outgoing-to-incoming operator

*Abstract*

Consider a time-harmonic acoustic point source incident on a bounded isotropic linearly elastic body immersed in a homogeneous compressible inviscid fluid. This paper is concerned with the inverse fluid-solid interaction (FSI) problem of recovering the elastic body from near-field data generated by infinitely many incident point source waves at a fixed energy. The incident point sources and the receivers for recording scattered signals are both located on a non-spherical closed surface, on which an outgoing-to-incoming (OtI) operator is appropriately defined. We provide a theoretical justification of the factorization method for precisely characterizing the scatterer by utilizing the spectrum of the near-field operator. This generalizes the imaging scheme developed in [G. Hu, J. Yang, B. Zhang, H. Zhang, Inverse Problems 30 (2014): 095005] to the case when near-field data are measured on non-spherical surfaces. Numerical examples in 2D are demonstrated to show the validity and accuracy of the inversion algorithm, even if limited aperture data are available on one or several line segments.

*Appeared in*

- Inverse Problems, 32 (2016) pp. 015003/1--015003/29.

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# Acoustic scattering from corners, edges and circular cones

*Authors*

- Elschner, Johannes
- Hu, Guanghui

*2010 Mathematics Subject Classification*

- 35R30 78A46

*Keywords*

- inverse medium scattering, Helmholtz equation, non-scattering wavenumbers

*Abstract*

Consider the time-harmonic acoustic scattering from a bounded penetrable obstacle imbedded in an isotropic homogeneous medium. The obstacle is supposed to possess a circular conic point or an edge point on the boundary in three dimensions and a planar corner point in two dimensions. The opening angles of cones and edges are allowed to be non-convex. We prove that such an obstacle scatters any incoming wave non-trivially (i.e., the far field patterns cannot vanish identically), leading to the absence of real non-scattering wavenumbers. Local and global uniqueness results for the inverse problem of recovering the shape of a penetrable scatterers are also obtained using a single incoming wave. Our approach relies on the singularity analysis of the inhomogeneous Laplace equation in a cone.

*Appeared in*

- Arch. Ration. Mech. Anal., 228:2 (2018) pp. 653--690.

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# Shape identification in inverse medium scattering problems with a single far-field pattern

*Authors*

- Hu, Guanghui
- Salo, Mikko
- Vesalainen, Esa V.

*2010 Mathematics Subject Classification*

- 35R30 74B05 78A46

*Keywords*

- uniqueness, inverse medium scattering, shape identification, Helmholtz equation

*Abstract*

Consider time-harmonic acoustic scattering from a bounded penetrable obstacle embedded in a homogeneous background medium. The index of refraction characterizing the material inside the obstacle is supposed to be Holder continuous near the corners. We prove that the shape and location of a convex polygon can be uniquely determined by the far-field pattern incited by a single incident wave at a fixed frequency. In three dimensions, the uniqueness applies to penetrable scatterers of rectangular type with additional assumptions on the smoothness of the contrast. Our arguments are motivated by recent studies on the absence of non-scattering wavenumbers in domains with corners. As a byproduct, we show that the smoothness conditions in previous corner scattering results are only required near the corners.

*Appeared in*

- SIAM J. Math. Anal., 48 (2016) pp. 152--165.

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# Boundary conditions for electrochemical interfaces

*Authors*

- Landstorfer, Manuel

*2010 Mathematics Subject Classification*

- 78A57 35Q35 34B15

*2008 Physics and Astronomy Classification Scheme*

- 82.45.Fk, 82.45.Gj, 68.43.Mn, 82.45.-h

*Keywords*

- Boundary conditions, double layer charging, charge transfer, current voltage relation, mixture theory, electrode/electrolyte interface

*DOI*

*Abstract*

Consistent boundary conditions for electrochemical interfaces, which cover double layer charging, pseudo-capacitive effects and transfer reactions, are of high demand in electrochemistry and adjacent disciplines. Mathematical modeling and optimization of electrochemical systems is a strongly emerging approach to reduce cost and increase efficiency of super-capacitors, batteries, fuel cells, and electrocatalysis. However, many mathematical models which are used to describe such systems lack a real predictive value. Origin of this shortcoming is the usage of oversimplified boundary conditions. In this work we derive the boundary conditions for some general electrode-electrolyte interface based on non-equilibrium thermodynamics for volumes and surfaces. The resulting equations are widely applicable and cover also tangential transport. The general framework is then applied to a specific material model which allows the deduction of a current-voltage relation and thus a comparison to experimental data. Some simplified 1D examples show the range of applicability of the new approach.

*Appeared in*

- J. Electrochem. Soc., 164 (2017) pp. 3671--3685.

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# Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D

*Authors*

- Farshbaf Shaker, Mohammad Hassan
- Heinemann, Christian

*2010 Mathematics Subject Classification*

- 49K20 35D35 35M33 35Q74 49J20 74A45 74D10 74F99 74P99

*Keywords*

- optimality conditions, optimal control, damage processes, phase-field model, viscoelasticity

*Abstract*

Controlling the growth of material damage is an important engineering task with plenty of real world applications. In this paper we approach this topic from the mathematical point of view by investigating an optimal boundary control problem for a damage phase-field model for viscoelastic media. We consider non-homogeneous Neumann data for the displacement field which describe external boundary forces and act as control variable. The underlying hyberbolic-parabolic PDE system for the state variables exhibit highly nonlinear terms which emerge in context with damage processes. The cost functional is of tracking type, and constraints for the control variable are prescribed. Based on recent results from [M. H. Farshbaf-Shaker, C. Heinemann: A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media. Math. Models Methods Appl. Sci. 25 (2015), 2749--2793], where global-in-time well-posedness of strong solutions to the lower level problem and existence of optimal controls of the upper level problem have been established, we show in this contribution differentiability of the control-to-state mapping, well-posedness of the linearization and existence of solutions of the adjoint state system. Due to the highly nonlinear nature of the state system which has by our knowledge not been considered for optimal control problems in the literature, we present a very weak formulation and estimation techniques of the associated adjoint system. For mathematical reasons the analysis is restricted here to the two-dimensional case. We conclude our results with first-order necessary optimality conditions in terms of a variational inequality together with PDEs for the state and adjoint state system.

*Appeared in*

- ESAIM: COCV, 24 (2018), pp. 579--603, DOI 10.1051/cocv/2017041.

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# On M-stationarity conditions in MPECs and the associated qualification conditions

*Authors*

- Adam, Lukáš

ORCID: 0000-0001-8748-4308 - Henrion, René
- Outrata, Jiří

*2010 Mathematics Subject Classification*

- 65K10 90C30 90C31 90C46

*Keywords*

- mathematical programs with equilibrium constraints, optimality conditions, constraint qualification, calmness, perturbation mapping

*Abstract*

Depending on whether a mathematical program with equilibrium constraints (MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed constraint qualifications (CQs) as well as the derived necessary optimality conditions may differ significantly. In this paper, we study this issue when imposing one of the weakest possible CQs, namely the calmness of the perturbation mapping associated with the respective generalized equations in both forms of the MPEC. It is well known that the calmness property allows one to derive so-called M-stationarity conditions. The strength of assumptions and conclusions in the two forms of the MPEC is strongly related with the CQs on the 'lower level' imposed on the set whose normal cone appears in the generalized equation. For instance, under just the Mangasarian-Fromovitz CQ (a minimum assumption required for this set), the calmness properties of the original and the enhanced perturbation mapping are drastically different. They become identical in the case of a polyhedral set or when adding the Full Rank CQ. On the other hand, the resulting optimality conditions are affected too. If the considered set even satisfies the Linear Independence CQ, both the calmness assumption and the derived optimality conditions are fully equivalent for the original and the enhanced form of the MPEC. A compilation of practically relevant consequences of our analysis in the derivation of necessary optimality conditions is provided in the main Theorem 4.3. The obtained results are finally applied to MPECs with structured equilibria.

*Appeared in*

- Math. Program., 168 (2018), pp. 229--259, DOI 10.1007/s10107-017-1146-3 .

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# H2-dependent attachment kinetics and shape evolution in chemical vapor deposition graphene growth

*Authors*

- Meca Álvarez, Esteban
- Shenoy, Vivek B.
- Lowengrub, John

*2008 Physics and Astronomy Classification Scheme*

- 68.43.Jk, 81.10.Aj, 81.15.Aa

*DOI*

*Abstract*

Experiments on graphene growth through chemical vapor deposition (CVD) involving methane (CH4) and hydrogen (H2) gases reveal a complex shape evolution and a nonmonotonic dependence on the partial pressure of H2 (pH2). To explain these intriguing observations, we develop a microkinetic model for the stepwise decomposition of CH4 into mobile radicals and consider two possible mechanisms of attachment to graphene crystals: CH radicals to hydrogen-decorated edges of the crystals and C radicals to bare crystal edges. We derive an effective mass flux and an effective kinetic coefficient, both of which depend on pH2, and incorporate these into a phase field model. The model reproduces both the non-monotonic dependence on pH2 and the characteristic shapes of graphene crystals observed in experiments. At small pH2, growth is limited by the kinetics of attachment while at large pH2 growth is limited because the effective mass flux is small. We also derive a simple analytical model that captures the non-monotone behavior, enables the two mechanisms of attachment to be distinguished and provides guidelines for CVD growth of defect-free 2D crystals.

*Appeared in*

- 2D Mater., 4 (2017), pp. 031010/1--031010/5, DOI 10.1088/2053-1583/aa74f1 .

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# Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics

*Authors*

- Farrell, Patricio

ORCID: 0000-0001-9969-6615 - Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434

*2010 Mathematics Subject Classification*

- 35Q99 82D37 65M08, 65N08, 74S10

*Keywords*

- finite volume method, flux discretization, Scharfetter--Gummel scheme, Fermi--Dirac statistics, degenerate semiconductors, van Roosbroeck system, semi-conductor device simulation, nonlinear diffusion, diffusion enhancement

*DOI*

*Abstract*

For a Voronoï finite volume discretization of the van Roosbroeck system with general charge carrier statistics we compare three thermodynamically consistent numerical fluxes known in the literature. We discuss an extension of the Scharfetter--Gummel scheme to non-Boltzmann (e.g. Fermi--Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.

*Appeared in*

- J. Comput. Phys., 346 (2017), pp. 497--513, DOI 10.1016/j.jcp.2017.06.023 .

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# Rate-independent elastoplasticity at finite strains and its numerical approximation

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Roubíček, Tomáš

*2010 Mathematics Subject Classification*

- 35K85 49S05 74S05 65M60 74A30 74C15 74M15

*Keywords*

- Plasticity, quasistatic evolution, energetic solutions, dissipation distance, hardening, polyconvexity, Ciarlet-Nečas condition, Signorini contact, finite-element approximation, Gamma-convergence, Lavrentiev phenomenon, 2nd-grade nonsimple materials

*Abstract*

Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation are approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The non-selfpenetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously by-passes the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme towards energetic solutions. In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.

*Appeared in*

- Math. Models Methods Appl. Sci., 26 (2016), pp. 2203--2236.

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# Scaling limit of the odometer in divisible sandpiles

*Authors*

- Cipriani, Alessandra
- Hazra, Rajat Subra
- Ruszel, Wioletta M.

*Keywords*

- divisible sandpile, bilaplacian Gaussian field, bilaplacian kernel, scaling limit

*DOI*

*Abstract*

In a recent work [LMPU] prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus.

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# Stochastic model for LFP-electrodes

*Authors*

- Dreyer, Wolfgang
- Friz, Peter

ORCID: 0000-0003-2571-8388 - Gajewski, Paul
- Guhlke, Clemens
- Maurelli, Mario

ORCID: 0000-0002-3028-1742

*2010 Mathematics Subject Classification*

- 35Q84 80A22 74N30

*2008 Physics and Astronomy Classification Scheme*

- 05.10.Gg, 05.70.Fh, 05.70.Ce, 82.47.Aa

*Keywords*

- lithium-ion batteries, lithium iron phospate, thermodynamics, phase transitions, many particle electode

*Abstract*

In the framework of non-equilibrium thermodynamics we derive a new model for porous electrodes. The model is applied to LiFePO4 (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithium-poor to a lithium-rich phase within LFP electrodes is controlled by surface fluctuations leading to a system of stochastic differential equations. The model is capable to derive an explicit relation between battery voltage and current that is controlled by thermodynamic state variables. This voltage-current relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltagecharge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates.

*Appeared in*

- Cont. Mech. Thermodyn., 30:3 (2018) pp. 593--628, changed title: Stochastic many-particle model for LFP electrodes

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# Constrained evolution for a quasilinear parabolic equation

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K59 35K20 34H05 80M50 93B52

*Keywords*

- feedback control, quasilinear parabolic equation, monotone nonlinearities, convex sets

*Abstract*

In the present contribution, a feedback control law is studied for a quasilinear parabolic equation. First, we prove the well-posedness and some regularity results for the Cauchy--Neumann problem for this equation, modified by adding an extra term which is a multiple of the subdifferential of the distance function from a closed convex set K of L^{2}(Ω). Then, we consider convex sets of obstacle or double-obstacle type, and we can act on the factor of the feedback control in order to be able to reach the convex set within a finite time, by proving rigorously this property.

*Appeared in*

- J. Optim. Theory Appl., 170 (2016), pp. 713--734.

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# Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems

*Authors*

- Neidhardt, Hagen
- Stephan, Artur
- Zagrebnov, Valentin A.

*2010 Mathematics Subject Classification*

- 47D06 34G10 34K30 47A55

*Keywords*

- Trotter product formula, convergence rate, approximation, evolution equations, solution operator, extension theory, perturbation theory, operator splitting

*DOI*

*Abstract*

In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space Lp(J,X), consisting of X-valued functions on the time-interval J. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in Lp(J,X). We show that the latter also allows to apply a full power of the operator-theoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces.

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# Detection of electromagnetic inclusions using topological sensitivity

*Authors*

- Wahab, Abdul
- Abbas, Tasawar
- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - Zaigham Zia, Qazi Muhammad

*2010 Mathematics Subject Classification*

- 35L05 35R30 74B05 47A52 65J20

*Keywords*

- Electromagnetic imaging, topological derivative, localization, resolution analysis, stability analysis, medium noise, measurement noise

*Abstract*

In this article a topological sensitivity framework for far field detection of a diametrically small electromagnetic inclusion is established. The cases of single and multiple measurements of the electric far field scattering amplitude at a fixed frequency are taken into account. The performance of the algorithm is analyzed theoretically in terms of its resolution and sensitivity for locating an inclusion. The stability of the framework with respect to measurement and medium noises is discussed. Moreover, the quantitative results for signal-to-noise ratio are presented. A few numerical results are presented to illustrate the detection capabilities of the proposed framework with single and multiple measurements.

*Appeared in*

- J. Comput. Math., 35 (2017), pp. 642--671, DOI 10.4208/jcm.1609-m2016-0498 .

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# A note on the Green's function for the transient random walk without killing on the half lattice, orthant and strip

*Authors*

- Chiarini, Alberto
- Cipriani, Alessandra

*2010 Mathematics Subject Classification*

- 60J10 60J45 60J55

*Keywords*

- Green's function, simple random walk, orthant, strip, half lattica

*DOI*

*Abstract*

In this note we derive an exact formula for the Green's function of the random walk on different subspaces of the discrete lattice (orthants, including the half space, and the strip) without killing on the boundary in terms of the Green's function of the simple random walk on $Z^d$, $dge 3$.

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# Density of convex intersections and applications

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Rautenberg, Carlos N.

ORCID: 0000-0001-9497-9296 - Rösel, Simon

*2010 Mathematics Subject Classification*

- 97N40 46E35 65K15 35J86 65N30 74C15 94A08

*Keywords*

- Density, convex constraints, variational inequalities, finite elements, image restoration, elasto-plasticity

*Abstract*

In this paper we address density properties of intersections of convex sets in several function spaces. Using the concept of Gamma-convergence, it is shown in a general framework, how these density issues naturally arise from the regularization, discretization or dualization of constrained optimization problems and from perturbed variational inequalities. A variety of density results (and counterexamples) for pointwise constraints in Sobolev spaces are presented and the corresponding regularity requirements on the upper bound are identified. The results are further discussed in the context of finite element discretizations of sets associated to convex constraints. Finally, two applications are provided, which include elasto-plasticity and image restoration problems.

*Appeared in*

- Proc. Roy. Soc. Edinburgh Sect. A, 473 (2017), pp. 20160919/1--20160919/28, DOI 10.1098/rspa.2016.0919 .

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# From adhesive to brittle delamination in visco-elastodynamics

*Authors*

- Rossi, Riccarda
- Thomas, Marita

*2010 Mathematics Subject Classification*

- 49J53 49J45 74H20 74C05 74C10 74M15 74R10

*Keywords*

- adhesive contact, brittle delamination, Kelvin-Voigt visco-elasticity, inertia, non-smooth brittle constraint, coupled rate-dependent/rate-independent evolution, energetic solutions

*Abstract*

In this paper we analyze a system for brittle delamination between two visco-elastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rate-independent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak solvability concept, generalizing the notion of energetic solution for rate-independent systems to the present mixed rate-dependent/rate-independent frame, is proposed. Via refined variational convergence techniques, existence of solutions is proved by passing to the limit in approximating systems which regularize the nonsmooth constraint by conditions for adhesive contact. The presence of the inertial term requires the design of suitable recovery spaces small enough to provide compactness but large enough to recover the information on the crack set in the limit.

*Appeared in*

- Math. Models Methods Appl. Sci., 27 (2017) pp. 1489--1546, DOI 10.1142/S0218202517500257 .

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# Tetrahedral mesh improvement using moving mesh smoothing and lazy searching flips

*Authors*

- Dassi, Franco
- Kamenski, Lennard
- Si, Hang

*2010 Mathematics Subject Classification*

- 65N50 65K10

*Keywords*

- mesh improvement,, moving mesh,, edge flipping,, mesh quality,, mesh smoothing

*Abstract*

In this paper we combine two new smoothing and flipping techniques. The moving mesh smoothing is based on the integration of an ordinary differential coming from a given functional. The lazy flip technique is a reversible edge removal algorithm to automatically search flips for local quality improvement. On itself, these strategies already provide good mesh improvement, but their combination achieves astonishing results which have not been reported so far. Provided numerical examples show that we can obtain final tetrahedral meshes with dihedral angles between 40 and 123 degrees. We compare the new method with other publicly available mesh improving codes.

*Appeared in*

- Procedia Engineering, 163 (2016) pp. 302--314.

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# Consistency results and confidence intervals for adaptive l1-penalized estimators of the high-dimensional sparse precision matrix

*Authors*

- Avanesov, Valeriy
- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Tabelow, Karsten

ORCID: 0000-0003-1274-9951

*2010 Mathematics Subject Classification*

- 62J07 62P10

*Keywords*

- adaptive l1 penalty, precision matrix, high-dimensional statistics, sparsity, confidence intervals, functional connectivity

*Abstract*

In this paper we consider the adaptive l1-penalized estimators for the precision matrix in a finite-sample setting. We show consistency results and construct confidence intervals for the elements of the true precision matrix. Additionally, we analyze the bias of these confidence intervals. We apply the estimator to the estimation of functional connectivity networks in functional Magnetic Resonance data and elaborate the theoretical results in extensive simulation experiments.

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# Neckpinch singularities in fractional mean curvature flows

*Authors*

- Cinti, Eleonora
- Sinestrari, Carlo
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 53C44 35R11

*Keywords*

- fractional perimeter, fractional mean curvature flow

*Abstract*

In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that, for any dimension n ≥ 2, there exist embedded hypersurfaces in R^{n} which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for n ≥ 3. Interestingly, when n=2, our result provides instead a counterexample in the nonlocal framework to the well known Grayson's Theorem [17], which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point. The essential step in our construction is an estimate which ensures that a suitably small perturbation of a thin strip has positive fractional curvature at every boundary point.

*Appeared in*

- Proc. Amer. Math. Soc., 146 (2018) pp. 2637--2646.

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# Regression based duality approach to optimal control with application to hydro electricity storage

*Authors*

- Hildebrand, Roland
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266 - Zhang, Jianing
- Dickmann, Fabian

*2010 Mathematics Subject Classification*

- 49J21 65C05

*Keywords*

- stochastic optimal control, dual martingale method, hydro electricity storage

*DOI*

*Abstract*

In this paper we consider the problem of optimal control of stochastic processes. We employ the dual martingale method brought forward in [Brown, Smith, and Sun, 2010]. The martingale constituting the solution of the dual problem is determined by linear regression within a Monte-Carlo approach. We apply the solution algorithm to a model of a hydro electricity storage and production system coupled with a model of the electricity wholesale market.

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# Reproducing kernel Hilbert spaces and variable metric algorithms in PDE constrained shape optimisation

*Authors*

- Eigel, Martin
- Sturm, Kevin

*2010 Mathematics Subject Classification*

- 35J15 46E22 49Q10 49K20 49K40

*Keywords*

- shape optimization, reproducing kernel Hilbert spaces, gradient method, variable metric, radial kernels

*DOI*

*Abstract*

In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE constrained shape optimisation problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so called smoothing parameter that allows a smoothness adjustment of the shape during the optimisation process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments.

*Appeared in*

- Optim. Methods Softw., 33 (2018), pp. 268--296 (published online on 03.05.2017), DOI 10.1080/10556788.2017.1314471 .

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# A fast solution method for time dependent multidimensional Schrödinger equations

*Authors*

- Lanzara, Flavia
- Maz'ya, Vladimir
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 65D32 35Q41 41A63

*Keywords*

- Schrödinger equation, higher dimensions, separated representations, error estimates

*Abstract*

In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schrödinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200.

*Appeared in*

- Appl. Anal., published online on 08.08.2017, DOI 10.1080/00036811.2017.1359571 .

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# The full Keller--Segel model is well-posed on fairly general domains

*Authors*

- Horstmann, Dirk
- Rehberg, Joachim
- Meinlschmidt, Hannes

*2010 Mathematics Subject Classification*

- 35A01 35K45 35K57 35Q92 92C17

*Keywords*

- Partial differential equations, Keller-Segel system, chemotaxis, reaction-crossdiffusion system, nonsmooth domains

*Abstract*

In this paper we prove the well-posedness of the full Keller-Segel system, a quasilinear strongly coupled reaction-crossdiffusion system, in the spirit that it always admits a unique local-in-time solution in an adequate function space, provided that the initial values are suitably regular. Apparently, there exists no comparable existence result for the full Keller-Segel system up to now. The proof is carried out for general source terms and is based on recent nontrivial elliptic and parabolic regularity results which hold true even on fairly general spatial domains, combined with an abstract solution theorem for nonlocal quasilinear equations by Amann.

*Appeared in*

- Nonlinearity, 31 (2018), pp. 1560--1592, DOI 10.1088/1361-6544/aaa2e1 .

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# Analytical aspects of spatially adapted total variation regularisation

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Papafitsoros, Konstantinos
- Rautenberg, Carlos N.

ORCID: 0000-0001-9497-9296

*2010 Mathematics Subject Classification*

- 26B30 49Q20 65J20

*Keywords*

- Total variation minimisation, weighted total variation, denoising, structure of solutions, regularisation

*Abstract*

In this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem, motivated by its application in signal recovery tasks. We study in depth the relationship between the weight function and the creation of new discontinuities in the solution. A partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed. We prove that the weighted total variation minimisation problem is well-posed even in the case of vanishing weight function, despite the lack of coercivity. This is based on the fact that the total variation of the solution is bounded by the total variation of the data, a result that it also shown here. Finally the relationship to the corresponding weighted fidelity problem is explored, showing that the two problems can produce completely different solutions even for very simple data functions.

*Appeared in*

- J. Math. Anal. Appl., 454 (2017), pp. 891--935, DOI 10.1016/j.jmaa.2017.05.025 .

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# Optimal distributed control of a diffuse interface model of tumor growth

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Rocca, Elisabetta
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K61 49J20 49K20 92C50

*Keywords*

- Distributed optimal control, first-order necessary optimality conditions, tumor growth, reaction-diffusion equations, Cahn--Hilliard equation

*Abstract*

In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins--Daruud et al. in citeHZO. The model consists of a Cahn--Hilliard equation for the tumor cell fraction $vp$ coupled to a reaction-diffusion equation for a function $s$ representing the nutrient-rich extracellular water volume fraction. The distributed control $u$ monitors as a right-hand side the equation for $s$ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

*Appeared in*

- Nonlinearity, 30 (2017), pp. 2518--2546.

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# Well-posedness for coupled bulk-interface diffusion with mixed boundary conditions

*Authors*

- Disser, Karoline

*2010 Mathematics Subject Classification*

- 35K20 35K59 47F05

*Keywords*

- Bulk-interface interaction, quasilinear parabolic equation, mixed boundary conditions, non-smooth domains

*Abstract*

In this paper, we consider a quasilinear parabolic system of equations describing coupled bulk and interface diusion, including mixed boundary conditions. The setting naturally includes non-smooth domains Ω. We show local well-posedness using maximal L^{s}-regularity in dual Sobolev spaces of type W^{-1,q}(Ω) for the associated abstract Cauchy problem.

*Appeared in*

- Analysis (Berlin), 35 (2015) pp. 309--317.

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# A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation

*Authors*

- Sprekels, Jürgen
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 49K21 35S11 49R05 47A60

*Keywords*

- Fractional operators, identification problems, first-order necessary and second-order sufficient optimality conditions, existence, uniqueness, regularity

*Abstract*

In this paper, we consider a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the power of a positive definite operator having a positive and discrete spectrum. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter. These results are then employed to derive existence as well as first-order necessary and second-order sufficient optimality conditions for a minimization problem, which is inspired by considerations in mathematical biology. In this problem, the fractional parameter $s$ serves as the ``control parameter'' that needs to be chosen in such a way as to minimize a given cost functional. This problem constitutes a new classof identification problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coefficient functions need to be identified, in the present case one has to determine the type of the differential operator itself. This problem exhibits the inherent analytical difficulty that with changing fractional parameter also the domain of definition, and thus the underlying function space, of the fractional operator changes.

*Appeared in*

- SIAM J. Control Optim., 55 (2017), pp. 70--93.

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# A multi-mode delay differential equation model for lasers with optical feedback

*Authors*

- Radziunas, Mindaugas

*2010 Mathematics Subject Classification*

- 78A60 78M34 35L40 58D30

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.-v 02.30.Jr 02.30.Ks

*Keywords*

- lasers, external cavity, optical feedback, cavity mode, modeling, traveling wave, Lang Kobayashi, multi-mode, delay differential equations

*DOI*

*Abstract*

In this paper, we discuss the relations between the spatially-distributed traveling wave, Lang-Kobayashi, and a new multi-mode delay differential equation models for Fabry-Perot type semiconductor diode lasers with an external optical feedback. All these models govern the dynamics of the slowly varying complex amplitudes of the optical fields and carrier density. To compare the models, we calculate the cavity modes determined by the threshold carrier density and optical frequency of the steady states in all three models. These calculations show that the Lang-Kobayashi type model is in good agreement with the traveling wave model only for the small feedback regimes, whereas newly derived multi-mode delay differential equation model remains correct even at moderate and large optical feedback regimes.

*Appeared in*

- Opt. Quantum Electron., 48 (2016), pp. 1--9, DOI 10.1007/s11082-016-0736-2; changed title: New multi-mode delay differential equation model for lasers with optical feedback.

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# Existence of weak solutions for the Cahn--Hilliard reaction model including elastic effects and damage

*Authors*

- Roggensack, Arne
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 35K61 35K86 49J40 35D30

*Keywords*

- Cahn-Hilliard reaction system, rate-dependent damage, phase separation, existence, non-linear Newton boundary condition, doubly non-linear differential inclusion

*Abstract*

In this paper, we introduce and study analytically a vectorial Cahn-Hilliard reaction model coupled with rate-dependent damage processes. The recently proposed Cahn-Hilliard reaction model can e.g. be used to describe the behavior of electrodes of lithium-ion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases. The coupling with the damage process allows considering simultaneously the evolution of a damage field, a second important physical effect occurring during the charging or discharging of lithium-ion batteries. Mathematically, this is realized by a Cahn-Larché system with a non-linear Newton boundary condition for the chemical potential and a doubly non-linear differential inclusion for the damage evolution. We show that this system possesses an underlying generalized gradient structure which incorporates the non-linear Newton boundary condition. Using this gradient structure and techniques from the field of convex analysis we are able to prove constructively the existence of weak solutions of the coupled PDE system.

*Appeared in*

- J. Partial Differ. Equ., 30 (2017) pp. 111-145, DOI doi: 10.4208/jpde.v30.n2.2 .

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# Time-periodic boundary layer solutions to singularly perturbed parabolic problems

*Authors*

- Omel'chenko, Oleh

ORCID: 0000-0003-0526-1878 - Recke, Lutz
- Butuzov, Valentin
- Nefedov, Nikolay

*2010 Mathematics Subject Classification*

- 35B25 35B10 35K20 35K58

*Keywords*

- Monotone and non-monotone boundary layers, two independent singular perturbation parameters, periodic-parabolic boundary value problem, implicit function theorem

*DOI*

*Abstract*

In this paper, we present a result of implicit function theorem type, which was designed for application to singularly perturbed problems. This result is based on fixed point iterations for contractive mappings, in particular, no monotonicity or sign preservation properties are needed. Then we apply our abstract result to time-periodic boundary layer solutions (which are allowed to be non-monotone with respect to the space variable) in semilinear parabolic problems with two independent singular perturbation parameters. We prove existence and local uniqueness of those solutions, and estimate their distance to certain approximate solutions.

*Appeared in*

- J. Differential Equations, 262 (2017) pp. 4823--4862.

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# Global existence for a nonstandard viscous Cahn--Hilliard system with dynamic boundary condition

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K61 35A05 35B40 74A15

*Keywords*

- Viscous Cahn-Hilliard system, phase field model, dynamic boundary conditions, well-posedness of solutions

*Abstract*

In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. **55** (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies.

*Appeared in*

- SIAM J. Math. Anal., 49 (2017), pp. 1732--1760, DOI 10.1137/16M1087539 .

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# Optimal boundary control of a nonstandard viscous Cahn--Hilliard system with dynamic boundary condition

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K61 49J20 49J50 49K20

*Keywords*

- Optimal control, viscous Cahn-Hilliard system, phase field model, dynamic boundary conditions, first-order necessary optimality conditions

*Abstract*

In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. **55** (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Fr&aecute;chenett differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality conditions in terms of a variational inequality and the adjoint state system.

*Appeared in*

- Nonlinear Analysis, 170 (2018), pp. 171--196, DOI 10.1016/j.na.2018.01.003 .

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# Apparent slip for an upper convected Maxwell fluid

*Authors*

- Münch, Andreas
- Wagner, Barbara
- Cook, L. Pamela
- Braun, Richard R.

*2010 Mathematics Subject Classification*

- 76A05 34E05 76A20

*Keywords*

- Wormlike micelle solutions, thin-film approximation, sharp-interface limit, matched asymptotic expansions

*Abstract*

In this study the flow field of a nonlocal, diffusive upper convected Maxwell (UCM) fluid with a polymer in a solvent undergoing shearing motion is investigated for pressure driven planar channel flow and the free boundary problem of a liquid layer on a solid substrate. For large ratios of the zero shear polymer viscosity to the solvent viscosity, it is shown that channel flows exhibit boundary layers at the channel walls. In addition, for increasing stress diffusion the flow field away from the boundary layers undergoes a transition from a parabolic to a plug flow. Using experimental data for the wormlike micelle solutions CTAB/NaSal and CPyCl/NaSal, it is shown that the analytic solution of the governing equations predicts these signatures of the velocity profiles. Corresponding flow structures and transitions are found for the free boundary problem of a thin layer sheared along a solid substrate. Matched asymptotic expansions are used to first derive sharp-interface models describing the bulk flow with expressions for an em apparent slip for the boundary conditions, obtained by matching to the flow in the boundary layers. For a thin film geometry several asymptotic regimes are identified in terms of the order of magnitude of the stress diffusion, and corresponding new thin film models with a slip boundary condition are derived.

*Appeared in*

- SIAM J. Appl. Math., 77 (2017) pp. 537--564, DOI 10.1137/16M1056869 .

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# Sharp-interface formation during lithium intercalation into silicon

*Authors*

- Meca Álvarez, Esteban
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 74N20 35Q74 35B25

*Keywords*

- Asymptotic analysis, phase-field model, interface dynamics, numerical methods

*Abstract*

In this study we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as a-Si. The governing equations couple a viscous Cahn-Hilliard-Reaction model with elasticity in the framework of the Cahn-Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple points where the sharp interface in the bulk of the layer intersects the free boundary. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.

*Appeared in*

- European J. Appl. Math., 29 (2018), pp. 118--145, DOI 10.1017/S0956792517000067 .

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# Thin-film electrodes for high-capacity lithium-ion batteries: Influence of phase transformations on stress

*Authors*

- Meca Álvarez, Esteban
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 74N20 35Q74 74S10

*Keywords*

- Phase-field Model, Interface Dynamics, Numerical Methods

*Abstract*

In this study we revisit experiments by Sethuraman et al. [J. Power Sources, 195, 5062 (2010)] on the stress evolution during the lithiation/delithiation cycle of a thin film of amorphous silicon. Based on recent work that show a two-phase process of lithiation of amorphous silicon, we formulate a phase-field model coupled to elasticity in the framework of Larché-Cahn. Using an adaptive nonlinear multigrid algorithm for the finite-volume discretization of this model, our two-dimensional numerical simulations show the formation of a sharp phase boundary between the lithiated and the amorphous silicon that continues to move as a front through the thin layer. We show that our model captures the non-monotone stress loading curve and rate dependence, as observed in experiments and connects characteristic features of the curve with the stucture formation within the layer. We take advantage of the thin film geometry and study the corresponding one-dimensional model to establish the dependence on the material parameters and obtain a comprehensive picture of the behaviour of the system.

*Appeared in*

- Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, 472 (2016) pp. 20160093/1--20160093/15.

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# Gradient flow structure for McKean--Vlasov equations on discrete spaces

*Authors*

- Erbar, Matthias
- Fathi, Max
- Laschos, Vaios
- Schlichting, André

*2010 Mathematics Subject Classification*

- 34A34 49J40 49J45 49Q20 60J25 60J27

*Keywords*

- Gradient flow structure, weakly interacting particles systems, nonlinear Markov chains, mean-field limit, evolutionary Gamma-convergence, transportation metric

*Abstract*

In this work, we show that a family of non-linear mean-field equations on discrete spaces, can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of N-particle dynamics, as N goes to infinity

*Appeared in*

- Discrete Contin. Dyn. Syst., 36 (2016), pp. 6799--6833.

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# Neurons' death and rebirth in sparse heterogeneous inhibitory networks

*Authors*

- Angulo-Garcia, David
- Luccioli, Stefano
- Olmi, Simona

ORCID: 0000-0002-8272-3493 - Torcini, Alessandro

*2008 Physics and Astronomy Classification Scheme*

- 87.19.lj, 05.45.Xt, 87.19.lm

*Keywords*

- Pulse-coupled heterogeneous inhibitory neural networks

*Abstract*

Inhibition is a key aspect of neural dynamics playing a fundamental role for the emergence of neural rhythms and the implementation of various information coding strategies. Inhibitory populations are present in several brain structures and the comprehension of their dynamics is strategical for the understanding of neural processing. In this paper, we discuss a general mechanism present in pulse-coupled heterogeneous inhibitory networks: inhibition can induce not only suppression of the neural activity, as expected, but it can also promote neural reactivation. In particular, for globally coupled systems, the number of firing neurons monotonically reduces upon increasing the strength of inhibition (neurons? death). The introduction of a sparse connectivity in the network is able to reverse the action of inhibition, i.e. a sufficiently strong synaptic strength can surprisingly promote, rather than depress, the activity of the neurons (neurons? rebirth). Specifically, for small synaptic strengths, one observes an asynchronous activity of nearly independent supra-threshold neurons. By increasing the inhibition, a transition occurs towards a regime where the neurons are all effectively sub-threshold and their irregular firing is driven by current fluctuations. We explain this transition from a mean-driven to a fluctuation-driven regime by deriving an analytic mean field approach able to provide the fraction of active neurons together with the first two moments of the firing time distribution. We show that, by varying the synaptic time scale, the mechanism underlying the reported phenomenon remains unchanged. However, for sufficiently slow synapses the effect becomes dramatic. For small synaptic coupling the fraction of active neurons is frozen over long times and their firing activity is perfectly regular. For larger inhibition the active neurons display an irregular bursting behaviour induced by the emergence of correlations in the current fluctuations. In this latter regime the model gives predictions consistent with experimental findings for a specific class of neurons, namely the medium spiny neurons in the striatum.

*Appeared in*

- New J. Phys. 19 (2017) 053011, changed title: Death and rebirth of neural activity in sparse inhibitory networks

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# Stronger wireless signals appear more Poisson

*Authors*

- Keeler, Paul

ORCID: 0000-0002-2063-1075 - Ross, Nathan
- Xia, Aihua
- Błaszczyszyn, Bartek

*2010 Mathematics Subject Classification*

- 60G55 60F99

*Keywords*

- Poisson process, wireless networks, approximation bounds

*DOI*

*Abstract*

Keeler, Ross and Xia [1] recently derived approximation and convergence results, which imply that the point process formed from the signal strengths received by an observer in a wireless network under a general statistical propagation model can be modelled by an inhomogeneous Poisson point process on the positive real line. The basic requirement for the results to apply is that there must be a large number of transmitters with different locations and random propagation effects. The aim of this note is to apply some of the main results of [1] in a less general but more easily applicable form to illustrate how the results can be applied in practice. New results are derived that show that it is the strongest signals, after being weakened by random propagation effects, that behave like a Poisson process, which supports recent experimental work. [1] P. Keeler, N. Ross, and A. Xia:``When do wireless network signals appear Poisson?? ''

*Appeared in*

- IEEE Wireless Comm. Letters, 5 (2016) pp. 572--575.

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# Copositive matrices with circulant zero pattern

*Authors*

- Hildebrand, Roland

*2010 Mathematics Subject Classification*

- 15B48 15A21

*Keywords*

- Copositive matrix, zero pattern, extreme ray

*Abstract*

Let n be an integer not smaller than 5 and let u1,...,un be nonnegative real n-vectors such that the indices of their positive elements form the sets 1,2,...,n-2,2,3,...,n-1,...,n,1,...,n-3, respectively. Here each index set is obtained from the previous one by a circular shift. The set of copositive forms which vanish on the vectors u1,...,un is a face of the copositive cone. We give an explicit semi-definite description of this face and of its subface consisting of positive semi-definite matrices, and study their properties. If the vectors u1,...,un and their positive multiples exhaust the zero set of an exceptional copositive form belonging to this face, then we call this form regular, otherwise degenerate. We show that degenerate forms are always extremal, and regular forms can be extremal only if n is odd. We construct explicit examples of extremal degenerate forms for any order n, and examples of extremal regular forms for any odd order n. The set of all degenerate forms, i.e., defined by different collections u1,...,un of zeros, is a submanifold of codimension 2n, the set of all regular forms a submanifold of codimension n.

*Appeared in*

- Linear Algebra Appl., 514 (2017) pp. 1--46, changed title: Copositive matrices with circulant zero support set.

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# Coalescence of Euclidean geodesics on the Poisson--Delaunay triangulation

*Authors*

- Coupier, David
- Hirsch, Christian

*2010 Mathematics Subject Classification*

- 60D05

*Keywords*

- coalescence, Burton-Keane argument, Delaunay triangulation, relative neighborhood graph, Poisson point process, first-passage percolation, sublinearity

*DOI*

*Abstract*

Let us consider Euclidean first-passage percolation on the Poisson-Delaunay triangulation. We prove almost sure coalescence of any two semi-infinite geodesics with the same asymptotic direction. The proof is based on an adapted Burton-Keane argument and makes use of the concentration property for shortest-path lengths in the considered graphs. Moreover, by considering the specific example of the relative neighborhood graph, we illustrate that our approach extends to further well-known graphs in computational geometry. As an application, we show that the expected number of semi-infinite geodesics starting at a given vertex and leaving a disk of a certain radius grows at most sublinearly in the radius.

*Appeared in*

- Bernoulli, 24 (2018), pp. 2721--2751.

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# A novel surface remeshing scheme via higher dimensional embedding and radial basis functions

*Authors*

- Dassi, Franco
- Farrell, Patricio

ORCID: 0000-0001-9969-6615 - Si, Hang

*2010 Mathematics Subject Classification*

- 65M50

*Keywords*

- Anisotropic Meshes, Radial Basis Function RBF, Mesh Optimization, Implicit Surfaces, Geometry Processing

*Abstract*

Many applications heavily rely on piecewise triangular meshes to describe complex surface geometries. High-quality meshes significantly improve numerical simulations. In practice, however, one often has to deal with several challenges. Some regions in the initial mesh may be overrefined, others too coarse. Additionally, the triangles may be too thin or not properly oriented. We present a novel mesh adaptation procedure which greatly improves the problematic input mesh and overcomes all of these drawbacks. By coupling surface reconstruction via radial basis functions with the higher dimensional embedding surface remeshing technique, we can automatically generate anisotropic meshes. Moreover, we are not only able to fill or coarsen certain mesh regions but also align the triangles according to the curvature of the reconstructed surface. This yields an acceptable trade-off between computational complexity and accuracy.

*Appeared in*

- SIAM J. Sci. Comput., 39 (2017) pp. B522--B547, DOI 10.1137/16M1077015 .

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# Uncertainty quantification for hysteresis operators and a model for magneto-mechanical hysteresis

*Authors*

- Klein, Olaf

ORCID: 0000-0002-4142-3603

*Keywords*

- Magneto-Mechanical Hysteresis, Random Variables, Uncertainty Quantification

*Abstract*

Many models for magneto-mechanical components involve hysteresis operators. The parameter within these operators have to be identified from measurements and are therefore subject to uncertainties. To quantify the influence of these uncertainties, the parameter in the hysteresis operator are considered as functions of random variables. Combining this with the hysteresis operator, we get new random variables and we can compute stochastic properties of the output of the model. For two hysteresis operators corresponding numerical results are presented in this paper. Moreover, the influence of the variation of the parameters in a model for a magneto-mechanical component is investigated.

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# Mathematical models: A research data category?

*Authors*

- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Tabelow, Karsten

ORCID: 0000-0003-1274-9951

*2010 Mathematics Subject Classification*

- 68U35 97M10

*Keywords*

- Research data, mathematical modeling and simulation, mathematical knowledge management

*Abstract*

Mathematical modeling and simulation (MMS) has now been established as an essential part of the scientific work in many disciplines and application areas. It is common to categorize the involved numerical data and to some extend the corresponding scientific software as research data. Both have their origin in mathematical models. In this contribution we propose a holistic approach to research data in MMS by including the mathematical models and discuss the initial requirements for a conceptual data model for this field.

*Appeared in*

- Mathematical Software -- ICMS 2016: 5th International Conference, Berlin, Germany, July 11--14, 2016, Proceedings, G.-M. Greuel, Th. Koch, P. Paule, A. Sommese, eds., Lecture Notes in Computer Science, Springer International Publishing AG Switzerland, Cham, 2016, pp. 423--428.

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# Are quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?

*Authors*

- Heitsch, Holger
- Leövey, Hernan
- Römisch, Werner

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- stochastic programming, two-stage, scenario, quasi-Monte Carlo, effective dimension, dimension reduction

*Abstract*

Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The two-stage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives. Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol' point sets accompanied with PCA for dimension reduction.

*Appeared in*

- Comput. Optim. Appl., 65 (2016) pp. 567--603.

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# Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier--Stokes equations

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Merdon, Christian

*2010 Mathematics Subject Classification*

- 76D05 65M60 65M12

*2008 Physics and Astronomy Classification Scheme*

- 47.11.Fg

*Keywords*

- incompressible Navier-Stokes, mixed finite element methods, a-priori error estimates, pressure-robustness, Helmholtz projector, irrotational forces

*Abstract*

Recently, it was understood how to repair a certain L2-orthogonality of discretely-divergence-free vector fields and gradient fields such that the velocity error of inf-sup stable discretizations for the incompressible Stokes equations becomes pressure-independent. These new 'pressure-robust' Stokes discretizations deliver a small velocity error, whenever the continuous velocity field can be well approximated on a given grid. On the contrary, classical inf-sup stable Stokes discretizations can guarantee a small velocity error only, when both the velocity and the pressure field can be approximated well, simultaneously.

In this contribution, 'pressure-robustness' is extended to the time-dependent Navier--Stokes equations. In particular, steady and time-dependent potential flows are shown to build an entire class of benchmarks, where pressure-robust discretizations can outperform classical approaches significantly. Speedups will be explained by a new theoretical concept, the 'discrete Helmholtz projector' of an inf-sup stable discretization. Moreover, different discrete nonlinear convection terms are discussed, and skew-symmetric pressure-robust discretizations are proposed.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 311 (2016) pp. 304--326.

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# Uniform second order convergence of a complete flux scheme on unstructured 1D grids for a singularly perturbed advection-diffusion equation and some multidimensional extensions

*Authors*

- Farrell, Patricio

ORCID: 0000-0001-9969-6615 - Linke, Alexander

ORCID: 0000-0002-0165-2698

*2010 Mathematics Subject Classification*

- 65L11 65L20 65N08 65N12

*Keywords*

- singularly perturbed advection-diffusion equation, uniform second-order convergence, finite-volume method, complete flux scheme

*DOI*

*Abstract*

The accurate and efficient discretization of singularly perturbed advection-diffusion equations on arbitrary 2D and 3D domains remains an open problem. An interesting approach to tackle this problem is the complete flux scheme (CFS) proposed by G. D. Thiart and further investigated by J. ten Thije Boonkkamp. For the CFS, uniform second order convergence has been proven on structured grids. We extend a version of the CFS to unstructured grids for a steady singularly perturbed advection-diffusion equation. By construction, the novel finite volume scheme is nodally exact in 1D for piecewise constant source terms. This property allows to use elegant continuous arguments in order to prove uniform second order convergence on unstructured one-dimensional grids. Numerical results verify the predicted bounds and suggest that by aligning the finite volume grid along the velocity field uniform second order convergence can be obtained in higher space dimensions as well.

*Appeared in*

- J. Sci. Comput., 72 (2017), pp. 373--395, DOI 10.1007/s10915-017-0361-7 .

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# A nonlocal concave-convex problem with nonlocal mixed boundary data

*Authors*

- Abdellaoui, Boumediene
- Dieb, Abdelrazek
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 35A15

*Keywords*

- integrodifferential operators, fractional Laplacian, weak solutions, mixed boundary condition, multiplicity of positive solution

*DOI*

*Abstract*

The aim of this paper is to study a nonlocal equation with mixed Neumann and Dirichlet external conditions. We prove existence, nonexistence and multiplicity of positive energy solutions and analyze the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.

*Appeared in*

- AIMS Commun. Pure Appl. Anal., 17:3 (2018), pp. 1103--1120.

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# A revisited Johnson--Mehl--Avrami--Kolmogorov model and the evolution of grain-size distributions in steel

*Authors*

- Hömberg, Dietmar
- Patacchini, Francesco Saverio
- Sakamoto, Kenichi
- Zimmer, Johannes

*2010 Mathematics Subject Classification*

- 35Q84 35Q74 35K10 74H40

*Keywords*

- grain size distribution, Fokker-Planck equation, nucleation and growth, phase transitions

*Abstract*

The classical Johnson-Mehl-Avrami-Kolmogorov approach for nucleation and growth models of diffusive phase transitions is revisited and applied to model the growth of ferrite in multiphase steels. For the prediction of mechanical properties of such steels, a deeper knowledge of the grain structure is essential. To this end, a Fokker-Planck evolution law for the volume distribution of ferrite grains is developed and shown to exhibit a log-normally distributed solution. Numerical parameter studies are given and confirm expected properties qualitatively. As a preparation for future work on parameter identification, a strategy is presented for the comparison of volume distributions with area distributions experimentally gained from polished micrograph sections.

*Appeared in*

- IMA J. Appl. Math., 82 (2017) pp. 763--780, DOI 10.1093/imamat/hxx012 .

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# Elastic scattering coefficients and enhancement of nearly elastic cloaking

*Authors*

- Abbas, Tasawar
- Ammari, Habib
- Hu, Guanghui
- Wahab, Abdul
- Ye, Jong Chul

*2010 Mathematics Subject Classification*

- 35L05 35R30 74B05 74J20 78A46

*Keywords*

- elastic scattering, scattering coefficients, elastic cloaking, inverse scattering

*Abstract*

The concept of scattering coefficients has played a pivotal role in a broad range of inverse scattering and imaging problems in acoustic and electromagnetic media. In view of their promising applications, we introduce the notion of scattering coefficients of an elastic inclusion in this article. First, we define elastic scattering coefficients and substantiate that they naturally appear in the expansions of elastic scattered field and far field scattering amplitudes corresponding to a plane wave incidence. Then an algorithm is developed and analyzed for extracting the elastic scattering coefficients from multi-static response measurements of the scattered field. Moreover, the estimate of the maximal resolving order is provided in terms of the signal-to-noise ratio. The decay rate and symmetry of the elastic scattering coefficients are also discussed. Finally, we design scattering-coefficients-vanishing structures and elucidate their utility for enhancement of nearly elastic cloaking.

*Appeared in*

- J. Elast., 128 (2017), pp. 203--243, DOI 10.1007/s10659-017-9624-7 .

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# Liquid-liquid dewetting: Morphologies and rates

*Authors*

- Bommer, Stefan
- Seemann, Ralf
- Jachalski, Sebastian
- Peschka, Dirk

ORCID: 0000-0002-3047-1140 - Wagner, Barbara

*2010 Mathematics Subject Classification*

- 68.05.Cf 02.60.Cb

*Keywords*

- Free boundary problems, interface dynamics, experimental methods, numerical simulations

*Abstract*

The dependence of the dissipation on the local details of the flow field of a liquid polymer film dewetting from a liquid polymer substrate is shown, solving the free boundary problem for a two-layer liquid system. As a key result we show that the dewetting rates of such a liquid bi-layer system can not be described by a single power law but shows transient behaviour of the rates, changing from increasing to decreasing behaviour. The theoretical predictions on the evolution of morphology and rates of the free surfaces and free interfaces are compared to measurements of the evolution of the polystyrene(PS)-air, the polymethyl methacrylate (PMMA)-air and the PS-PMMA interfaces using in situ atomic force microscopy (AFM), and they show excellent agreement.

*Appeared in*

- Sci. Rep., 8 (2018), pp. 13295/1--13295/11, DOI 10.1038/s41598-018-31418-1 .

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# The partial molar volume and area of solvated ions and some aspects of partial charge transfer

*Authors*

- Landstorfer, Manuel

*2010 Mathematics Subject Classification*

- 78A57 35Q35 34B15

*2008 Physics and Astronomy Classification Scheme*

- 82.45.Gj, 68.43.-h, 68.35.Md

*Keywords*

- Double layer, adsorption, solvation, surface mixture theory, partial charge, transfer, electrode/electrolyte enterface

*DOI*

*Abstract*

The double layer capacity is one of the central quantities in theoretical and experimental electrochemistry of metal/electrolyte interfaces. It turns out that the capacity is related to two central thermodynamic quantities, i.e. the partial molar volume of an ionic constituent and the partial molar area of the respective adsorbate. Since ions in solution (or on the surface) accumulated solvent molecules in their solvation shell, the partial molar volume and area are effected by this phenomena. In this work we discuss several aspects of the relationship between the molar volume and area of an ion, the solvation number and the charge number. In addition, we account for partial charge transfer on the metal surface which explains naturally the difference of the capacity maxima between ceF- and ceClO4- on silver. We provide simple yet validated analytical expressions for the partial molar volume and area of multi-valent ions and parameter values for aqueous solutions.

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# A local projection stabilization/continuous Galerkin--Petrov method for incompressible flow problems

*Authors*

- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - John, Volker

ORCID: 0000-0002-2711-4409 - Matthies, Gunar
- Novo, Julia

*2010 Mathematics Subject Classification*

- 65M12 65M15 65M80

*Keywords*

- Evolutionary Oseen problem, inf-sup stable pairs of finite element spaces, local projection stabilization (LPS) methods, continuous Galerkin--Petrov (cGP) methods

*DOI*

*Abstract*

The local projection stabilization (LPS) method in space is consid-ered to approximate the evolutionary Oseen equations. Optimal error bounds independent of the viscosity parameter are obtained in the continuous-in-time case for the approximations of both velocity and pressure. In addition, the fully discrete case in combination with higher order continuous Galerkin--Petrov (cGP) methods is studied. Error estimates of order k + 1 are proved, where k denotes the polynomial degree in time, assuming that the convective term is time-independent. Numerical results show that the predicted order is also achieved in the general case of time-dependent convective terms.

*Appeared in*

- Appl. Math. Comput., 333 (2018), pp. 304--324, DOI 10.1016/j.amc.2018.03.088 .

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# On the mesh nonsingularity of the moving mesh PDE method

*Authors*

- Huang, Weizhang
- Kamenski, Lennard

*2010 Mathematics Subject Classification*

- 65N50 65K10

*Keywords*

- variational mesh generation, mesh adaptation, moving mesh PDE, mesh nonsingularity, limiting mesh

*Abstract*

The moving mesh PDE (MMPDE) method for variational mesh generation and adaptation is studied theoretically at the discrete level, in particular the nonsingularity of the obtained meshes. Meshing functionals are discretized geometrically and the MMPDE is formulated as a modified gradient system of the corresponding discrete functionals for the location of mesh vertices. It is shown that if the meshing functional satisfies a coercivity condition, then the mesh of the semi-discrete MMPDE is nonsingular for all time if it is nonsingular initially. Moreover, the altitudes and volumes of its elements are bounded below by positive numbers depending only on the number of elements, the metric tensor, and the initial mesh. Furthermore, the value of the discrete meshing functional is convergent as time increases, which can be used as a stopping criterion in computation. Finally, the mesh trajectory has limiting meshes which are critical points of the discrete functional. The convergence of the mesh trajectory can be guaranteed when a stronger condition is placed on the meshing functional. Two meshing functionals based on alignment and equidistribution are known to satisfy the coercivity condition. The results also hold for fully discrete systems of the MMPDE provided that the time step is sufficiently small and a numerical scheme preserving the property of monotonically decreasing energy is used for the temporal discretization of the semi-discrete MMPDE. Numerical examples are presented

*Appeared in*

- Math. Comp., 87 (2018), pp. 1887--1911 (published online on 02.10.2017), DOI 10.1090/mcom/3271 .

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# Scattering of general incident beams by diffraction gratings

*Authors*

- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 78A45 78A40 65D30

*Keywords*

- diffraction, periodic structure, beam incidence, adaptive algorithm, cubature

*DOI*

*Abstract*

The paper is devoted to the electromagnetic scattering of arbitrary time-harmonic fields by periodic structures. The Floquet-Fourier transform converts the full space Maxwell problem to a two-parameter family of diffraction problems with quasiperiodic incidence waves, for which conventional grating methods become applicable. The inverse transform is given by integrating with respect to the parameters over a infinite strip in ℝ². For the computation of the scattered fields we propose an algorithm, which extends known adaptive methods for the approximate calculation of multiple integrals. The novel adaptive approach provides autonomously the expansion of the incident field into quasiperiodic waves in order to approximate the scattered fields within a prescribed error tolerance. Some application examples are numerically examined.

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# Corrector estimates for a thermo-diffusion model with weak thermal coupling

*Authors*

- Muntean, Adrian
- Reichelt, Sina

*2010 Mathematics Subject Classification*

- 35B27 35Q79 74A15 78A48

*Keywords*

- Homogenization, corrector estimates, periodic unfolding, gradient folding operator, perforated domain, thermo-diffusion, composite media

*Abstract*

The present work deals with the derivation of corrector estimates for the two-scale homogenization of a thermo-diffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged high-contrast microstructures. The terminology ``weak thermal coupling'' refers here to the variable scaling in terms of the small homogenization parameter ε of the heat conduction-diffusion interaction terms, while the ``high-contrast'' is thought particularly in terms of the heat conduction properties of the composite material. As main target, we justify the first-order terms of the multiscale asymptotic expansions in the presence of coupled fluxes, induced by the joint contribution of Sorret and Dufour-like effects. The contrasting heat conduction combined with cross coupling lead to the main mathematical difficulty in the system. Our approach relies on the method of periodic unfolding combined with ε-independent estimates for the thermal and concentration fields and for their coupled fluxes

*Appeared in*

- Multiscale Model. Simul., 16 (2018), pp. 807--832, DOI 10.1137/16M109538X .

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# Shape optimisation for a class of semilinear variational inequalities with applications to damage models

*Authors*

- Heinemann, Christian
- Sturm, Kevin

*2010 Mathematics Subject Classification*

- 49K40 49J27 49J40 49Q10 35J61 49K20 74R05 74B99

*Keywords*

- shape optimisation, semilinear elliptic variational inequalities, optimisation problems in Banach spaces, obstacle problems, damage phase field models, elasticity

*Abstract*

The present contribution investigates shape optimisation problems for a class of semilinear elliptic variational inequalities with Neumann boundary conditions. Sensitivity estimates and material derivatives are firstly derived in an abstract operator setting where the operators are defined on polyhedral subsets of reflexive Banach spaces. The results are then refined for variational inequalities arising from minimisation problems for certain convex energy functionals considered over upper obstacle sets in $H^1$. One particularity is that we allow for dynamic obstacle functions which may arise from another optimisation problems. We prove a strong convergence property for the material derivative and establish state-shape derivatives under regularity assumptions. Finally, as a concrete application from continuum mechanics, we show how the dynamic obstacle case can be used to treat shape optimisation problems for time-discretised brittle damage models for elastic solids. We derive a necessary optimality system for optimal shapes whose state variables approximate desired damage patterns and/or displacement fields.

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# Linearized elasticity as Mosco-limit of finite elasticity in the presence of cracks

*Authors*

- Gussmann, Pascal
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35J85 49J40 74B20 74A45

*Keywords*

- Gamma convergence, Mosco convergence, finite-strain elasticity, rigidity estimate for, crack domains, global injectivity, local non-interpenetration condition

*DOI*

*Abstract*

The small-deformation limit of finite elasticity is considered in presence of a given crack. The rescaled finite energies with the constraint of global injectivity are shown to Gamma converge to the linearized elastic energy with a local constraint of noninterpenetrability along the crack.

*Appeared in*

- Adv. Calculus Variations, (2017), published online on 17.10.2017, DOI 10.1515/acv-2017-0010 .

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# Adiabatic Floquet model for the optical response in femtosecond filaments

*Authors*

- Hofmann, Michael
- Brée, Carsten

*2008 Physics and Astronomy Classification Scheme*

- 31.15.A-, 32.80.Rm, 42.50.Ct, 42.65.An

*Keywords*

- light-matter interaction, ab initio simulations, metastable states, strong-field ionization

*Abstract*

The standard model of femtosecond filamentation is based on phenomenological assumptions which suggest that the ionization-induced carriers can be treated as free according to the Drude model, while the nonlinear response of the bound carriers follows the all-optical Kerr effect. Here, we demonstrate that the additional plasma generated at a multiphoton resonance dominates the saturation of the nonlinear refractive index. Since resonances are not captured by the standard model, we propose a modification of the latter in which ionization enhancements can be accounted for by an ionization rate obtained from non-Hermitian Floquet theory. In the adiabatic regime of long pulse envelopes, this augmented standard model is in excellent agreement with direct quantum mechanical simulations. Since our proposal maintains the structure of the standard model, it can be easily incorporated into existing codes of filament simulation.

*Appeared in*

- J. Phys. B, 49 (2016), pp. 205004/1--205004/12, DOI 10.1088/0953-4075/49/20/205004 .

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# Bayesian inversion with a hierarchical tensor representation

*Authors*

- Eigel, Martin
- Marschall, Manuel

ORCID: 0000-0003-0648-1936 - Schneider, Reinhold

*2010 Mathematics Subject Classification*

- 35R60 47B80 60H35 65C20 65N12 65N22 65J10

*Keywords*

- partial differential equations with random coefficients, tensor representation, tensor train, uncertainty quantification, stochastic finite element methods, operator equations, adaptive methods, low-rank

*DOI*

*Abstract*

The statistical Bayesian approach is a natural setting to resolve the ill-posedness of inverse problems by assigning probability densities to the considered calibration parameters. Based on a parametric deterministic representation of the forward model, a sampling-free approach to Bayesian inversion with an explicit representation of the parameter densities is developed. The approximation of the involved randomness inevitably leads to several high dimensional expressions, which are often tackled with classical sampling methods such as MCMC. To speed up these methods, the use of a surrogate model is beneficial since it allows for faster evaluation with respect to calibration parameters. However, the inherently slow convergence can not be remedied by this. As an alternative, a complete functional treatment of the inverse problem is feasible as demonstrated in this work, with functional representations of the parametric forward solution as well as the probability densities of the calibration parameters, determined by Bayesian inversion. The proposed sampling-free approach is discussed in the context of hierarchical tensor representations, which are employed for the adaptive evaluation of a random PDE (the forward problem) in generalized chaos polynomials and the subsequent high-dimensional quadrature of the log-likelihood. This modern compression technique alleviates the curse of dimensionality by hierarchical subspace approximations of the involved low rank (solution) manifolds. All required computations can be carried out efficiently in the low-rank format. A priori convergence is examined, considering all approximations that occur in the method. Numerical experiments demonstrate the performance and verify the theoretical results.

*Appeared in*

- Inverse Problems, 34 (2018) pp. 035010/1--035010/29, DOI 10.1088/1361-6420/aaa998; changed title: Sampling-free Bayesian inversion with adaptive hierarchical tensor representations.

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# Numerical methods for drift-diffusion models

*Authors*

- Farrell, Patricio

ORCID: 0000-0001-9969-6615 - Rotundo, Nella
- Doan, Duy Hai
- Kantner, Markus
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Koprucki, Thomas

ORCID: 0000-0001-6235-9412

*2010 Mathematics Subject Classification*

- 25N08 35K55

*Keywords*

- Scharfetter-Gummel scheme, thermodynamic consistency, Drift-diffusion equations, non-Boltzmann statistic distributions, diffusion enhancement

*Abstract*

The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume discretization scheme and recent efforts to generalize this approach to general statistical distribution functions.

*Appeared in*

- P. Farrell, N. Rotundo, D.H. Doan, M. Kantner, J. Fuhrmann, Th. Koprucki, Chapter 50 ``Drift-Diffusion Models'' in Volume 2 of Handbook of Optoelectronic Device Modeling and Simulation: Fundamentals, Materials, Nanostructures, LEDs, and Amplifiers , J. Piprek, ed., Series in Optics and Optoelectronics, CRC Press Taylor & Francis Group, 2017, pp. 733--771.

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# Error estimates for elliptic equations with not exactly periodic coefficients

*Authors*

- Reichelt, Sina

*2010 Mathematics Subject Classification*

- 35B10 35B27 35B40 35D30 35J70

*Keywords*

- homogenization, error estimates, periodic unfolding, gradient folding operator

*Abstract*

This note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not exactly ε-periodic and the ellipticity constant may degenerate for vanishing ε. Here ε>0 denotes the ratio between the microscopic and the macroscopic length scale. It is shown that for degenerating and non-degenerating coefficients the error between the original solution and the effective solution is of order √ε. Therefore suitable test functions are constructed via the periodic unfolding method and a gradient folding operator making only minimal additional assumptions on the given data and the effective solution with respect to the macroscopic scale.

*Appeared in*

- Adv. Math. Sci. Appl., 25 (2016), pp. 117--131.

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# Planelike interfaces in long-range Ising models and connections with nonlocal minimal surfaces

*Authors*

- Cozzi, Matteo
- Dipierro, Serena
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 82C20 82B05 35R11

*Keywords*

- planelike minimizers, phase transitions, spin models, Ising models, long-range interactions, nonlocal minimal surfaces

*Abstract*

This paper contains three types of results: the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane, the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane, the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces. In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other).

In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.

*Appeared in*

- J. Stat. Phys., 167:6 (2017) pp. 1401--1451.

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# Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K55 49K20 74A15

*Keywords*

- Distributed optimal control, phase field systems, double obstacle potentials, nonlocal operators, first-order necessary optimality conditions

*Abstract*

This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. The *local* model has been investigated in a series of papers by P. Podio-Guidugli and the present authors the *nonlocal* model studied here consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interval in which the order parameter of the phase segregation attains its values. It also contains a nonlocal term modeling long-range interactions. Due to the strong nonlinear couplings between the state variables (which even involve products with time derivatives), the analysis of the state system is difficult. In addition, the presence of the differential inclusion is the reason that standard arguments of optimal control theory cannot be applied to guarantee the existence of Lagrange multipliers. In this paper, we employ recent results proved for smooth logarithmic potentials and perform a so-called `deep quench' approximation to establish existence and first-order necessary optimality conditions for the nonsmooth case of the double obstacle potential.

*Appeared in*

- Evol. Equ. Control Theory, 6 (2017), pp. 35--58.

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# A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn--Hilliard--Navier--Stokes system

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Hinze, Michael
- Kahle, Christian
- Keil, Tobias

*2010 Mathematics Subject Classification*

- 49K20 49M29 65K15 76T10 90C33

*Keywords*

- Cahn-Hilliard, C-stationarity, mathemathical programming with equilibrium constraints, Navier-Stokes, non-matched densities, non-smooth potentials, optimal control, adaptive finite element method, goal-oriented error estimation

*Abstract*

This paper is concerned with the development and implementation of an adaptive solution algorithm for the optimal control of a time-discrete Cahn--Hilliard--Navier--Stokes system with variable densities. The free energy density associated to the Cahn--Hilliard system incorporates the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier--Stokes equation. A dual-weighed residual approach for goal-oriented adaptive finite elements is presented which is based on the concept of C-stationarity. The overall error representation depends on primal residual weighted by approximate dual quantities and vice versa as well as various complementary mismatch errors. Details on the numerical realization of the adaptive concept and a report on numerical tests are given.

*Appeared in*

- Optim. Eng., (2018), published online on 23.06.2018, DOI 10.1007/s11081-018-9393-6 .

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# Optimal control of the thermistor problem in three spatial dimensions

*Authors*

- Meinlschmidt, Hannes
- Meyer, Christian
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 33K55 35M10 49J20 49K20

*Keywords*

- partial differential equations, optimal control problems, state constraints

*Abstract*

This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasilinear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions are addressed, which includes analysis of the linearized state system via maximal parabolic regularity, and existence of optimal controls is shown if the temperature gradient is under control. The adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem in form of a qualified optimality system. The theoretical findings are illustrated by numerical results.

*Appeared in*

- SIAM J. Control Optim., Part 1 (Existence of optimal solutions) 55:5 (2017) pp. 2876--2904, Part 2 (Optimality Conditions) 55:4 (2017) pp. 2368--2392.

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# The divisible sandpile with heavy-tailed variables

*Authors*

- Cipriani, Alessandra
- Hazra, Rajat Subra
- Ruszel, Wioletta M.

*2010 Mathematics Subject Classification*

- 60G52 60J45 60G15 82C20

*Keywords*

- divisible sandpile, heavy-tailed variables, alpha-stable random distribution

*DOI*

*Abstract*

This work deals with the divisible sandpile model when an initial configuration sampled from a heavy-tailed distribution. Extending results of Levine et al. (2015) and Cipriani et al. (2016) we determine sufficient conditions for stabilization and non-stabilization on infinite graphs. We determine furthermore that the scaling limit of the odometer on the torus is an α-stable random distribution.

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# Assessment of reduced order Kalman filter for parameter identification in one-dimensional blood flow models using experimental data

*Authors*

- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Caforio, Federica
- Montecinos, Gino
- Müller, Lucas O.
- Blanco, Pablo J.
- Toro, Eleuterio F.

*2008 Physics and Astronomy Classification Scheme*

- 07.05.Tp,47.11.-j,47.63.Cb

*Keywords*

- blood flow, one-dimensional model, Kalman filter, parameter estimation, finite volume method

*Abstract*

This work presents a detailed investigation of a parameter estimation approach based on the reduced order unscented Kalman filter (ROUKF) in the context of one-dimensional blood flow models. In particular, the main aims of this study are (i) to investigate the effect of using real measurements vs. synthetic data (i.e., numerical results of the same in silico model, perturbed with white noise) for the estimation and (ii) to identify potential difficulties and limitations of the approach in clinically realistic applications in order to assess the applicability of the filter to such setups. For these purposes, our numerical study is based on the in vitro model of the arterial network described by [Alastruey et al. 2011, J. Biomech. bf 44], for which experimental flow and pressure measurements are available at few selected locations. In order to mimic clinically relevant situations, we focus on the estimation of terminal resistances and arterial wall parameters related to vessel mechanics (Young's modulus and thickness) using few experimental observations (at most a single pressure or flow measurement per vessel). In all cases, we first perform a theoretical identifiability analysis based on the generalized sensitivity function, comparing then the results obtained with the ROUKF, using either synthetic or experimental data, to results obtained using reference parameters and to available measurements.

*Appeared in*

- Internat. J. Numer. Methods Biomedical Engrg., 33 (2017), pp. e2843/1--e2843/26, DOI 10.1002/cnm.2843 .

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# Optimal sensor placement: A robust approach

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Rautenberg, Carlos N.

ORCID: 0000-0001-9497-9296 - Mohammadi, Masoumeh
- Kanitsar, Martin

*2010 Mathematics Subject Classification*

- 35K86 47J20 49J40 49M15 65J15 65K10

*Keywords*

- Sensor placement, trace class operators, sensitivities, Riccati equation

*Abstract*

We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. The paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem and finalizes with a range of numerical tests.

*Appeared in*

- SIAM J. Control Optim., 55 (2017), pp. 3609--3639.

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# Chimera states in pulse coupled neural networks: The influence of dilution and noise

*Authors*

- Olmi, Simona

ORCID: 0000-0002-8272-3493 - Torcini, Alessandro

*2008 Physics and Astronomy Classification Scheme*

- 05.45-a, 05.45Xt, 84.35.+i

*Keywords*

- Nonlinear dynamics and chaos, synchronization, pulse-coupled oscillators, neural networks, chimera states

*Abstract*

We analyse the possible dynamical states emerging for two symmetrically pulse coupled populations of leaky integrate-and-fire neurons. In particular, we observe broken symmetry states in this set-up: namely, breathing chimeras, where one population is fully synchronized and the other is in a state of partial synchronization (PS) as well as generalized chimera states, where both populations are in PS, but with different levels of synchronization. Symmetric macroscopic states are also present, ranging from quasi-periodic motions, to collective chaos, from splay states to population anti-phase partial synchronization. We then investigate the influence disorder, random link removal or noise, on the dynamics of collective solutions in this model. As a result, we observe that broken symmetry chimeralike states, with both populations partially synchronized, persist up to 80% of broken links and up to noise amplitudes ≃ 8% of threshold-reset distance. Furthermore, the introduction of disorder on symmetric chaotic state has a constructive effect, namely to induce the emergence of chimera-like states at intermediate dilution or noise level. 1 Introduction

*Appeared in*

- Corinto F., Torcini A. (eds.), Nonlinear Dynamics in Computational Neuroscience, PoliTO Springer Series, Springer, Cham, 2019, pp. 65--79. https://doi.org/10.1007/978-3-319-71048-8_5.

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# Joint dynamic probabilistic constraints with projected linear decision rules

*Authors*

- Guigues, Vincent
- Henrion, René

*2010 Mathematics Subject Classification*

- 90C15 90C90 90C30

*Keywords*

- dynamic probabilistic constraints, multistage stochastic linear programs, linear decision rules

*Abstract*

We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized linear model for the underlying stochastic process with noises that are Gaussian or truncated Gaussian, we show that the value and gradient of the objective and constraint functions of the approximating problems can be computed analytically.

*Appeared in*

- Optim. Methods Softw., 32 (2017) pp. 1006--1032.

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# Systems describing electrothermal effects with p(x)-Laplacian like structure for discontinuous variable exponents

*Authors*

- Bulíček, Miroslav
- Glitzky, Annegret
- Liero, Matthias

ORCID: 0000-0002-0963-2915

*2010 Mathematics Subject Classification*

- 35J92 35Q79 35J57 80A20

*Keywords*

- Sobolev spaces with variable exponent, existence of weak solution, thermistor system, p(x)-Laplacian, heat transfer

*Abstract*

We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the p(x)-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a priori L^{1} term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles - the discontinuous variable exponent (which limits the use of standard methods) and the L^{1} right hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.

*Appeared in*

- SIAM J. Math. Anal., 48 (2016), pp. 3496--3514.

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# Asymptotic expansions of the contact angle in nonlocal capillarity problems

*Authors*

- Dipierro, Serena
- Maggi, Francesco
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 76B45 76D45 45M05

*Keywords*

- nonlocal surface tension, contact angle, asymptotics

*Abstract*

We consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting a family of fractional interaction kernels The fractional Young's law (contact angle condition) predicted by these models coincides, in the limit, with the classical Young's law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for s close to 1, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient is negative, and larger if it is positive. In addition, we address the asymptotics of the fractional Young's law in the limit case s close to 0 of interaction kernels with heavy tails. Interestingly, forsmall s, the dependence of the contact angle from the relative adhesion coefficient becomes linear.

*Appeared in*

- J. Nonlin. Sci., 27:5 (2017) pp. 1531--1550.

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# Global existence, uniqueness and stability for nonlinear dissipative systems of bulk-interface interaction

*Authors*

- Disser, Karoline

*2010 Mathematics Subject Classification*

- 35K20 35K51 35R01 35R05 35B50 35B40

*Keywords*

- Bulk-interface interaction, bulk-surface interaction, gradient structure, fast diffusion, porous media equation, nonlinear parabolic system, maximum principle, Poincaré inequality, exponential stability, maximal parabolic regularity, Schaefer's fixed point theorem

*Abstract*

We consider a general class of nonlinear parabolic systems corresponding to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes non-smooth geometries and e.g. slow, fast and "entropic'' diffusion processes under mass conservation. The main results are global well-posedness and exponential stability of equilibria. As a part of the proof, we show bulk-interface maximum principles and a bulk-interface Poincaré inequality. The method of proof for global existence is a simple but very versatile combination of maximal parabolic regularity of the linearization, a priori L^{∞}-bounds and a Schaefer's fixed point argument. This allows us to extend the setting e.g. to Allen-Cahn dissipative dynamics and to include large classes of inhomogeneous boundary conditions and external forces.

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# Study of the bifurcation of a multiple limit cycle of the second kind by means of a Dulac--Cherkas function: A case study

*Authors*

- Schneider, Klaus R.
- Grin, Alexander

*2010 Mathematics Subject Classification*

- 34C05 34C23

*Keywords*

- generalized pendulum equation, limit cycles of the second kind, limit cycle of multiplicity three, bifurcation behavior, Dulac--Cherkas function

*Abstract*

We consider a generalized pendulum equation depending on the scalar parameter $mu$ having for $mu=0$ a limit cycle $Gamma$ of the second kind and of multiplicity three. We study the bifurcation behavior of $Gamma$ for $-1 le mu le (sqrt5+3)/2$ by means of a Dulac-Cherkas function.

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# Strong solutions to nonlocal 2D Cahn--Hilliard--Navier--Stokes systems with nonconstant viscosity, degenerate mobility and singular potential

*Authors*

- Frigeri, Sergio Petro
- Gal, Ciprian G.
- Grasselli, Maurizio
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35Q30 37L30 45K05 76D03 76T99

*Keywords*

- Incompressible binary fluids, Navier-Stokes equations, nonlocal Cahn-Hilliard equations, time discretization schemes, strong solutions, regularization, global attractors

*Abstract*

We consider a nonlinear system which consists of the incompressible Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We suppose that the viscosity depends smoothly on the order parameter as well as the mobility. Moreover, we assume that the mobility is degenerate at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with no-slip boundary condition for the (average) velocity and homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weak-strong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal Cahn-Hilliard equation, with a given velocity field, in the three dimensional case as well.

*Appeared in*

- Nonlinearity, 32 (2019), pp. 678--727, DOI 10.1088/1361-6544/aaedd0 .

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# On fractional elliptic equations in Lipschitz sets and epigraphs: Regularity, monotonicity and rigidity results

*Authors*

- Dipierro, Serena
- Soave, Nicola
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 26A33 60G22 35J91

*Keywords*

- nonlocal semilinear equations, sliding methods, monotonicity, regularity, symmetry

*Abstract*

We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we deal with halfspaces, coercive epigraphs and epigraphs that are flat at infinity.

These results can be seen as the nonlocal counterpart of the celebrated article [4].

*Appeared in*

- Math. Anal., 369:3-4 (2017) pp. 1283--1326.

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# Space-time large deviations in capacity-constrained relay networks

*Authors*

- Hirsch, Christian
- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065 - Patterson, Robert I. A.

*2010 Mathematics Subject Classification*

- 60F10 60K35

*Keywords*

- large deviation, measure-valued differential equations, entropy, capacity, relay

*DOI*

*Abstract*

We consider a single-cell network of random transmitters and fixed relays in a bounded domain of Euclidean space. The transmitters arrive over time and select one relay according to a spatially inhomogeneous preference kernel. Once a transmitter is connected to a relay, the connection remains and the relay is occupied. If an occupied relay is selected by another transmitters with later arrival time, this transmitter becomes frustrated. We derive a large deviation principle for the space-time evolution of frustrated transmitters in the high-density regime.

*Appeared in*

- ALEA Lat. Am. J. Probab. Math. Stat., 15 (2018), pp. 587--615, DOI 10.30757/ALEA.v15-24 .

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# A logistic equation with nonlocal interactions

*Authors*

- Caffarelli, Luis
- Dipierro, Serena
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35Q92 46N60 35R11 60G22

*Keywords*

- Mathematical models for biology, local and nonlocal dispersals, spectral analysis, existence of nontrivial solutions

*Abstract*

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a Lévy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: beginitemize item bounded domains, item periodic environments, item transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. enditemize In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.

*Appeared in*

- Kinet. Relat. Models, 10:1 (2017) pp. 141--170.

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# Anisotropic nonlocal operators regularity and rigidity theorems for a class of anisotropic nonlocal operators

*Authors*

- Farina, Alberto
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 35B53 35R09

*Keywords*

- Nonlocal anisotropic integro-differential equations, regularity result

*Abstract*

We consider here operators which are sum of (possibly) fractional derivatives, with (possibly different) order. The main constructive assumption is that the operator is of order $2$ in one variable. By constructing an explicit barrier, we prove a Lipschitz estimate which controls the oscillation of the solutions in such direction with respect to the oscillation of the nonlinearity in the same direction. As a consequence, we obtain a rigidity result that, roughly speaking, states that if the nonlinearity is independent of a coordinate direction, then so is any global solution (provided that the solution does not grow too much at infinity). A Liouville type result then follows as a byproduct.

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# On maximal parabolic regularity for non-autonomous parabolic operators

*Authors*

- Disser, Karoline
- ter Elst, A. F. M.
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35B65 47A07 35K20 35B45 46B70

*Keywords*

- Non-autonomous evolution equations, parabolic initial boundary value problems, maximal parabolic regularity, extrapolation of maximal parabolic regularity

*DOI*

*Abstract*

We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents r ≠ 2. This allows us to prove maximal parabolic Lr-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.

*Appeared in*

- J. Differential Equations, 262 (2017), pp. 2039--2072.

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# (Sub-) Gradient formulae for probability functions of random inequality systems under Gaussian distribution

*Authors*

- van Ackooij, Wim
- Henrion, René

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- stochastic optimization, gradients of probability functions, spheric radial decomposition, multivariate Gaussian distribution, Clarke subdifferential, Mordukhovich subdifferential, probabilistic constraint

*Abstract*

We consider probability functions of parameter-dependent random inequality systems under Gaussian distribution. As a main result, we provide an upper estimate for the Clarke subdifferential of such probability functions without imposing compactness conditions. A constraint qualification ensuring continuous differentiability is formulated. Explicit formulae are derived from the general result in case of linear random inequality systems. In the case of a constant coefficient matrix an upper estimate for even the smaller Mordukhovich subdifferential is proven.

*Appeared in*

- SIAM ASA J. Uncertain. Quantif., 5 (2017) pp. 63--87.

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# Uniform exponential decay for reaction-diffusion systems with complex-balanced mass-action kinetics

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35K57 35B40 35Q79 92E20

*Keywords*

- Reaction-diffusion systems, mass-action law, log-Sobolev inequality, exponential decay of relative entropy, energy-dissipation estimate, complex balance condition, detailed balance condition, convexity method

*Abstract*

We consider reaction-diffusion systems on a bounded domain with no-flux boundary conditions. All reactions are given by the mass-action law and are assumed to satisfy the complex-balance condition. In the case of a diagonal diffusion matrix, the relative entropy is a Liapunov functional. We give an elementary proof for the Liapunov property as well a few explicit examples for the condition of complex or detailed balancing. We discuss three methods to obtain energy-dissipation estimates, which guarantee exponential decay of the relative entropy, all of which rely on the log-Sobolev estimate and suitable handling of the reaction terms as well as the mass-conservation relations. The three methods are (i) a convexification argument based on the author's joint work with Haskovec and Markowich, (ii) a series of analytical estimates derived by Desvillettes, Fellner, and Tang, and (iii) a compactness argument of developed by Glitzky and Hünlich.

*Appeared in*

- Patterns of Dynamics, P. Gurevich, J. Hell, B. Sandstede, A. Scheel, eds., Proceedings in Mathematics & Statistics, Springer, 2018, pp. 149--171, DOI 10.1007/978-3-319-64173-7_10 .

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# Adiabatic theory of champion solitons

*Authors*

- Pickartz, Sabrina
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X

*2010 Mathematics Subject Classification*

- 78A60 35Q60 70H11

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k, 42.81.Dp, 05.45.Yv, 03.65.Nk

*Keywords*

- Champion solitons, All-optical switching, Extreme events, Soliton perturbation theory, Event horizons

*DOI*

*Abstract*

We consider scattering of small-amplitude dispersive waves at an intense optical soliton which constitutes a nonlinear perturbation of the refractive index. Specifically, we consider a single-mode optical fiber and a group velocity matched pair: an optical soliton and a nearly perfectly reflected dispersive wave, a fiber-optical analogue of the event horizon. By combining (i) an adiabatic approach that is used in soliton perturbation theory and (ii) scattering theory from Quantum Mechanics, we give a quantitative account for the evolution of all soliton parameters. In particular, we quantify the increase in the soliton peak power that may result in spontaneous appearance of an extremely large, so-called champion soliton. The presented adiabatic theory agrees well with the numerical solutions of the pulse propagation equation. Moreover, for the first time we predict the full frequency band of the scattered dispersive waves and explain an emerging caustic structure in the space-time domain.

*Appeared in*

- Phys. Rev. A, 94 (2016) pp. 033811/1--033811/12.

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# Nonlocal minimal surfaces: Interior regularity, quantitative estimates and boundary stickiness

*Authors*

- Dipierro, Serena
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 49Q05 53A10 58E12 35R11

*Keywords*

- nonlocal minimal surfaces, regularity theory and applications

*Abstract*

We consider surfaces which minimize a nonlocal perimeter functional and we discuss their interior regularity and rigidity properties, in a quantitative and qualitative way, and their (perhaps rather surprising) boundary behavior. We present at least a sketch of the proofs of these results, in a way that aims to be as elementary and self contained as possible, referring to the papers [CRS10, SV13, CV13, BFV14, FV, DSV15, CSV16] for full details.

*Appeared in*

- Recent Developments in Nonlocal Theory, G. Palatucci, T. Kuusi, eds., Sciendo, de Gruyter, 2018, pp. 165--209.

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# The Widom--Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality

*Authors*

- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065 - Külske, Christof

*2010 Mathematics Subject Classification*

- 82C21 60K35

*Keywords*

- Gibbsianness, non-Gibbsianness, point processes, Widom-Rowlinson model, spin-flip dynamics, quasilocality, non almost-sure quasilocality, -topology

*DOI*

*Abstract*

We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs-non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices. We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-a.s. quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time tG> 0 the model is a.s. quasilocal. For the colorsymmetric model there is no reentrance. On the constructive side, for all t > tG , we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary conditions.

*Appeared in*

- Ann. Appl. Probab., 27 (2017), pp. 3845--3892, DOI 10.1214/17-AAP1298 .

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# A class of unstable free boundary problems

*Authors*

- Dipierro, Serena
- Karakhanyan, Aram
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R35

*Keywords*

- Free boundary problems

*Abstract*

We consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter. The main difference with the existing literature is that the total energy is here a nonlinear superposition of the either local or nonlocal surface tension effect with the elastic energy. In sharp contrast with the linear case, the problem considered in this paper is unstable, namely a minimizer in a given domain is not necessarily a minimizer in a smaller domain. We provide an explicit example for this instability. We also give a free boundary condition, which emphasizes the role played by the domain in the geometry of the free boundary. In addition, we provide density estimates for the free boundary and regularity results for the minimal solution. As far as we know, this is the first case in which a nonlinear function of the perimeter is studied in this type of problems. Also, the results obtained in this nonlinear setting are new even in the case of the local perimeter, and indeed the instability feature is not a consequence of the possibly nonlocality of the problem, but it is due to the nonlinear character of the energy functional.

*Appeared in*

- Anal. PDE, 10:6 (2017) pp. 1317--1359.

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# Parameter identification in a semilinear hyperbolic system

*Authors*

- Egger, Herbert
- Kugler, Thomas
- Strogies, Nikolai

*2010 Mathematics Subject Classification*

- 35R30 49J20 49N45 65J22 74J25

*Keywords*

- Parameter identification, semilinear wave equation, nonlinear inverse problem, Tikhonov regularization, approximate source condition, conditional stability

*Abstract*

We consider the identification of a nonlinear friction law in a one-dimensional damped wave equation from additional boundary measurements. Well-posedness of the governing semilinear hyperbolic system is established via semigroup theory and contraction arguments. We then investigate the inverse problem of recovering the unknown nonlinear damping law from additional boundary measurements of the pressure drop along the pipe. This coefficient inverse problem is shown to be ill-posed and a variational regularization method is considered for its stable solution. We prove existence of minimizers for the Tikhonov functional and discuss the convergence of the regularized solutions under an approximate source condition. The meaning of this condition and some arguments for its validity are discussed in detail and numerical results are presented for illustration of the theoretical findings.

*Appeared in*

- Inverse Problems, 33 (2017), pp. 055022/1--055022/25, DOI 10.1088/1361-6420/aa648c .

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# Fast decay of covariances under delta-pinning in the critical and supercritical membrane model

*Authors*

- Bolthausen, Erwin
- Cipriani, Alessandra
- Kurt, Noemi

*2010 Mathematics Subject Classification*

- 31B30 39A12 60K35 60K37 82B41

*Keywords*

- membrane model, pinning, bilaplacian, decay of covariances

*DOI*

*Abstract*

We consider the membrane model, that is the centered Gaussian field on Z^d whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a delta-pinning condition, giving a reward of strength epsilon for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions larger than 4 covariances of the pinned field decay at least stretched-exponentially, as opposed to the field without pinning, where the decay is polynomial in dimensions larger than 5 and logarithmic in 4 dimensions. The proof is based on estimates for certain discrete Sobolev norms, and on a Bernoulli domination result.

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# Exponential decay of covariances for the supercritical membrane model

*Authors*

- Bolthausen, Erwin
- Cipriani, Alessandra
- Kurt, Noemi

*2010 Mathematics Subject Classification*

- 60K35 31B30 39A12 60K37 82B41

*Keywords*

- membrane model, pinning, bilaplacian, decay of covariances

*DOI*

*Abstract*

We consider the membrane model, that is the centered Gaussian field on Zd whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a δ-pinning condition, giving a reward of strength ε for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions d≥5 covariances of the pinned field decay at least exponentially, as opposed to the field without pinning, where the decay is polynomial. The proof is based on estimates for certain discrete weighted norms, a percolation argument and on a Bernoulli domination result.

*Appeared in*

- Comm. Math. Phys., 353:3 (2017) pp. 1217--1240.

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# The parabolic Anderson model on the hypercube

*Authors*

- Avena, Luca
- Gün, Onur
- Hesse, Marion

*2010 Mathematics Subject Classification*

- 60H25 82C27 92D25 82D30 60K37

*Keywords*

- parabolic Anderson model, mutation-selection model, localisation, random energy model

*DOI*

*Abstract*

We consider the parabolic Anderson model (PAM) on the n-dimensional hypercube with random i.i.d. potentials. We parametrize time by volume and study the solution at the location of the k-th largest potential. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales it behaves like a system without diffusion, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the Anderson operator, for which we give precise asymptotics. Moreover, the transition time depends only on the difference between the largest and k-th largest potential. One of our main motivations in this article is to investigate the mutation-selection model of population genetics on a random fitness landscape, which is given by the ratio of the solution of PAM to its total mass, with the field corresponding to the fitness landscape. We show that the phase transition of the solution translates to the mutation-selection model as follows: a population initially concentrated at the site of the k-th best fitness value moves completely to the site of the best fitness on time scales where the transition of growth rates happens. The class of potentials we consider involve the Random Energy Model (REM) of statistical physics which is studied as one of the main examples of a random fitness landscape.

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# Efficient all-optical control of solitons

*Authors*

- Pickartz, Sabrina
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X

*2010 Mathematics Subject Classification*

- 78A60 35Q60 70H11

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k, 42.81.Dp, 05.45.Yv, 03.65.Nk

*Keywords*

- Champion solitons, All-optical switching, Extreme events, Soliton perturbation theory, Event horizons

*Abstract*

We consider the phenomenon of an optical soliton controlled (eg. amplified) by a much weaker second pulse which is efficiently scattered at the soliton. An important problem in this context is to quantify the small range of parameters at which the interaction takes place. This has been achieved by using adiabatic ODEs for the soliton characteristics, which is much faster than an empirical scan of the full propagation equations for all parameters in question.

*Appeared in*

- Opt. Quantum Electron., (2016) pp. 503/1--503/7.

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# Smoothing the payoff for efficient computation of basket option prices

*Authors*

- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Siebenmorgen, Markus
- Tempone, Raul

*2010 Mathematics Subject Classification*

- 91G60 65D30 65C20

*Keywords*

- Computational Finance, European Option Pricing, Multivariate approximation and integration, Sparse grids, Stochastic Collocation methods, Monte Carlo and Quasi Monte Carlo methods

*Abstract*

We consider the problem of pricing basket options in a multivariate Black Scholes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster compared to Monte Carlo or Quasi Monte Carlo in dimensions up to 25.

*Appeared in*

- Quant. Finance, (2017), appeared online, DOI 10.1080/14697688.2017.1308003 .

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# Traffic flow densities in large transport networks

*Authors*

- Hirsch, Christian
- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065 - Keeler, Paul

ORCID: 0000-0002-2063-1075 - Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857

*2010 Mathematics Subject Classification*

- 60K30 60F15 90B20

*Keywords*

- traffic density, routing, navigation, transport network, sub-ballisticity

*DOI*

*Abstract*

We consider transport networks with nodes scattered at random in a large domain. At certain local rates, the nodes generate traffic flowing according to some navigation scheme in a given direction. In the thermodynamic limit of a growing domain, we present an asymptotic formula expressing the local traffic flow density at any given location in the domain in terms of three fundamental characteristics of the underlying network: the spatial intensity of the nodes together with their traffic generation rates, and of the links induced by the navigation. This formula holds for a general class of navigations satisfying a link-density and a sub-ballisticity condition. As a specific example, we verify these conditions for navigations arising from a directed spanning tree on a Poisson point process with inhomogeneous intensity function.

*Appeared in*

- Adv. Appl. Probab., 49 (2017), pp. 1091--1115, DOI 10.1017/apr.2017.35 .

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# Analysis of improved Nernst--Planck--Poisson models of isothermal compressible electrolytes subject to chemical reactions: The case of a degenerate mobility matrix

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35Q35 76T30 78A57 35Q30 76N10 35M33 35D30 35B45

*Keywords*

- Electrolyte, electrochemical interface, chemical reaction, compressible fluid, Navier-Stokes equations, advection-diffusion-reaction equations, PDE system of mixed-type, a-priori estimates, weak solution

*Abstract*

We continue our investigations of the improved Nernst-Planck-Poisson model introduced by Dreyer, Guhlke and Müller 2013. In the paper by Dreyer, Druet, Gajewski and Guhlke 2016, the analysis relies on the hypothesis that the mobility matrix has maximal rank under the constraint of mass conservation (rank N-1 for the mixture of N species). In this paper we allow for the case that the positive eigenvalues of the mobility matrix tend to zero along with the partial mass densities of certain species. In this approach the mobility matrix has a variable rank between zero and N-1 according to the number of locally available species. We set up a concept of weak solution able to deal with this scenario, showing in particular how to extend the fundamental notion of emphdifferences of chemical potentials that supports the modelling and the analysis in Dreyer, Druet, Gajewski and Guhlke 2016. We prove the global-in-time existence in this solution class.

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# Gradient structure for optoelectronic models of semiconductors

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Peschka, Dirk

ORCID: 0000-0002-3047-1140 - Rotundo, Nella
- Thomas, Marita

*2010 Mathematics Subject Classification*

- 82D37 78A35 35K57 80A17

*Keywords*

- Gradient flow, optoelectronic semiconductor model, dual dissipation potential, reaction-diffusion systems

*Abstract*

We derive an optoelectronic model based on a gradient formulation for the relaxation of electron-, hole- and photon- densities to their equilibrium state. This leads to a coupled system of partial and ordinary differential equations, for which we discuss the isothermal and the non-isothermal scenario separately

*Appeared in*

- Quintela P. et al. (eds) Progress in Industrial Mathematics at ECMI 2016. ECMI 2016. Mathematics in Industry, vol 26. Springer, Cham, pp. 291-298, changed title: On some extension of energy-drift-diffusion models: Gradient structure for optoelectronic models of semiconductors.

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# Decay to equilibrium for energy-reaction-diffusion systems

*Authors*

- Haskovec, Jan
- Hittmeir, Sabine
- Markowich, Peter
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35K57 35B40 35Q79

*Keywords*

- Energy-reaction-diffusion systems, entropy functional, dissipation functional, log-Sobolev inequality, entropy entropy-production balance

*Abstract*

We derive thermodynamically consistent models of reaction-diffusion equations coupled to a heat equation. While the total energy is conserved, the total entropy serves as a driving functional such that the full coupled system is a gradient flow. The novelty of the approach is the Onsager structure, which is the dual form of a gradient system, and the formulation in terms of the densities and the internal energy. In these variables it is possible to assume that the entropy density is strictly concave such that there is a unique maximizer (thermodynamical equilibrium) given linear constraints on the total energy and suitable density constraints. We consider two particular systems of this type, namely, a diffusion-reaction bipolar energy transport system, and a drift-diffusion-reaction energy transport system with confining potential. We prove corresponding entropy-entropy production inequalities with explicitely calculable constants and establish the convergence to thermodynamical equilibrium, at first in entropy and further in L^{1} using Cziszar-Kullback-Pinsker type inequalities.

*Appeared in*

- SIAM J. Math. Anal., (2018), DOI 10.1137/16M1062065 .

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# Long-time behavior for crystal dislocation dynamics

*Authors*

- Patrizi, Stefania
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 82D25 35R09 74E15 35R11 47G20

*Keywords*

- Peierls-Nabarro model, nonlocal integro-differential equations, dislocation dynamics, attractive/repulsive potentials, collisions

*Abstract*

We describe the asymptotic states for the solutions of a nonlocal equation of evolutionary type, which have the physical meaning of the atom dislocation function in a periodic crystal.

More precisely, we can describe accurately the ``smoothing effect'' on the dislocation function occurring slightly after a ``particle collision'' (roughly speaking, two opposite transitions layers average out) and, in this way, we can trap the atom dislocation function between a superposition of transition layers which, as time flows, approaches either a constant function or a single heteroclinic (depending on the algebraic properties of the orientations of the initial transition layers).The results are endowed of explicit and quantitative estimates and, as a byproduct, we show that the ODE systems of particles that governs the evolution of the transition layers does not admit stationary solutions (i.e., roughly speaking, transition layers always move).

*Appeared in*

- Math. Models Methods Appl. Sci., 27:12 (2017) pp. 2185--2228.

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# Optimal entropy-transport problems and a new Hellinger--Kantorovich distance between positive measures

*Authors*

- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Savaré, Giuseppe

*2010 Mathematics Subject Classification*

- 28A33 54E35 49Q20 49J35 49J40 49K35 46G99

*Keywords*

- Entropy-transport problem, Hellinger-Kantorovich distance, relative entropy, optimality conditions, cone over metric space

*Abstract*

We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.

*Appeared in*

- Inv. Math., 211 (2018), pp. 969--1117 (published online on 14.12.2017), DOI 10.1007/s00222-017-0759-8 .

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# Convergence to equilibrium in energy-reaction-diffusion systems using vector-valued functional inequalities

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Mittnenzweig, Markus

*2010 Mathematics Subject Classification*

- 35K57 35B40 35Q79 92E20

*Keywords*

- Energy-reaction-diffusion systems, vector-valued inequalities, cross diffusion, log-Sobolev inequality, entropy functional, exponential decay of relative entropy, convexity method

*Abstract*

We discuss how the recently developed energy-dissipation methods for reactiondi usion systems can be generalized to the non-isothermal case. For this we use concave entropies in terms of the densities of the species and the internal energy, where the importance is that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method.

*Appeared in*

- J. Nonlinear Sci., 28 (2018), pp. 765--806 (published online on 11.11.2017), DOI 10.1007/s00332-017-9427-9 .

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# Nonlocal phase transitions in homogeneous and periodic media

*Authors*

- Cozzi, Matteo
- Dipierro, Serena
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 82B26

*Keywords*

- nonlocal Ginzburg-Landau-Allen-Cahn equation, De Giorgi conjecture, planelike minimizers, chaotic orbits

*Abstract*

We discuss some results related to a phase transition model in which the potential energy induced by a double-well function is balanced by a fractional elastic energy. In particular, we present asymptotic results (such as Gamma-convergence, energy bounds and density estimates for level sets), flatness and rigidity results, and the construction of planelike minimizers in periodic media.

Finally, we consider a nonlocal equation with a multiwell potential, motivated by models arising in crystal dislocations, and we construct orbits exhibiting symbolic dynamics, inspired by some classical results by Paul Rabinowitz.

*Appeared in*

- J. Fixed Point Theory Appl., 19:1 (2017) pp 387--405.

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# Is there an impact of small phase lags in the Kuramoto model?

*Authors*

- Omel'chenko, Oleh

ORCID: 0000-0003-0526-1878 - Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 34C15 37N20 37N25

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt, 89.75.Kd

*Keywords*

- coupled oscillators, synchronization transition, Kuramoto model

*Abstract*

We discuss the influence of small phase lags on the synchronization transitions in the Kuramoto model for a large inhomogeneous population of globally coupled phase oscillators. Without a phase lag, all unimodal distributions of the natural frequencies give rise to a classical synchronization scenario, where above the onset of synchrony at the Kuramoto threshold there is an increasing synchrony for increasing coupling strength. We show that already for arbitrarily small phase lags there are certain unimodal distributions of natural frequencies such that for increasing coupling strength synchrony may decrease and even complete incoherence may regain stability. Moreover, our example allows a qualitative understanding of the mechanism for such non-universal synchronization transitions.

*Appeared in*

- Chaos, 26 (2016) pp. 094806/1--094806/6.

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# Quantitative flatness results and $BV$-estimates for stable nonlocal minimal surfaces

*Authors*

- Cinti, Eleonora
- Serra, Joaquim
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 49Q05 35R11 53A10

*Keywords*

- Nonlocal minimal surfaces, existence and regularity results

*Abstract*

We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case. On the one hand, we establish universal $BV$-estimates in every dimension $nge 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has finite classical perimeter in $B_1/2$, with a universal bound. This nonlocal result is new even in the case of $s$-perimeters and its local counterpart (for classical stable minimal surfaces) was known only for simply connected two-dimensional surfaces immersed in $R^3$. On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions $n=2,3$. More precisely, we show that a stable set in $B_R$, with $R$ large, is very close in measure to being a half space in $B_1$ ---with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane.

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# Capillarity problems with nonlocal surface tension energies

*Authors*

- Maggi, Francesco
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 76B45 35R11 49K21

*Keywords*

- nonlocal perimeter functional, long range particle interactions, capillarity phenomena, Young's law

*Abstract*

We explore the possibility of modifying the classical Gauss free energy functional used in capillarity theory by considering surface tension energies of nonlocal type. The corresponding variational principles lead to new equilibrium conditions which are compared to the mean curvature equation and Young's law found in classical capillarity theory. As a special case of this family of problems we recover a nonlocal relative isoperimetric problem of geometric interest.

*Appeared in*

- Commun. Partial Differential Equations, 42:9 (2017) pp. 1403--1446.

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# Definition of fractional Laplacian for functions with polynomial growth

*Authors*

- Dipierro, Serena
- Savin, Ovidiu
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11

*Keywords*

- fractional operators, growth conditions, conditions at infinity

*Abstract*

We introduce a notion of fractional Laplacian for functions which grow more than linearly at infinity. In such case, the operator is not defined in the classical sense: nevertheless, we can give an ad-hoc definition which can be useful for applications in various fields, such as blowup and free boundary problems. In this setting, when the solution has a polynomial growth at infinity, the right hand side of the equation is not just a function, but an equivalence class of functions modulo polynomials of a fixed order. We also give a sharp version of the Schauder estimates in this framework, in which the full smooth Hölder norm of the solution is controlled in terms of the seminorm of the nonlinearity. Though the method presented is very general and potentially works for general nonlocal operators, for clarity and concreteness we focus here on the case of the fractional Laplacian.

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# Higher-order discontinuous Galerkin time stepping and local projection stabilization techniques for the transient Stokes problem

*Authors*

- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - Becher, Simon
- Matthies, Gunar

*2010 Mathematics Subject Classification*

- 65M12 65M15 65M60

*Keywords*

- Stabilized finite elements, transient Stokes equation, equal-order elements, local projection, discontinuous Galerkin method

*DOI*

*Abstract*

We introduce and analyze discontinuous Galerkin time discretizations coupled with continuous finite element methods based on equal-order interpolation in space for velocity and pressure in transient Stokes problems. Spatial stability of the pressure is ensured by adding a stabilization term based on local projection. We present error estimates for the semi-discrete problem after discretization in space only and for the fully discrete problem. The fully discrete pressure shows an instability in the limit of small time step length. Numerical tests are presented which confirm our theoretical results including the pressure instability.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 313 (2017) pp. 28--52.

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# Distributed optimal control of a nonstandard nonlocal phase field system

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K55 49K20 74A15

*Keywords*

- Distributed optimal control, nonlinear phase field systems, nonlocal operators, first-order necessary optimality conditions

*Abstract*

We investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.

*Appeared in*

- AIMS Math., 1 (2016), pp. 246--281.

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# Discretisation and error analysis for a mathematical model of milling processes

*Authors*

- Hömberg, Dietmar
- Rott, Oliver
- Sturm, Kevin

*2010 Mathematics Subject Classification*

- 35Q74 65M15 74H15

*Keywords*

- error estimates, high speed milling, finite elements

*DOI*

*Abstract*

We investigate a mathematical model for milling where the cutting tool dynamics is considered together with an elastic workpiece model. Both are coupled by the cutting forces consisting of two dynamic components representing vibrations of the tool and of the workpiece, respectively, at the present and previous tooth periods. We develop a numerical solution algorithm and derive error estimates both for the semi-discrete and the fully discrete numerical scheme. Numerical computations in the last section support the analytically derived error estimates.

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# On the evolutionary Gamma-convergence of gradient systems modeling slow and fast chemical reactions

*Authors*

- Disser, Karoline
- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Zinsl, Jonathan

*2010 Mathematics Subject Classification*

- 34E15 49J40 49J45 80A30 92E20

*Keywords*

- Gradient systems, mass-action law, dissipation potential, energy dissipation balance, multiscale evolution problems, reversible reaction kinetics, Gamma-convergence

*Abstract*

We investigate the limit passage for a system of ordinary differential equations modeling slow and fast chemical reaction of mass-action type, where the rates of fast reactions tend to infinity. We give an elementary proof of convergence to a reduced dynamical system acting in the slow reaction directions on the manifold of fast reaction equilibria. Then we study the entropic gradient structure of these systems and prove an E-convergence result via Γ-convergence of the primary and dual dissipation potentials, which shows that this structure carries over to the fast reaction limit. We recover the limit dynamics as a gradient flow of the entropy with respect to a pseudo-metric.

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# Rigidity of critical points for a nonlocal Ohta--Kawasaki energy

*Authors*

- Dipierro, Serena
- Novaga, Matteo
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 49Q10 49Q20 35B38 58J70

*Keywords*

- Otha-Kawasaki functional, long-range interactions, symmetry results, critical point

*Abstract*

We investigate the shape of critical points for a free energy consisting of a nonlocal perimeter plus a nonlocal repulsive term. In particular, we prove that a volume-constrained critical point is necessarily a ball if its volume is sufficiently small with respect to its isodiametric ratio, thus extending a result previously known only for global minimizers.

We also show that, at least in one-dimension, there exist critical points with arbitrarily small volume and large isodiametric ratio. This example shows that a constraint on the diameter is, in general, necessary to establish the radial symmetry of the critical points.

*Appeared in*

- Nonlinearity, 30:4 (2017) pp. 1523--1535.

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# Modeling and efficient simulations of broad-area edge-emitting semiconductor lasers and amplifiers

*Authors*

- Radziunas, Mindaugas

*2010 Mathematics Subject Classification*

- 35Q60 65Y05 68W10 68W15 78-04

*Keywords*

- Traveling wave model, numerical scheme, simulations, parallel computations, MPI, semiconductor device, broad area

*DOI*

*Abstract*

We present a (2+1)-dimensional partial differential equation model for spatial-lateral dynamics of edge-emitting broad-area semiconductor devices and several extensions of this model describing different physical effects. MPI-based parallelization of the resulting middlesize numerical problem is implemented and tested on the blade cluster and separate multi-core computers at the Weierstrass Institute in Berlin. It was found, that an application of 25-30 parallel processes on all considered platforms was guaranteeing a nearly optimal performance of the algorithm with the speedup around 20-25 and the efficiency of 0.7-0.8. It was also shown, that a simultaneous usage of several in-house available multi-core computers allows a further increase of the speedup without a significant loss of the efficiency. Finally, an importance of the considered problem and the efficient numerical simulations of this problem were illustrated by a few examples occurring in real world applications.

*Appeared in*

- Int. J. High Perform. Comput. Appl., 32 (2018), pp. 512--522, DOI 10.1177/1094342016677086 .

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# Modeling and simulation of non-isothermal rate-dependent damage processes in inhomogeneous materials using the phase-field approach

*Authors*

- Kraus, Christiane
- Radszuweit, Markus

*2010 Mathematics Subject Classification*

- 74F05 74F20 74N20 74R20 74S05 80A17

*2008 Physics and Astronomy Classification Scheme*

- 02.70.Dh 62.20.mt 64.70.kd 64.75.Op 65.40.De

*Keywords*

- Damage, Fracture, Phase field model, Binary alloys, Thermo-mechanics, Spinodal decomposition, Finite Element

*Abstract*

We present a continuum model that incorporates rate-dependent damage and fracture, a material order parameter field and temperature. Different material characteristics throughout the medium yield a strong inhomogeneity and affect the way fracture propagates. The phasefield approach is employed to describe degradation. For the material order parameter we assume a Cahn Larché-type dynamics, which makes the model in particular applicable to binary alloys. We give thermodynamically consistent evolution equations resulting from a unified variational approach. Diverse coupling mechanisms can be covered within the model, such as heat dissipation during fracture, thermal-expansion-in- duced failure and elastic-inhomogeneity effects. We furthermore present an adaptive Finite Element code in two space dimensions, that is capable of solving such a highly nonlinear and non-convex system of partial differential equations. With the help of this tool we conduct numerical experiments of different complexity in order to investigate the possibilities and limitations of the presented model. A main feature of our model is that we can describe the process of micro-crack nucleation in regions of partial damage to form macro-cracks in a unifying approach.

*Appeared in*

- Computational Mechanics, 60 (2017) pp. 163--179, DOI 10.1007/s00466-017-1393-4 .

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# Analysis of an operator-differential model for magnetostrictive energy harvesting

*Authors*

- Davino, Daniele
- Krejčí, Pavel
- Pimenov, Alexander
- Rachinskii, Dmitrii
- Visone, Ciro

*2010 Mathematics Subject Classification*

- 34C55 47J40 74F15, 37N15

*Keywords*

- magnetostrictive materials, hysteresis, energy harvesting, optimization problems

*Abstract*

We present a model of, and analysis of an optimization problem for, a magnetostrictive harvesting device which converts mechanical energy of the repetitive process such as vibrations of the smart material to electrical energy that is then supplied to an electric load. The model combines a lumped differential equation for a simple electronic circuit with an operator model for the complex constitutive law of the magnetostrictive material. The operator based on the formalism of the phenomenological Preisach model describes nonlinear saturation effects and hysteresis losses typical of magnetostrictive materials in a thermodynamically consistent fashion. We prove well-posedness of the full operator-differential system and establish global asymptotic stability of the periodic regime under periodic mechanical forcing that represents mechanical vibrations due to varying environmental conditions. Then we show the existence of an optimal solution for the problem of maximization of the output power with respect to a set of controllable parameters (for the periodically forced system). Analytical results are illustrated with numerical examples of an optimal solution.

*Appeared in*

- Comm. Nonl. Sc. Num. Sim., 39 (2016) pp. 504--519.

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# The Phillip Island penguin parade (a mathematical treatment)

*Authors*

- Dipierro, Serena
- Lombardini, Luca
- Miraglio, Pietro
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 92B05 92B25 37N25

*Keywords*

- population dynamics, Eudyptula minor, Phillip Island, mathematical models

*DOI*

*Abstract*

We present a simple mathematical formulation to describe the little penguins parade in Phillip Island. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account "natural parameters", such as the sight of the penguins, their cruising speed and the possible "fear" of animals. On the one hand, this favors the formation of rafts of penguins but, on the other hand, this may lead to the panic of isolated and exposed individuals. The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behavior of the speed of the penguins, the mathematical treatment (to get existence and uniqueness of the solution) is based on a ßtop-and-go" procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home (there are also cases in which some penguins freeze due to panic). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguins parade.

*Appeared in*

- The ANZIAM Journal, published online on August 8, 2018, https://doi.org/10.1017/S1446181118000147.

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# Dispersive time-delay dynamical systems

*Authors*

- Pimenov, Alexander
- Slepneva, Svetlana
- Huyet, Guillaume
- Vladimirov, Andrei G.

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Mi, 42.65.Sf, 42.55.Ah, 42.55.Px

*Keywords*

- Time-delay systems, chromatic dispersion, laser dynamics, FDML, modulational instability

*DOI*

*Abstract*

We present a theoretical approach to model the dynamics of a dispersive nonlinear system using a set of delay differential equations with distributed delay term. We illustrate the use of this approach by considering a frequency swept laser comprimising a semiconductor optical amplifier (SOA), a tunable bandpass filter and a long dispersive fiber delay line. We demonstrate that this system exhibits a rich spectrum of dynamical behaviors which are in agreement with the experimental observations. In particular, the multimode modulational instability observed experimentally in the laser in the anomalous dispersion regime and leading to a turbulent laser output was found analytically in the limit of large delay time.

*Appeared in*

- Phys. Rev. Lett., 118 (2017) 193901.

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# Numerical studies of higher order variational time stepping schemes for evolutionary Navier--Stokes equations

*Authors*

- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - Matthies, Gunar

*2010 Mathematics Subject Classification*

- 76D05 65M20 65M60

*Keywords*

- ransient incompressible Navier--Stokes equations, inf-sup stable pairs of finite element spaces, discontinuous Galerkin methods, continuous Galerkin--Petrov methods

*DOI*

*Abstract*

We present in this paper numerical studies of higher order variational time stepping schemes com-bined with finite element methods for simulations of the evolutionary Navier--Stokes equations. In particular, conforming inf-sup stable pairs of finite element spaces for approximating velocity and pressure are used as spatial discretization while continuous Galerkin--Petrov methods (cGP) and discontinuous Galerkin (dG) methods are applied as higher order variational time discretizations. Numerical results for the well-known problem of incompressible flows around a circle will be presented.

*Appeared in*

- Huang Z., Stynes M., Zhang Z. (eds) Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016. Lecture Notes in Computational Science and Engineering, vol 120. Springer, Cham, pp 19--33.

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# Single logarithmic conditional stability in determining unknown boundaries

*Authors*

- Elschner, Johannes
- Hu, Guanghui
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 35R30 74B05 78A46

*Keywords*

- inverse problems, stability estimate, elliptic equations, Carleman estimate

*DOI*

*Abstract*

We prove a conditional stability estimate of log-type for determining unknown boundaries from a single Cauchy data taken on an accessible subboundary. Our approach relies on new interior and boundary estimates derived from the Carleman estimate for elliptic equations. A local stability result for target identification of an acoustic sound-soft scatterer from a single far-field pattern is also obtained.

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# Quenched large deviations for simple random walks on percolation clusters including long-range correlations

*Authors*

- Berger, Noam
- Mukherjee, Chiranjib
- Okamura, Kazuki

*2010 Mathematics Subject Classification*

- 60J65 60J55 60F10 60K37

*Keywords*

- Large deviations, random walk on percolation clusters, long-range correlations, random interlacements, Gaussian free field, random cluster model

*DOI*

*Abstract*

We prove a quenched large deviation principle (LDP)for a simple random walk on a supercritical percolation cluster (SRWPC) on the lattice.The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and its vacant set and the level sets of the Gaussian free field. Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan ([KRV06]) for proving quenched LDP for elliptic diffusions with a random drift, and by Yilmaz ([Y08]) and Rosenbluth ([R06]) for similar results regarding elliptic random walks in random environment, we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the pair empirical measures if the environment Markov chain in the non-elliptic case of SRWPC. Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas. The main approach of our proofs are based on exploiting coercivity properties of the relative entropy in the context of convex variational analysis, combined with input from ergodic theory and invoking geometric properties of the percolation cluster under supercriticality.

*Appeared in*

- Commun. Math. Phys., 358 (2018), pp. 633--673.

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# Improvement of flatness for nonlocal phase transitions

*Authors*

- Dipierro, Serena
- Serra, Joaquim
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 60G22 82B26

*Keywords*

- nonlocal phase transitions, rigidity results, sliding methods

*DOI*

*Abstract*

We prove an improvement of flatness result for nonlocal phase transitions. For a class of nonlocal equations, we obtain a result in the same spirit of a celebrated theorem of Savin for the classical case. The results presented are completely new even for the case of the fractional Laplacian, but the robustness of the proofs allows us to treat also more general, possibly anisotropic, integro-differential operators.

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# Existence of solutions to an anisotropic degenerate Cahn--Hilliard-type equation

*Authors*

- Dziwnik, Marion
- Jachalski, Sebastian

*2010 Mathematics Subject Classification*

- 74Gxx 74Hxx 35K55 35K65 49Jxx 82C26

*Keywords*

- degenerate Cahn--Hilliard equation, anisotropic parabolic equations, existence of solutions, boundedness of solutions

*Abstract*

We prove existence of solutions to an anisotropic Cahn-Hilliard-type equation with degenerate diffusional mobility. In particular, the mobility vanishes at the pure phases, which is typically used to model motion by surface diffusion. The main difficulty of the present existence result is the strong non-linearity given by the fourth-order anisotropic operator. Imposing particular assumptions on the domain and assuming that the strength of the anisotropy is sufficiently small enables to establish appropriate auxiliary results which play an essential part in the present existence proof. In addition to the existence we show that the absolute value of the corresponding solutions is bounded by 1.

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# A coupled ligand-receptor bulk-surface system on a moving domain: Well posedness, regularity and convergence to equilibrium

*Authors*

- Alphonse, Amal
- Elliott, Charles M.
- Terra, Joana

*2010 Mathematics Subject Classification*

- 35K57 35K5 35Q92 35R01 35R37 92C37

*Keywords*

- Parabolic equations, advection-diffusion, moving domain, bulk-surface coupling, ligand-receptor

*DOI*

*Abstract*

We prove existence, uniqueness, and regularity for a reaction-diffusion system of coupled bulk-surface equations on a moving domain modelling receptor-ligand dynamics in cells. The nonlinear coupling between the three unknowns is through the Robin boundary condition for the bulk quantity and the right hand sides of the two surface equations. Our results are new even in the non-moving setting, and in this case we also show exponential convergence to a steady state. The primary complications in the analysis are indeed the nonlinear coupling and the Robin boundary condition. For the well posedness and essential boundedness of solutions we use several De Giorgi-type arguments, and we also develop some useful estimates to allow us to apply a Steklov averaging technique for time-dependent operators to prove that solutions are strong. Some of these auxiliary results presented in this paper are of independent interest by themselves.

*Appeared in*

- SIAM J. Math. Anal., 50 (2018), pp. 1544--1592, DOI 10.1137/16M110808X .

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# Nonlocal phase transitions: Rigidity results and anisotropic geometry

*Authors*

- Dipierro, Serena
- Serra, Joaquim
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 60G22 82B26

*Keywords*

- nonlocal phase transitions, rigidity results, sliding methods

*DOI*

*Abstract*

We provide a series of rigidity results for a nonlocal phase transition equation. The results that we obtain are an improvement of flatness theorem and a series of theorems concerning the one-dimensional symmetry for monotone and minimal solutions, in the research line dictaded by a classical conjecture of E. De Giorgi. Here, we collect a series of pivotal results, of geometric type, which are exploited in the proofs of the main results in a companion paper.

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# Ocean rogue waves and their phase space dynamics in the limit of a linear interference model

*Authors*

- Birkholz, Simon
- Brée, Carsten
- Veselić, Ivan
- Demircan, Ayhan
- Steinmeyer, Günter

*2010 Mathematics Subject Classification*

- 76B15

*2008 Physics and Astronomy Classification Scheme*

- 47.35.Bb, 47.27.Sd

*Keywords*

- Extreme Wave Statistics

*DOI*

*Abstract*

We reanalyse the probability for formation of extreme waves using the simple model of linear interference of a finite number of elementary waves with fixed amplitude and random phase fluctuations. Under these model assumptions no rogue waves appear when less than 10 elementary waves interfere with each other. Above this threshold rogue wave formation becomes increasingly likely, with appearance frequencies that may even exceed long-term observations by an order of magnitude. For estimation of the effective number of interfering waves, we suggest the Grassberger-Procaccia dimensional analysis of individual time series. For the ocean system, it is further shown that the resulting phase space dimension may vary, such that the threshold for rogue wave formation is not always reached. Time series analysis as well as the appearance of particular focusing wind conditions may enable an effective forecast of such rogue-wave prone situations. In particular, extracting the dimension from ocean time series allows much more specific estimation of the rogue wave probability.

*Appeared in*

- Sci. Rep., 6 (2016) pp. 35207/1--35207/8.

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# Asymptotically stable compensation of soliton self-frequency shift

*Authors*

- Pickartz, Sabrina
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X

*2010 Mathematics Subject Classification*

- 78A60 35Q60 70H11

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k, 42.81.Dp, 05.45.Yv, 03.65.Nk

*Keywords*

- Soliton self-frequency shift, Raman scattering, Soliton perturbation theory, Event horizons

*DOI*

*Abstract*

We report the cancellation of the soliton self-frequency shift in nonlinear optical fibers. A soliton which interacts with a group velocity matched low intensity dispersive pump pulse, experiences a continuous blue-shift in frequency, which counteracts the soliton selffrequency shift due to Raman scattering. The soliton self-frequency shift can be fully compensated by a suitably prepared dispersive wave.We quantify this kind of soliton-dispersive wave interaction by an adiabatic approach and demonstrate that the compensation is stable in agreement with numerical simulations.

*Appeared in*

- Opt. Lett., 42 (2017), pp. 1416--1419, DOI 10.1364/OL.42.001416 .

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# An entropic gradient structure for Lindblad equations and GENERIC for quantum systems coupled to macroscopic models

*Authors*

- Mittnenzweig, Markus
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 82B10 82C10 47D07 49S05

*Keywords*

- Quantum Markov semigroups, open quantum systems, Lindblad operator, detailed-balance condition, relative entropy, Onsager, operators, general equations for non-equilibrium reversible irreversible coupling, GENERIC

*Abstract*

We show that all Lindblad operators (i.e. generators of quantum semigroups) on a finite-dimensional Hilbert space satisfying the detailed balance condition with respect to the thermal equilibrium state can be written as a gradient system with respect to the relative entropy. We discuss also thermodynamically consistent couplings to macroscopic systems, either as damped Hamiltonian systems with constant temperature or as GENERIC systems.

*Appeared in*

- J. Statist. Phys., 167 (2017), pp. 205--233, DOI 10.1007/s10955-017-1756-4 .

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# Local approximation of arbitrary functions by solutions of nonlocal equations

*Authors*

- Dipierro, Serena
- Savin, Ovidiu
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 60G22 35A35 34A08

*Keywords*

- density properties, approximation, s-caloric functions

*Abstract*

We show that any function can be locally approximated by solutions of prescribed linear equations of nonlocal type. In particular, we show that every function is locally s-caloric, up to a small error. The case of non-elliptic and non-parabolic operators is taken into account as well.

*Appeared in*

- J. Geom. Anal. (2018), https://doi.org/10.1007/s12220-018-0045-z.

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# Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions

*Authors*

- Bulíček, Miroslav
- Glitzky, Annegret
- Liero, Matthias

ORCID: 0000-0002-0963-2915

*2010 Mathematics Subject Classification*

- 35J92 35Q79 35J57 80A20

*Keywords*

- Sobolev spaces with variable exponent, existence of weak solution, entropy solution, thermistor system, p(x)-Laplacian, heat transfer

*Abstract*

We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of *p(x)*-Laplacian-type with discontinuous exponent *p*, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori *L ^{1}* term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.

*Appeared in*

- PDE 2015: Theory and Applications of Partial Differential Equations (PDE 2015), H.-Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., vol. 10 of Discrete and Continuous Dynamical Systems, Series S, American Institute of Mathematical Sciences, Springfield, 2017, pp. 697--713.

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# A nonlinear free boundary problem with a self-driven Bernoulli condition

*Authors*

- Dipierro, Serena
- Karakhanyan, Aram
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R35 35B65

*Keywords*

- nonlinear energy superposition, free boundary, regularity theory, Bernoulli condition

*DOI*

*Abstract*

We study a Bernoulli type free boundary problem with two phases and a nonlinear energy superposition. We show that, for this problem, the Bernoulli constant, which determines the gradient jump condition across the free boundary, is of global type and it is indeed determined by the weighted volumes of the phases. In particular, the Bernoulli condition that we obtain can be seen as a pressure prescription in terms of the volume of the two phases of the minimizer itself (and therefore it depends on the minimizer itself and not only on the structural constants of the problem). Another property of this type of problems is that the minimizer in a given domain is not necessarily a minimizer in a smaller subdomain, due to the nonlinear structure of the problem. Due to these features, this problem is highly unstable as opposed to the classical case studied by Alt, Caffarelli and Friedman. It also interpolates the classical case, in the sense that the blow-up limits are minimizers of the Alt-Caffarelli-Friedman functional. Namely, the energy of the problem somehow linearizes in the blow-up limit. We also develop a detailed optimal regularity theory for the minimizers and for their free boundaries.

*Appeared in*

- J. Funct. Anal., 273:11 (2017) pp. 3549-3615.

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# Cohesive zone-type delamination in visco-elasticity

*Authors*

- Thomas, Marita
- Zanini, Chiara

*2010 Mathematics Subject Classification*

- 35A15 35Q74 74H20 74C10 49J53 49J45 74C05

*Keywords*

- Cohesive zone delamination, weak formulation, rate-independent processes, semistable energetic solutions, non-smooth constraint, gradient systems, dynamics, irreversibility

*DOI*

*Abstract*

We study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [Ortiz&Pandoli99Int.J.Numer.Meth.Eng.], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for different responses upon loading and unloading. Due to the presence of multivalued and unbounded operators featuring non-penetration and the "memory"-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [Roubicek09M2AS] and refined in [Rossi&Thomas15WIAS-Preprint2113].

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), pp. 1487--1517, DOI 10.3934/dcdss.2017077 .

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# Projected particle methods for solving McKean--Vlaslov equations

*Authors*

- Belomestny, Denis
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60H10 60K35

*Keywords*

- McKean-Vlaslov equations, particle systems, projection estimators, explicit solutions

*DOI*

*Abstract*

We study a novel projection-based particle method to the solution of the corresponding McKean-Vlasov equation. Our approach is based on the projection-type estimation of the marginal density of the solution in each time step. The projection-based particle method can profit from additional smoothness of the underlying density and leads in many situation to a significant reduction of numerical complexity compared to kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The case of linearly growing coefficients of the McKean-Vlasov equation turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKean-Vlasov equations with affine drift.

*Appeared in*

- SIAM J. Numer. Anal., 56 (2018), pp. 3169--3195, DOI 10.1137/17M1111024 .

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# The space of bounded variation with infinite-dimensional codomain

*Authors*

- Heida, Martin
- Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857 - Renger, D. R. Michiel

ORCID: 0000-0003-3557-3485

*2010 Mathematics Subject Classification*

- 26A45 26A24 28B99 46G05 46G10

*Keywords*

- Bounded variation, infinite-dimensional codomain, metric spaces, non-metric topologies, Banach spaces, vector measures, Aubin-Lions, compactness

*DOI*

*Abstract*

We study functions of bounded variation with values in a Banach or in a metric space. We provide several equivalent notions of variations and provide the notion of a time derivative in this abstract setting. We study four distinct topologies on the space of bounded variations and provide some insight into the structure of these topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical Aubin-Lions theorem. We finally provide some useful applications to stochastic processes.

*Appeared in*

- J. Evol. Equ., 19 (2018), pp. 111--152, DOI 10.1007/s00028-018-0471-1 .

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# Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model

*Authors*

- Flegel, Franziska

*2010 Mathematics Subject Classification*

- 47B80 47A75 60J27

*Keywords*

- random conductance model, Dirichlet spectrum, eigenfunction localization, heavy tails

*DOI*

*Abstract*

We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of Zd (d ≥ 2) with zero Dirichlet condition. We assume that the conductances w are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If γ = sup q ≥ 0; E [w^-q]<∞ <¼, then we show that for almost every environment the principal Dirichlet eigenvector asymptotically concentrates in a single site and the corresponding eigenvalue scales subdiffusively. The threshold γrm c = ¼ is sharp. Indeed, other recent results imply that for γ>¼ the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the local speed measure, Borel-Cantelli arguments, the Rayleigh-Ritz formula, results from percolation theory, and path arguments.

*Appeared in*

- Electron. J. Probab., 23 (2018), pp. 1--43, DOI doi:10.1214/18-EJP160 .

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# Averaging of time-periodic dissipation potentials in rate-independent processes

*Authors*

- Heida, Martin
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 34C55 47J20 49J40 74N30

*Keywords*

- Rate-independent systems, play operator with time-dependent thresholds, energetic solutions, locomotion

*Abstract*

We study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period ε. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit ε → 0 and show that the effective dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equi-continuity of the solutions in the limit ε → 0.

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), pp. 1303--1327.

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# Infinite hierarchy of nonlinear Schrödinger equations and their solutions

*Authors*

- Ankiewicz, Adrian
- Kedziora, David Jacob
- Chowdury, Amdad
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Akhmediev, Nail

*2010 Mathematics Subject Classification*

- 35Q55 37K10 35C08

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Yv, 42.65.Tg, 42.81.qb

*Keywords*

- Nonlinear Schrödinger Equations, Infinite Hierarchy, Solitons, Breathers, Rogue Waves

*Abstract*

We study the infinite integrable nonlinear Schrödinger equation (NLSE) hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, AB breathers, Kuznetsov-Ma breathers, periodic solutions and rogue wave solutions for this infinite order hierarchy. We find that 'even' order equations in the set affect phase and 'stretching factors' in the solutions, while 'odd' order equations affect the velocities. Hence 'odd' order equation solutions can be real functions, while 'even' order equation solutions are always complex.

*Appeared in*

- Phys. Rev. E, 93 (2016) pp. 012206/1--012206/10.

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# Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails

*Authors*

- Biskup, Marek
- König, Wolfgang
- Soares dos Santos, Renato

*2010 Mathematics Subject Classification*

- 60J55 60F10 60K35 60K37 82B44

*Keywords*

- heat equation with random coefficients, random Schrödinger operator, Feynman-Kac formula, Anderson localisation, mass concentration, spectral expansion, eigenvalue order statistics

*DOI*

*Abstract*

We study the solutions to the Cauchy problem on the with random potential and localised initial data. Here we consider the random Schr?dinger operator, i.e., the Laplace operator with random field, whose upper tails are doubly exponentially distributed in our case. We prove that, for large times and with large probability, a majority of the total mass of the solution resides in a bounded neighborhood of a site that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian and the distance to the origin. The processes of mass concentration and the rescaled total mass are shown to converge in distribution under suitable scaling of space and time. Aging results are also established. The proof uses the characterization of eigenvalue order statistics for the random Schrödinger operator in large sets recently proved by the first two authors.

*Appeared in*

- Probab. Theory Related Fields, 171 (2018), pp. 251--331 (published online on 27.05.2017), DOI 10.1007/s00440-017-0777-x .

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# Stochastic homogenization of rate-independent systems

*Authors*

- Heida, Martin

*2010 Mathematics Subject Classification*

- 74QXX 74C05 34E13

*Keywords*

- Rate-independent, stochastic homogenization, convex functionals, two-scale

*Abstract*

We study the stochastic and periodic homogenization 1-homogeneous convex functionals. We proof some convergence results with respect to stochastic two-scale convergence, which are related to classical Gamma-convergence results. The main result is a general liminf-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rateindependent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandlt-Reuss plasticity, Coulomb friction on a macroscopic surface and Coulomb friction on microscopic fissure.

*Appeared in*

- Contin. Mech. Thermodyn., 29 (2017), pp. 853--894, DOI 10.1007/s00161-017-0564-z .

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# Error estimates in weighted Sobolev norms for finite element immersed interface methods

*Authors*

- Heltai, Luca
- Rotundo, Nella

*2010 Mathematics Subject Classification*

- 74S05 65M15 46E35 46E39

*Keywords*

- Finite element method, immersed interface method, immersed boundary method, weighted Sobolev spaces, error estimates

*Abstract*

When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using a uniform background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation.

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# Balanced truncation and singular perturbation approximation model order reduction for stochastically controlled linear systems

*Authors*

- Redmann, Martin

ORCID: 0000-0001-5182-9773 - Freitag, Melina A.

*2010 Mathematics Subject Classification*

- 93A15 93A30 15A24

*Keywords*

- model order reduction, balanced truncation, singular perturbation approximation, stochastic systems, Lévy process, Gramians, Lyapunov equations

*Abstract*

When solving linear stochastic differential equations numerically, usually a high order spatial discretisation is used. Balanced truncation (BT) and singular perturbation approximation (SPA) are well-known projection techniques in the deterministic framework which reduce the order of a control system and hence reduce computational complexity. This work considers both methods when the control is replaced by a noise term. We provide theoretical tools such as stochastic concepts for reachability and observability, which are necessary for balancing related model order reduction of linear stochastic differential equations with additive Lévy noise. Moreover, we derive error bounds for both BT and SPA and provide numerical results for a specific example which support the theory.

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