# Finite element error analysis of a mantle convection model

*Authors*

- John, Volker

ORCID: 0000-0002-2711-4409 - Kaya, Songul
- Novo, Julia

*2010 Mathematics Subject Classification*

- 65M60 65N30

*Keywords*

- Mantel convection, Stokes problem with variable viscosity, temperature problem, with variable thermal convection, inf-sup stable finite elements, SUPG stabilization

*DOI*

*Abstract*

A mantle convection model consisting of the stationary Stokes equations and a time-dependent convection-diffusion equation for the temperature is studied. The Stokes problem is discretized with a conforming inf-sup stable pair of finite element spaces and the temperature equation is stabilized with the SUPG method. Finite element error estimates are derived which show the dependency of the error of the solution of one problem on the error of the solution of the other equation. The dependency of the error bounds on the coefficients of the problem is monitored.

*Appeared in*

- Int. J. Numer. Anal. Model., 15 (2018), pp. 677--698, DOI 10.20347/WIAS.PREPRINT.2403 .

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# Duality and fixation in $Xi$-Wright--Fisher processes with frequency-dependent selection

*Authors*

- González Casanova Soberón, Adrián
- Spanò, Dario

*2010 Mathematics Subject Classification*

- 60G99 60K35 92D10 92D11 92D25 92D10 92D11 92D25

*Keywords*

- Cannings models, frequency-dependent selection, moment duality, ancestral processes, branching-coalescing stochastic processes, fixation probability, $Xi$-Fleming-Viot processes

*DOI*

*Abstract*

A two-types, discrete-time population model with finite, constant size is constructed, allowing for a general form of frequency-dependent selection and skewed offspring distribution. Selection is defined based on the idea that individuals first choose a (random) number of emphpotential parents from the previous generation and then, from the selected pool, they inherit the type of the fittest parent. The probability distribution function of the number of potential parents per individual thus parametrises entirely the selection mechanism. Using duality, weak convergence is then proved both for the allele frequency process of the selectively weak type and for the population's ancestral process. The scaling limits are, respectively, a two-types Ξ--Fleming-Viot jump-diffusion process with frequency-dependent selection, and a branching-coalescing process with general branching and simultaneous multiple collisions. Duality also leads to a characterisation of the probability of extinction of the selectively weak allele, in terms of the ancestral process' ergodic properties.

*Appeared in*

- Annals of Appl. Probab., 28 (2018), pp. 250-284.

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# Simultaneous adaptive smoothing of relaxometry and quantitative magnetization transfer mapping

*Authors*

- Mohammadi, Siawoosh
- D'Alonzo, Chiara
- Ruthotto, Lars
- Polzehl, Jörg

ORCID: 0000-0001-7471-2658 - Ellerbrock, Isabel
- Callaghan, Martina F.
- Weiskopf, Nikolaus
- Tabelow, Karsten

ORCID: 0000-0003-1274-9951

*2010 Mathematics Subject Classification*

- 62P10 62G05

*Keywords*

- Quantitative MRI, Multi-Paramter Mapping, structural adaptive smoothing

*DOI*

*Abstract*

Attempts for in-vivo histology require a high spatial resolution that comes with the price of a decreased signal-to-noise ratio. We present a novel iterative and multi-scale smoothing method for quantitative Magnetic Resonance Imaging (MRI) data that yield proton density, apparent transverse and longitudinal relaxation, and magnetization transfer maps. The method is based on the propagation-separation approach. The adaptivity of the procedure avoids the inherent bias from blurring subtle features in the calculated maps that is common for non-adaptive smoothing approaches. The characteristics of the methods were evaluated on a high-resolution data set (500 μ isotropic) from a single subject and quantified on data from a multi-subject study. The results show that the adaptive method is able to increase the signal-to-noise ratio in the calculated quantitative maps while largely avoiding the bias that is otherwise introduced by spatially blurring values across tissue borders. As a consequence, it preserves the intensity contrast between white and gray matter and the thin cortical ribbon.

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# Geometric properties of cones with applications on the Hellinger--Kantorovich space, and a new distance on the space of probability measures

*Authors*

- Laschos, Vaios
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 51Fxx

*Keywords*

- Geometry on cones, local angle condition, K-semiconcavity, Hellinger-Kantorovich, Spherical Hellinger-Kantorovich

*DOI*

*Abstract*

By studying general geometric properties of cone spaces, we prove the existence of a distance on the space of Probability measures that turns the Hellinger--Kantorovich space into a cone space over the space of probabilities measures. Here we exploit a natural two-parameter scaling property of the Hellinger-Kantorovich distance. For the new space, we obtain a full characterization of the geodesics. We also provide new geometric properties for the original space, including a two-parameter rescaling and reparametrization of the geodesics, local-angle condition and some partial K-semiconcavity of the squared distance, that it will be used in a future paper to prove existence of gradient flows.

*Appeared in*

- J. Funct. Anal., 276 (2019), pp. 3529--3576, DOI 10.1016/j.jfa.2018.12.013 .

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# Optimal design of the tweezer control for chimera states

*Authors*

- Omelchenko, Iryna
- Omel'chenko, Oleh E.

ORCID: 0000-0003-0526-1878 - Zakharova, Anna
- Schöll, Eckehard

*2010 Mathematics Subject Classification*

- 34H10 34C15

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt 05.45.Ra 89.75.-k

*Keywords*

- Nonlinear systems, dynamical networks, spatial chaos, chimera state, feedback control

*DOI*

*Abstract*

Chimera states are complex spatio-temporal patterns, which consist of coexisting domains of spatially coherent and incoherent dynamics in systems of coupled oscillators. In small networks, chimera states usually exhibit short lifetimes and erratic drifting of the spatial position of the incoherent domain. A tweezer feedback control scheme can stabilize and fix the position of chimera states. We analyse the action of the tweezer control in small nonlocally coupled networks of Van der Pol and FitzHugh--Nagumo oscillators, and determine the ranges of optimal control parameters. We demonstrate that the tweezer control scheme allows for stabilization of chimera states with different shapes, and can be used as an instrument for controlling the coherent domains size, as well as the maximum average frequency difference of the oscillators.

*Appeared in*

- Phys. Rev. E, 97 (2018), published online on 25.01.2018, DOI https://doi.org/10.1103/PhysRevE.97.012216 .

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# The mathematics behind chimera states

*Authors*

- Omel'chenko, Oleh E.

ORCID: 0000-0003-0526-1878

*2010 Mathematics Subject Classification*

- 34C15 35B36 35B32 35Q83 35Q84 34H10

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt 89.75.Kd

*Keywords*

- Coupled oscillators, pattern formation, spatial chaos, chimera states, coherence-incoherence patterns

*DOI*

*Abstract*

Chimera states are self-organized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, Ott--Antonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed.

*Appeared in*

- Nonlinearity, 31 (2018), published online on 04.04.2018, DOI https://doi.org/10.1088/1361-6544/aaaa07 .

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# Adaptive regularization for image reconstruction from subsampled data

*Authors*

- Hintermüller, Michael

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

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

*2010 Mathematics Subject Classification*

- 68U10 94A08 49K20 49K40 49M37 47A52 65K10

*Keywords*

- image restoration, spatially adaptive regularization, partial Fourier data, wavelet inpainting

*DOI*

*Abstract*

Choices of regularization parameters are central to variational methods for image restoration. In this paper, a spatially adaptive (or distributed) regularization scheme is developed based on localized residuals, which properly balances the regularization weight between regions containing image details and homogeneous regions. Surrogate iterative methods are employed to handle given subsampled data in transformed domains, such as Fourier or wavelet data. In this respect, this work extends the spatially variant regularization technique previously established in [15], which depends on the fact that the given data are degraded images only. Numerical experiments for the reconstruction from partial Fourier data and for wavelet inpainting prove the efficiency of the newly proposed approach.

*Appeared in*

- Imaging, Vision and Learning Based on Optimization and PDEs IVLOPDE, Bergen, Norway, August 29 - September 2, 2016, X.-Ch. Tai, E. Bae, M. Lysaker, eds., Mathematics and Visualization, Springer International Publishing, Berlin, 2017, pp. Part 1, Chapter 1, DOI 10.1007/978-3-319-91274-5 .

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# An assessment of solvers for saddle point problems emerging from the incompressible Navier--Stokes equations

*Authors*

- Ahmed, Naveed

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

ORCID: 0000-0002-2711-4409 - Wilbrandt, Ulrich

*2010 Mathematics Subject Classification*

- 65N22

*Keywords*

- Linear saddle point problems, inf-sup stable pairs of finite element spaces, UMFPACK, flexible GMRES, coupled multigrid preconditioners with Vanka smoother, Least Squares Commutator preconditioners

*DOI*

*Abstract*

Efficient incompressible flow simulations, using inf-sup stable pairs of finite element spaces, require the application of efficient solvers for the arising linear saddle point problems. This paper presents an assessment of different solvers: the sparse direct solver UMFPACK, the flexible GMRES (FGMRES) method with different coupled multigrid preconditioners, and FGMRES with Least Squares Commutator (LSC) preconditioners. The assessment is performed for steady-state and time-dependent flows around cylinders in 2d and 3d. Several pairs of inf-sup stable finite element spaces with second order velocity and first order pressure are used. It turns out that for the steady-state problems often FGMRES with an appropriate multigrid preconditioner was the most efficient method on finer grids. For the time-dependent problems, FGMRES with LSC preconditioners that use an inexact iterative solution of the velocity subproblem worked best for smaller time steps.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 331 (2018), pp. 492--513, DOI 10.1016/j.cma.2017.12.004 .

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# Dissipative and non-dissipative evolutionary quasi-variational inequalities with gradient constraints

*Authors*

- Hintermüller, Michael

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

ORCID: 0000-0001-9497-9296 - Strogies, Nikolai

*2010 Mathematics Subject Classification*

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

*Keywords*

- Quasi-variational inequality, gradient constraint, dissipative and non-dissipative processes, variable splitting solver

*DOI*

*Abstract*

Evolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A semi-discretization in time is employed for the study of the problems and the derivation of a numerical solution scheme, respectively. Convergence of the discretization procedure is proven and properties of the original infinite dimensional problem, such as existence, extra regularity and non-decrease in time, are derived. The proposed numerical solver reduces to a finite number of gradient-constrained convex optimization problems which can be solved rather efficiently. The paper ends with a report on numerical tests obtained by a variable splitting algorithm involving different nonlinearities and types of constraints.

*Appeared in*

- Set-Valued Var. Anal., 27 (2019), pp. 433--468 (published online on 14.07.2018), DOI 10.1007/s11228-018-0489-0 .

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# Estimation of the infinitesimal generator by square-root approximation

*Authors*

- Donati, Luca
- Heida, Martin
- Weber, Marcus
- Keller, Bettina

*2008 Physics and Astronomy Classification Scheme*

- 02.50.Ga 05.10Gg

*Keywords*

- molecular simulation, Markov state models, transfer operator, molecular kinetics

*DOI*

*Abstract*

For the analysis of molecular processes, the estimation of time-scales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is -- from a mathematical point of view -- an invariant subspace projection problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is approximated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the geometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correlation between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide.

*Appeared in*

- J. Phys.: Condens. Matter, 30 (2018), pp. 425201/1--425201/14, DOI 10.1088/1361-648X/aadfc8 .

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# Tetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction

*Authors*

- Dassi, Franco
- Kamenski, Lennard
- Farrell, Patricio

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

*2010 Mathematics Subject Classification*

- 65N50 65M50 65L50 65K10

*Keywords*

- mesh improvement, mesh quality, edge flipping, mesh smoothing, moving mesh, radial basis functions

*DOI*

*Abstract*

Given a tetrahedral mesh and objective functionals measuring the mesh quality which take into account the shape, size, and orientation of the mesh elements, our aim is to improve the mesh quality as much as possible. In this paper, we combine the moving mesh smoothing, based on the integration of an ordinary differential equation coming from a given functional, with the lazy flip technique, a reversible edge removal algorithm to modify the mesh connectivity. Moreover, we utilize radial basis function (RBF) surface reconstruction to improve tetrahedral meshes with curved boundary surfaces. Numerical tests show that the combination of these techniques into a mesh improvement framework achieves results which are comparable and even better than the previously reported ones.

*Appeared in*

- Comput. Aided Des., 102 (2018), pp. 2--13 (published online on 8.12.2017), DOI 10.1016/j.cad.2017.11.010 .

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# An analogue of grad-div stabilization in nonconforming methods for incompressible flows

*Authors*

- Akbas, Mine
- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Rebholz, Leo G.
- Schroeder, Philipp W.

*2010 Mathematics Subject Classification*

- 35Q30 65M15 65M60 76M10

*Keywords*

- Incompressible Navier--Stokes equations, mixed finite element methods, grad-div stabilization, Discontinuous Galerkin method, nonconforming finite elements

*DOI*

*Abstract*

Grad-div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the relaxation of the divergence constraint in classical mixed methods, and are excited whenever the spacial discretization has to deal with comparably large and complicated pressures. In this contribution, an analogue of grad-div stabilization is presented for nonconforming flow discretizations of Discontinuous Galerkin or nonconforming finite element type. Here the key is the penalization of the jumps of the normal velocities over facets of the triangulation, which controls the measure-valued part of the distributional divergence of the discrete velocity solution. Furthermore, we characterize the limit for arbitrarily large penalization parameters, which shows that the proposed nonconforming Discontinuous Galerkin methods remain robust and accurate in this limit. Several numerical examples illustrate the theory and show their relevance for the simulation of practical, nontrivial flows.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 341 (2018), pp. 917--938, DOI 10.20347/WIAS.PREPRINT.2448 .

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# On the consistency of Runge--Kutta methods up to order three applied to the optimal control of scalar conservation laws

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Strogies, Nikolai

*2010 Mathematics Subject Classification*

- 49J15 49J20 35L65 65L06 65M06 65M12

*Keywords*

- optimal control, conservation laws, discretization methods, RK methods, TVD-RK

*DOI*

*Abstract*

Higher-order Runge-Kutta (RK) time discretization methods for the optimal control of scalar conservation laws are analyzed and numerically tested. The hyperbolic nature of the state system introduces specific requirements on discretization schemes such that the discrete adjoint states associated with the control problem converge as well. Moreover, conditions on the RK-coefficients are derived that coincide with those characterizing strong stability preserving Runge-Kutta methods. As a consequence, the optimal order for the adjoint state is limited, e.g., to two even in the case where the conservation law is discretized by a third-order method. Finally, numerical tests for controlling Burgers equation validate the theoretical results.

*Appeared in*

- Numerical Analysis and Optimization, M. Al-Baali, L. Grandinetti, A. Purnama, eds., vol. 235 of Springer Proceedings in Mathematics & Statistics, Springer Nature Switzerland AG, Cham, 2019, pp. 119--154.

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# A gradient system with a wiggly energy and relaxed EDP-convergence

*Authors*

- Dondl, Patrick
- Frenzel, Thomas
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 35K55 35B27 35A15 49S05 49J40 49J45

*Keywords*

- Variational evolution, energy functional, dissipation potential, gradient flows, Gamma convergence, EDP-convergence, energy-dissipation, balance, homogenization

*DOI*

*Abstract*

If gradient systems depend on a microstructure, we want to derive a macroscopic gradient structure describing the effective behavior of the microscopic system. We introduce a notion of evolutionary Gamma-convergence that relates the microscopic energy and the microscopic dissipation potential with their macroscopic limits via Gamma-convergence. We call this notion relaxed EDP-convergence since the special structure of the dissipation functional may not be preserved under Gamma-convergence. However, by investigating the kinetic relation we derive the macroscopic dissipation potential.

*Appeared in*

- ESAIM Control Optim. Calc. Var., 25 (2019), pp. 68/1--68/45 (published on 05.11.2019), DOI 10.1051/cocv/2018058 .

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# Benchmark problems for numerical treatment of backflow at open boundaries

*Authors*

- Bertoglio, Cristóbal
- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Bazilevs, Yuri
- Braack, Malte
- Esmaily-Moghadam, Mahdi
- Gravemeier, Volker
- Marsden, Alison L.
- Pironneau, Olivier
- Vignon-Clementel, Irene E.
- Wall, Wolfgang A.

*2008 Physics and Astronomy Classification Scheme*

- 02.70.-c,07.05.Tp,47.11.-j,47.11.Fg,47.63.-b

*Keywords*

- Navier--Stokes Equations, backflow stabilization, benchmarking, blood flows, respiratory flows

*DOI*

*Abstract*

In computational fluid dynamics, incoming velocity at open boundaries, or backflow, often yields to unphysical instabilities already for moderate Reynolds numbers. Several treatments to overcome these backflow instabilities have been proposed in the literature. However, these approaches have not yet been compared in detail in terms of accuracy in different physiological regimes, in particular due to the difficulty to generate stable reference solutions apart from analytical forms. In this work, we present a set of benchmark problems in order to compare different methods in different backflow regimes (with a full reversal flow and with propagating vortices after a stenosis). The examples are implemented in FreeFem++ and the source code is openly available, making them a solid basis for future method developments.

*Appeared in*

- Internat. J. Numer. Methods Biomedical Engrg., 34 (2018), e2918, DOI 10.1002/cnm.2918 .

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# A threshold model for local volatility: Evidence of leverage and mean reversion effects on historical data

*Authors*

- Lejay, Antoine
- Pigato, Paolo

*2010 Mathematics Subject Classification*

- 62F07 91G08

*Keywords*

- Oscillating Brownian motion, leverage effect, realized volatility, mean-reversion, selfexciting threshold autoregressive model, regime-switch

*DOI*

*Abstract*

In financial markets, low prices are generally associated with high volatilities and vice-versa, this well known stylized fact usually being referred to as leverage effect. We propose a local volatility model, given by a stochastic differential equation with piecewise constant coefficients, which accounts of leverage and mean-reversion effects in the dynamics of the prices. This model exhibits a regime switch in the dynamics accordingly to a certain threshold. It can be seen as a continuous time version of the Self-Exciting Threshold Autoregressive (SETAR) model. We propose an estimation procedure for the volatility and drift coefficients as well as for the threshold level. Tests are performed on the daily prices of 21 assets. They show empirical evidence for leverage and mean-reversion effects, consistent with the results in the literature.

*Appeared in*

- Int. J. Theor. Appl. Finance, 22 (2019), published online on 29.05.2019, DOI 10.1142/S0219024919500171 .

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# Towards time-limited $H_2$-optimal model order reduction

*Authors*

- Goyal, Pawan
- Redmann, Martin

ORCID: 0000-0001-5182-9773

*2010 Mathematics Subject Classification*

- 15A16 15A24 93A15

*Keywords*

- model order reduction, linear systems, H_2-optimality, Gramians, Slyvester equations

*DOI*

*Abstract*

In order to solve partial differential equations numerically and accurately, a high order spatial discretization is usually needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular class of MOR techniques are H_2-optimal methods such as the iterative rational Krylov subspace algorithm (IRKA) and related schemes. However, these methods are used to obtain good approximations on a infinite time-horizon. Thus, in this work, our main goal is to discuss MOR schemes for time-limited linear systems. For this, we propose an alternative time-limited H_2-norm and show its connection with the time-limited Gramians. We then provide first-order optimality conditions for an optimal reduced order model (ROM) with respect to the time-limited H_2-norm. Based on these optimality conditions, we propose an iterative scheme which upon convergences aims at satisfying these conditions. Then, we analyze how far away the obtained ROM is from satisfying the optimality conditions. We test the efficiency of the proposed iterative scheme using various numerical examples and illustrate that the newly proposed iterative method can lead to a better reduced-order compared to unrestricted IRKA in the time interval of interest.

*Appeared in*

- Appl. Math. Comput., 355 (2019), pp. 184--197, DOI 10.1016/j.amc.2019.02.065 .

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# Unsaturated deformable porous media flow with phase transition

*Authors*

- Krejčí, Pavel
- Rocca, Elisabetta
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 76S05 80A22 35Q74

*Keywords*

- Porous media flow, phase transition, initial-boundary value problem

*DOI*

*Abstract*

In the present paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquid-solid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the Clausius-Duhem inequality for entropy. It couples the evolution of the displacement in the matrix material, of the capillary pressure, of the absolute temperature, and of the phase fraction. Mathematical results are proved under the additional hypothesis that inertia effects and shear stresses can be neglected. For the resulting highly nonlinear system of two PDEs, one ODE and one ordinary differential inclusion with natural initial and boundary conditions, existence of global in time solutions is proved by means of cut-off techniques and suitable Moser-type estimates.

*Appeared in*

- Math. Models Methods Appl. Sci., 27 (2017), pp. 2675--2710, DOI 10.1142/S0218202517500555 .

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# Towards pressure-robust mixed methods for the incompressible Navier--Stokes equations

*Authors*

- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - Linke, Alexander

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

*2010 Mathematics Subject Classification*

- 65M12 665M30 65M15 76D07 76M10

*Keywords*

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

*DOI*

*Abstract*

In this contribution, classical mixed methods for the incompressible Navier-Stokes equations that relax the divergence constraint and are discretely inf-sup stable, are reviewed. Though the relaxation of the divergence constraint was claimed to be harmless since the beginning of the 1970ies, Poisson locking is just replaced by another more subtle kind of locking phenomenon, which is sometimes called poor mass conservation. Indeed, divergence-free mixed methods and classical mixed methods behave qualitatively in a different way: divergence-free mixed methods are pressure-robust, which means that, e.g., their velocity error is independent of the continuous pressure. The lack of pressure-robustness in classical mixed methods can be traced back to a consistency error of an appropriately defined discrete Helmholtz projector. Numerical analysis and numerical examples reveal that really locking-free mixed methods must be discretely inf-sup stable and pressure-robust, simultaneously. Further, a recent discovery shows that locking-free, pressure-robust mixed methods do not have to be divergence-free. Indeed, relaxing the divergence constraint in the velocity trial functions is harmless, if the relaxation of the divergence constraint in some velocity test functions is repaired, accordingly.

*Appeared in*

- Comput. Methods Appl. Math., 18 (2018), pp. 353--372 (published online on 18.11.2017), DOI 10.1515/cmam-2017-0047 .

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# Regression on particle systems connected to mean-field SDEs with applications

*Authors*

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

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60H10 60K35 62L15

*Keywords*

- McKean-Vlasov equations, optimal stopping, regression, Bellman principle

*DOI*

*Abstract*

In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKean-Vlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered.

*Appeared in*

- SIAM J. Control Optim., 58 (2020), pp. 529--550; changed title: Optimal stopping of McKean-Vlasov diffusions via regression on particle systems, DOI 10.1137/18M1195590 .

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# Change-point detection in high-dimensional covariance structure

*Authors*

- Avanesov, Valeriy
- Buzun, Nazar

*2010 Mathematics Subject Classification*

- 62M10 62H15

*DOI*

*Abstract*

In this paper we introduce a novel approach for an important problem of break detection. Specifically, we are interested in detection of an abrupt change in the covariance structure of a high-dimensional random process ? a problem, which has applications in many areas e.g., neuroimaging and finance. The developed approach is essentially a testing procedure involving a choice of a critical level. To that end a non-standard bootstrap scheme is proposed and theoretically justified under mild assumptions. Theoretical study features a result providing guaranties for break detection. All the theoretical results are established in a high-dimensional setting (dimensionality p n). Multiscale nature of the approach allows for a trade-off between sensitivity of break detection and localization. The approach can be naturally employed in an on-line setting. Simulation study demonstrates that the approach matches the nominal level of false alarm probability and exhibits high power, outperforming a recent approach.

*Appeared in*

- Electron. J. Stat., 12 (2018), pp. 3254--3294, DOI 10.1214/18-EJS1484 .

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# Global--in--time solvability of thermodynamically motivated parabolic systems

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35K40 35K51 35K57, 35K59, 35D30, 35B65

*Keywords*

- doubly nonlinear parabolic systems, quasilinear parabolic equations, advection--diffusion--reaction equations, a-priori estimates, generalised solutions, smoothness of solutions

*DOI*

*Abstract*

In this paper, doubly non linear parabolic systems in divergence form are investigated form the point of view of their global--in--time weak solvability. The non--linearity under the time derivative is given by the gradient of a strictly convex, globally Lipschitz continuous potential, multiplied by a position--dependent weight. This weight admits singular values. The flux under the spatial divergence is also of monotone gradient type, but it is not restricted to polynomial growth. It is assumed that the elliptic operator generates some equi--coercivity on the spatial derivatives of the unknowns. The paper introduces some original techniques to deal with the case of nonlinear purely Neumann boundary conditions. In this respect, it generalises or complements the results by Alt and Luckhaus (1983) and Alt (2012). A field of application of the theory are the multi species diffusion systems driven by entropy.

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# Total variation diminishing schemes in optimal control of scalar conservation laws

*Authors*

- Hajian, Soheil
- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Ulbrich, Stefan

*2010 Mathematics Subject Classification*

- 49J20 65M12 65K10

*Keywords*

- optimal control of PDEs, adjoint equation, scalar conservation laws, TVD Runge-Kutta methods

*DOI*

*Abstract*

In this paper, optimal control problems subject to a nonlinear scalar conservation law are studied. Such optimal control problems are challenging both at the continuous and at the discrete level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore discretization of the underlying optimal control problem should be designed with care. Here the discretize-then-optimize approach is employed where first the full discretization of the objective function as well as the underlying PDE is considered. Then, the derivative of the reduced objective is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta (TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods, also called strong stability preserving (SSP), are originally designed to preserve total variation of the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is too strong. Also approximation properties that the discrete adjoint inherits from the discretization of the state equation are studied. Moreover order conditions are derived. In addition, optimal choices with respect to CFL constant are discussed and numerical experiments are presented.

*Appeared in*

- IMA J. Numer. Anal., 39 (2019), pp. 105--140 (published online on 14.12.2017), DOI 10.1093/imanum/drx073 .

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# Optimal velocity control of a viscous Cahn--Hilliard system with convection and dynamic boundary conditions

*Authors*

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

*2010 Mathematics Subject Classification*

- 49J20 49K20 35K61 35K25 76R05 82C26 80A22

*Keywords*

- Cahn-Hilliard system, convection term, dynamic boundary conditions, optimal velocity control, optimality conditions, adjoint state system

*DOI*

*Abstract*

In this paper, we investigate a distributed optimal control problem for a convective viscous Cahn--Hilliard system with dynamic boundary conditions. Such systems govern phase separation processes between two phases taking place in an incompressible fluid in a container and, at the same time, on the container boundary. The cost functional is of standard tracking type, while the control is exerted by the velocity of the fluid in the bulk. In this way, the coupling between the state (given by the associated order parameter and chemical potential) and control variables in the governing system of nonlinear partial differential equations is bilinear, which presents an additional difficulty for the analysis. The nonlinearities in the bulk and surface free energies are of logarithmic type, which entails that the thermodynamic forces driving the phase separation process may become singular. We show existence for the optimal control problem under investigation, prove the Fréchet differentiability of the associated control-to-state mapping in suitable Banach spaces, and derive the first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint system. Due to the strong nonlinear couplings between state variables and control, the corresponding proofs require a considerable analytical effort.

*Appeared in*

- SIAM J. Control Optim., 56 (2018), pp. 1665--1691, DOI 10.1137/17M1146786 .

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# Optimal velocity control of a convective Cahn--Hilliard system with double obstacles and dynamic boundary conditions: A `deep quench' approach

*Authors*

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

*2010 Mathematics Subject Classification*

- 49J20 49K20 74M15 35K86 76R05 82C26 80A22

*Keywords*

- Cahn-Hilliard system, convection term, dynamic boundary conditions, double obstacle potentials, optimal velocity control, optimality conditions

*DOI*

*Abstract*

In this paper, we investigate a distributed optimal control problem for a convective viscous Cahn-Hilliard system with dynamic boundary conditions. Such systems govern phase separation processes between two phases taking place in an incompressible fluid in a container and, at the same time, on the container boundary. The cost functional is of standard tracking type, while the control is exerted by the velocity of the fluid in the bulk. In this way, the coupling between the state (given by the associated order parameter and chemical potential) and control variables in the governing system of nonlinear partial differential equations is bilinear, which presents a difficulty for the analysis. In contrast to the previous paper *Optimal velocity control of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions* by the same authors, the bulk and surface free energies are of double obstacle type, which renders the state constraint nondifferentiable. It is well known that for such cases standard constraint qualifications are not satisfied so that standard methods do not apply to yield the existence of Lagrange multipliers. In this paper, we overcome this difficulty by taking advantage of results established in the quoted paper for logarithmic nonlinearities, using a so-called `deep quench approximation'. We derive results concerning the existence of optimal controls and the first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint system.

*Appeared in*

- J. Convex Anal., 26 (2019), pp. 485--514 .

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# Optimal boundary control of a nonstandard Cahn--Hilliard system with dynamic boundary condition and double obstacle inclusions

*Authors*

- Colli, Pierluigi
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74M15 49J20 49J50 35K61

*Keywords*

- Optimal control, parabolic obstacle problems, MPECs, dynamic boundary conditions, optimality conditions

*DOI*

*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 P.Podio-Guidugli in Ric. Mat. 55 (2006), pp.105-118. The model consists of a strongly coupled system of nonlinear parabolic differential inclusions, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically; the system is complemented by initial and boundary conditions. For the order parameter of the phase separation process, a dynamic boundary condition involving the Laplace-Beltrami operator is assumed, which models an additional nonconserving phase transition occurring on the surface of the domain. We complement in this paper results that were established in the recent contribution appeared in Evol. Equ. Control Theory 6 (2017), pp. 35-58, by the two authors and Gianni Gilardi. In contrast to that paper, in which differentiable potentials of logarithmic type were considered, we investigate here the (more difficult) case of nondifferentiable potentials of double obstacle type. For such nonlinearities, the standard techniques of optimal control theory to establish the existence of Lagrange multipliers for the state constraints are known to fail. To overcome these difficulties, we employ the following line of approach: we use the results contained in the preprint arXiv:1609.07046 [math.AP] (2016), pp. 1-30, for the case of (differentiable) logarithmic potentials and perform a so-called "deep quench limit". Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (nondifferentiable) double obstacle potentials.

*Appeared in*

- Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs: In Honour of Prof. Gianni Gilardi, P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels, eds., vol. 22 of Springer INdAM Series, Springer International Publishing, 2017, pp. 151--182, DOI 10.1007/978-3-319-64489-9 .

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# Thin film models for an active gel

*Authors*

- Kitavtsev, Georgy
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 76A15 80M35

*2008 Physics and Astronomy Classification Scheme*

- 61.30.Jf 87.17.Aa 87.17.Jj

*Keywords*

- Liquid crystals, asymptotic analysis, cell locomotion

*DOI*

*Abstract*

In this study we present a free-boundary problem for an active liquid crystal based on the Beris-Edwards theory that uses a tensorial order parameter and includes active contributions to the stress tensor to analyse the rich defect structure observed in applications such as the Adenosinetriphosphate (ATP) driven motion of a thin film of an actin filament network. The small aspect ratio of the film geometry allows for an asymptotic approximation of the free-boundary problem in the limit of weak elasticity of the network and strong active terms. The new thin film model captures the defect dynamics in the bulk as well as wall defects and thus presents a significant extension of previous models based on the Leslie-Erickson-Parodi theory. Analytic expressions are derived that reveal the interplay of anchoring conditions, film thickness and active terms and their control of transitions of flow structure.

*Appeared in*

- Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., published online on 19.12.2018, DOI 10.1098/rspa.2017.0828 .

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# On the dissociation degree of ionic solutions considering solvation effects

*Authors*

- Landstorfer, Manuel

*2010 Mathematics Subject Classification*

- 78A57 35Q35 34B15

*Keywords*

- ion pairs, dissociation degree, double layer, solvation shell, mixture theory

*DOI*

*Abstract*

In this work the impact of solvation effects on the dissociation degree of strong electrolytes and salts is discussed. The investigation is based on a thermodynamic model which is capable to predict qualitatively and quantitatively the double layer capacity of various electrolytes. A remarkable relationship between capacity maxima, partial molar volume of ions in solution, and solvation numbers, provides an experimental access to determine the number of solvent molecules bound to a specific ion in solution. This shows that the Stern layer is actually a saturated solution of 1 mol L-1 solvated ions, and we point out some fundamental similarities of this state to a saturated bulk solution. Our finding challenges the assumption of complete dissociation, even for moderate electrolyte concentrations, whereby we introduce an undissociated ion-pair in solution. We re-derive the equilibrium conditions for a two-step dissociation reaction, including solvation effects, which leads to a new relation to determine the dissociation degree. A comparison to Ostwald's dilution law clearly shows the shortcomings when solvation effects are neglected and we emphasize that complete dissociation is questionable beyond 0.5 mol L-1 for aqueous, mono-valent electrolytes.

*Appeared in*

- Electrochem. Commun., 92 (2018), pp. 56--59, DOI 10.1016/j.elecom.2018.05.011 .

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# Stochastic homogenization of rate-dependent models of monotone type in plasticity

*Authors*

- Heida, Martin
- Nesenenko, Sergiy

*2010 Mathematics Subject Classification*

- 74Q15 74C05 74C10 74D10 35J25 34G20 34G25 47H04 47H05

*Keywords*

- Stochastic homogenization, random measures, plasticity, stochastic two-scale convergence, Gamma-convergence, monotone operator method, Fitzpatrick's function, Palm measures, random microstructure

*DOI*

*Abstract*

In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Gamma-convergence theory.

*Appeared in*

- Asymptot. Anal., 112 (2019), pp. 185--212, DOI 10.3233/ASY-181502 .

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# A function space framework for structural total variation regularization with applications in inverse problems

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Holler, Martin
- Papafitsoros, Kostas

ORCID: 0000-0001-9691-4576

*2010 Mathematics Subject Classification*

- 47J30 49J45 49N15 49N45

*Keywords*

- inverse problems, structural total variation, relaxation, duality theory, MRI guided PET

*DOI*

*Abstract*

In this work, we introduce a function space setting for a wide class of structural/weighted total variation (TV) regularization methods motivated by their applications in inverse problems. In particular, we consider a regularizer that is the appropriate lower semi-continuous envelope (relaxation) of a suitable total variation type functional initially defined for sufficiently smooth functions. We study examples where this relaxation can be expressed explicitly, and we also provide refinements for weighted total variation for a wide range of weights. Since an integral characterization of the relaxation in function space is, in general, not always available, we show that, for a rather general linear inverse problems setting, instead of the classical Tikhonov regularization problem, one can equivalently solve a saddle-point problem where no a priori knowledge of an explicit formulation of the structural TV functional is needed. In particular, motivated by concrete applications, we deduce corresponding results for linear inverse problems with norm and Poisson log-likelihood data discrepancy terms. Finally, we provide proof-of-concept numerical examples where we solve the saddle-point problem for weighted TV denoising as well as for MR guided PET image reconstruction.

*Appeared in*

- Inverse Problems, 34 (2018), pp. 064002/1--064002/39, DOI 10.1088/1361-6420/aab586 .

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# Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier--Stokes equations

*Authors*

- Schroeder, Philipp W.
- Lehrenfeld, Christoph
- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Lube, Gerd

*2010 Mathematics Subject Classification*

- 35Q30 65M15 65M60 76D17 76M10

*Keywords*

- Time-dependent incompressible flow, Re-semi-robust error estimates, pressure-robustness, inf-sup stable methods, exactly divergence-free FEM

*DOI*

*Abstract*

Inf-sup stable FEM applied to time-dependent incompressible Navier--Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on an essential regularity assumption for the gradient of the velocity, which is discussed in detail. In the sense of best practice, we review and establish pressure- and Re-semi-robust estimates for pointwise divergence-free H1-conforming FEM (like Scott--Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.

*Appeared in*

- SeMA J., 75 (2018), pp. 629--653, DOI 10.1007/s40324-018-0157-1 .

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# On really locking-free mixed finite element methods for the transient incompressible Stokes equations

*Authors*

- Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - Linke, Alexander

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

*2010 Mathematics Subject Classification*

- 65M12 65M30 65M15, 76D07, 76M10

*Keywords*

- transient incompressible Stokes equations, mixed finite element methods, locking phenomenon, pressure-robustness, a priori error analysis

*DOI*

*Abstract*

Inf-sup stable mixed methods for the steady incompressible Stokes equations that relax the divergence constraint are often claimed to deliver locking-free discretizations. However, this relaxation leads to a pressure-dependent contribution in the velocity error, which is proportional to the inverse of the viscosity, thus giving rise to a (different) locking phenomenon. However, a recently proposed modification of the right hand side alone leads to a discretization that is really locking-free, i.e., its velocity error converges with optimal order and is independent of the pressure and the smallness of the viscosity. In this contribution, we extend this approach to the transient incompressible Stokes equations, where besides the right hand side also the velocity time derivative requires an improved space discretization. Semi-discrete and fully-discrete a-priori velocity and pressure error estimates are derived, which show beautiful robustness properties. Two numerical examples illustrate the superior accuracy of pressure-robust space discretizations in the case of small viscosities.

*Appeared in*

- SIAM J. Numer. Anal., 56 (2018), pp. 185--209.

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# Fractal homogenization of a multiscale interface problem

*Authors*

- Heida, Martin
- Kornhuber, Ralf
- Podlesny, Joscha

*2010 Mathematics Subject Classification*

- 74B05 74Qxx 74Q05 65M22 65M55

*Keywords*

- Multiscale interfaces, fractal homogenization, elasticity, jump conditions, multiscale numerics

*DOI*

*Abstract*

Inspired from geological problems, we introduce a new geometrical setting for homogenization of a well known and well studied problem of an elliptic second order differential operator with jump condition on a multiscale network of interfaces. The geometrical setting is fractal and hence neither periodic nor stochastic methods can be applied to the study of such kind of multiscale interface problem. Instead, we use the fractal nature of the geometric structure to introduce smoothed problems and apply methods from a posteriori theory to derive an estimate for the order of convergence. Computational experiments utilizing an iterative homogenization approach illustrate that the theoretically derived order of convergence of the approximate problems is close to optimal.

*Appeared in*

- Multiscale Model. Simul., 18 (2020), pp. 294--314, DOI 10.1137/18M1204759 .

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# Surfing the edge: Finding nonlinear solutions using feedback control

*Authors*

- Willis, Ashley P.
- Duguet, Yohann
- Omel'chenko, Oleh E.

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

*2010 Mathematics Subject Classification*

- 76D05 93B52

*Keywords*

- Pipe flow, edge states, feedback control

*DOI*

*Abstract*

Many transitional wall-bounded shear flows are characterised by the coexistence in state-space of laminar and turbulent regimes. Probing the edge boundarz between the two attractors has led in the last decade to the numerical discovery of new (unstable) solutions to the incompressible Navier--Stokes equations. However, the iterative bisection method used to achieve this can become prohibitively costly for large systems. Here we suggest a simple feedback control strategy to stabilise edge states, hence accelerating their numerical identification by several orders of magnitude. The method is illustrated for several configurations of cylindrical pipe flow. Traveling waves solutions are identified as edge states, and can be isolated rapidly in only one short numerical run. A new branch of solutions is also identified. When the edge state is a periodic orbit or chaotic state, the feedback control does not converge precisely to solutions of the uncontrolled system, but nevertheless brings the dynamics very close to the original edge manifold in a single run. We discuss the opportunities offered by the speed and simplicity of this new method to probe the structure of both state space and parameter space.

*Appeared in*

- J. Fluid Mech., 831 (2017) pp. 579--591, DOI 10.20347/WIAS.PREPRINT.2389 .

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# Mathematical models as research data via flexiformal theory graphs

*Authors*

- Kohlhase, Michael
- Koprucki, Thomas

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

ORCID: 0000-0003-1274-9951

*2010 Mathematics Subject Classification*

- 86T30 35Q68 68Q55

*Keywords*

- Mathematical models, OMDoc/MMT, research data, mathematical knowledge management

*DOI*

*Abstract*

Mathematical modeling and simulation (MMS) has now been established as an essential part of the scientific work in many disciplines. It is common to categorize the involved numerical data and to some extent the corresponding scientific software as research data. But both have their origin in mathematical models, therefore any holistic approach to research data in MMS should cover all three aspects: data, software, and models. While the problems of classifying, archiving and making accessible are largely solved for data and first frameworks and systems are emerging for software, the question of how to deal with mathematical models is completely open. In this paper we propose a solution -- to cover all aspects of mathematical models: the underlying mathematical knowledge, the equations, boundary conditions, numeric approximations, and documents in a flexiformal framework, which has enough structure to support the various uses of models in scientific and technology workflows. Concretely we propose to use the OMDoc/MMT framework to formalize mathematical models and show the adequacy of this approach by modeling a simple, but non-trivial model: van Roosbroeck's drift-diffusion model for one-dimensional devices. This formalization -- and future extensions -- allows us to support the modeler by e.g. flexibly composing models, visualizing Model Pathway Diagrams, and annotating model equations in documents as induced from the formalized documents by flattening. This directly solves some of the problems in treating MMS as "research data'' and opens the way towards more MKM services for models.

*Appeared in*

- Intelligent Computer Mathematics: 10th International Conference, CICM 2017, Edinburgh, UK, July 17-21, 2017, Proceedings, H. Geuvers, M. England, O. Hasan, F. Rabe , O. Teschke, eds., Springer International Publishing, Cham, 2017, pp. 224--238, DOI 10.1007/978-3-319-62075-6_16 .

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# Model pathway diagrams for the representation of mathematical models

*Authors*

- Koprucki, Thomas

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

ORCID: 0000-0003-1274-9951 - Müller, Dennis
- Rabe, Florian

*2010 Mathematics Subject Classification*

- 86T30 00A71 68Q55

*Keywords*

- Mathematical models, research data, Model Pathway Diagrams, drift-diffusion equations

*DOI*

*Abstract*

Mathematical models are the foundation of numerical simulation of optoelectronic devices. We present a concept for a machine-actionable as well as human-understandable representation of the mathematical knowledge they contain and the domain-specific knowledge they are based on. We propose to use theory graphs to formalize mathematical models and model pathway diagrams to visualize them. We illustrate our approach by application to the van Roosbroeck system describing the carrier transport in semiconductors by drift and diffusion. We introduce an approach for the block-based composition of models from simpler components.

*Appeared in*

- Opt. Quantum Electron., 50 (2018), pp. 70/1--70/9, DOI 10.1007/s11082-018-1321-7 .

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# Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes problem

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Merdon, Christian
- Neilan, Michael
- Neumann, Felix

*2010 Mathematics Subject Classification*

- 65N12 65N30 65N15 76D07 76M10

*Keywords*

- incompressible Stokes equations, mixed finite element methods, nonconforming discretizations, pressure-robustness, a-priori error estimates, Helmholtz projector

*DOI*

*Abstract*

Nearly all classical inf-sup stable mixed finite element methods for the incompressible Stokes equations are not pressure-robust, i.e., the velocity error is dependent on the pressure. However, recent results show that pressure-robustness can be recovered by a non-standard discretization of the right hand side alone. This variational crime introduces a consistency error in the method which can be estimated in a straightforward manner provided that the exact velocity solution is sufficiently smooth. The purpose of this paper is to analyze the pressure-robust scheme with low regularity. The numerical analysis applies divergence-free H¹-conforming Stokes finite element methods as a theoretical tool. As an example, pressure-robust velocity and pressure a-priori error estimates will be presented for the (first order) nonconforming Crouzeix--Raviart element. A key feature in the analysis is the dependence of the errors on the Helmholtz projector of the right hand side data, and not on the entire data term. Numerical examples illustrate the theoretical results.

*Appeared in*

- Math. Comp., 87 (2018), pp. 1543--1566, DOI 10.1090/mcom/3344 .

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# Exact firing time statistics of neurons driven by discrete inhibitory noise

*Authors*

- Olmi, Simona

ORCID: 0000-0002-8272-3493 - Angulo-Garcia, David
- Imparato, Alberto
- Torcini, Alessandro

*2008 Physics and Astronomy Classification Scheme*

- 87.19.lj 05.45.Xt 87.19.lm

*Keywords*

- Pulse-coupled heterogeneous inhibitory neural networks

*DOI*

*Abstract*

Neurons in the intact brain receive a continuous and irregular synaptic bombardment from excitatory and inhibitory pre-synaptic neurons, which determines the firing activity of the stimulated neuron. In order to investigate the influence of inhibitory stimulation on the firing time statistics, we consider Leaky Integrate-and-Fire neurons subject to inhibitory instantaneous post-synaptic potentials. In particular, we report exact results for the firing rate, the coefficient of variation and the spike train spectrum for various synaptic weight distributions. Our results are not limited to stimulations of infinitesimal amplitude, but they apply as well to finite amplitude post-synaptic potentials, thus being able to capture the effect of rare and large spikes. The developed methods are able to reproduce also the average firing properties of heterogeneous neuronal populations.

*Appeared in*

- Sci. Rep., 7 (2017), pp. 1577/1--1577/15, DOI 10.1038/s41598-017-01658-8 .

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# 3D electrothermal simulations of organic LEDs showing negative differential resistance

*Authors*

- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Fuhrmann, Jürgen

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

ORCID: 0000-0001-6235-9412 - Fischer, Axel
- Reineke, Sebastian

*2010 Mathematics Subject Classification*

- 35Q79 35J92 80M12

*2008 Physics and Astronomy Classification Scheme*

- 72.20.Pa, 72.80.Le, 81.05.Fb, 85.80.Fi, 02.60.Cb

*Keywords*

- organic semiconductors, self-heating, negative differential resistance, p-Laplacian, thermistor model, hybrid finite-volume/finite-element scheme

*DOI*

*Abstract*

Organic semiconductor devices show a pronounced interplay between temperature-activated conductivity and self-heating which in particular causes inhomogeneities in the brightness of large-area OLEDs at high power. We consider a 3D thermistor model based on partial differential equations for the electrothermal behavior of organic devices and introduce an extension to multiple layers with nonlinear conductivity laws, which also take the diode-like behavior in recombination zones into account. We present a numerical simulation study for a red OLED using a finite-volume approximation of this model. The appearance of S-shaped current-voltage characteristics with regions of negative differential resistance in a measured device can be quantitatively reproduced. Furthermore, this simulation study reveals a propagation of spatial zones of negative differential resistance in the electron and hole transport layers toward the contact.

*Appeared in*

- Opt. Quantum Electron., 49 (2017), pp. 330/1--330/8, DOI 10.1007/s11082-017-1167-4 .

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# Analysis and simulations for a phase-field fracture model at finite strains based on modified invariants

*Authors*

- Thomas, Marita
- Bilgen, Carola
- Weinberg, Kerstin

*2010 Mathematics Subject Classification*

- 35K85 49J40 74B20 74R10 74R20

*Keywords*

- Phase-field model for fracture, Ambrosio-Tortorelli, viscous evolution, finite strains, modified invariants

*DOI*

*Abstract*

Phase-field models have already been proven to predict complex fracture patterns in two and three dimensions for brittle fracture at small strains. In this paper we discuss a model for phase-field fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. We here present a phase-field model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split and the modified invariants of the right Cauchy-Green strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions and we show that the time-discrete solutions converge in a weak sense to a solution of the time-continuous formulation of the model. Numerical examples in two and three space dimensions are carried out in the range of validity of the analytical results.

*Appeared in*

- ZAMM Z. Angew. Math. Mech., published online on 22.07.2020, DOI 10.1002/zamm.201900288 .

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# A PDE-constrained optimization approach for topology optimization of strained photonic devices

*Authors*

- Adam, Lukáš
- Hintermüller, Michael

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

ORCID: 0000-0003-2473-4984

*2010 Mathematics Subject Classification*

- 49J20 35Q93 35Q74 90C46

*Keywords*

- semiconductor lasers, germanium, topology optimization, optimization with PDE constraints, elasticity, phase-field

*DOI*

*Abstract*

Recent studies have demonstrated the potential of using tensile-strained, doped Germanium as a means of developing an integrated light source for (amongst other things) future microprocessors. In this work, a multi-material phase-field approach to determine the optimal material configuration within a so-called Germanium-on-Silicon microbridge is considered. Here, an ``optimal" configuration is one in which the strain in a predetermined minimal optical cavity within the Germanium is maximized according to an appropriately chosen objective functional. Due to manufacturing requirements, the emphasis here is on the cross-section of the device; i.e. a socalled aperture design. Here, the optimization is modeled as a non-linear optimization problem with partial differential equation (PDE) and manufacturing constraints. The resulting problem is analyzed and solved numerically. The theory portion includes a proof of existence of an optimal topology, differential sensitivity analysis of the displacement with respect to the topology, and the derivation of first and second-order optimality conditions. For the numerical experiments, an array of first and second-order solution algorithms in function-space are adapted to the current setting, tested, and compared. The numerical examples yield designs for which a significant increase in strain (as compared to an intuitive empirical design) is observed.

*Appeared in*

- Optim. Eng., 19 (2018), pp. 521--557, DOI 10.1007/s11081-018-9394-5 .

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# Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

*Authors*

- Lederer, Philip L.
- Merdon, Christian
- Schöberl, Joachim

*2010 Mathematics Subject Classification*

- 65N15 65N30 76D07 76M10

*Keywords*

- Incompressible Navier--Stokes equations, mixed finite elements, pressure robustness, a posteriori error estimators, adaptive mesh refinement

*DOI*

*Abstract*

Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure-independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness.

*Appeared in*

- J. Numer. Math., 142 (2019), pp. 713--748.

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# A stochastic algorithm without time discretization error for the Wigner equation

*Authors*

- Muscato, Orazio
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 65C05 60J25 81Q05

*Keywords*

- Wigner equation, stochastic algorithms, numerical experiments

*DOI*

*Abstract*

Stochastic particle methods for the numerical treatment of the Wigner equation are considered. The approximation properties of these methods depend on several numerical parameters. Such parameters are the number of particles, a time step (if transport and other processes are treated separately) and the grid size (used for the discretization of the position and the wave-vector). A stochastic algorithm without time discretization error is introduced. Its derivation is based on the theory of piecewise deterministic Markov processes. Numerical experiments are performed in a one-dimensional test case. Approximation properties with respect to the grid size and the number of particles are studied. Convergence of a time-splitting scheme to the no-splitting algorithm is demonstrated. The no-splitting algorithm is shown to be more efficient in terms of computational effort.

*Appeared in*

- Kinetic and Related Models, 12 (2019), pp. 59-77.

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# The Bouchaud--Anderson model with double-exponential potential

*Authors*

- Muirhead, Stephen
- Pymar, Richard
- Soares dos Santos, Renato

*2010 Mathematics Subject Classification*

- 60H25 82C44

*Keywords*

- Parabolic Anderson model, Bouchaud trap model, intermittency, localisation

*DOI*

*Abstract*

The Bouchaud--Anderson model (BAM) is a generalisation of the parabolic Anderson model (PAM) in which the driving simple random walk is replaced by a random walk in an inhomogeneous trapping landscape; the BAM reduces to the PAM in the case of constant traps. In this paper we study the BAM with double-exponential potential. We prove the complete localisation of the model whenever the distribution of the traps is unbounded. This may be contrasted with the case of constant traps (i.e. the PAM), for which it is known that complete localisation fails. This shows that the presence of an inhomogeneous trapping landscape may cause a system of branching particles to exhibit qualitatively distinct concentration behaviour.

*Appeared in*

- Ann. Appl. Probab., 29:1 (2019), pp. 264-325.

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# Phase sensitive excitability of a limit cycle

*Authors*

- Franović, Igor
- Omel'chenko, Oleh E.

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

*2008 Physics and Astronomy Classification Scheme*

- 89.75.Fb 05.40.Ca

*Keywords*

- Excitability, coherence resonance

*DOI*

*Abstract*

The classical notion of excitability refers to an equilibrium state that shows under the influence of perturbations a nonlinear threshold-like behavior. Here, we extend this concept by demonstrating how periodic orbits can exhibit a specific form of excitable behavior where the nonlinear threshold-like response appears only after perturbations applied within a certain part of the periodic orbit, i.e the excitability happens to be phase sensitive. As a paradigmatic example of this concept we employ the classical FitzHugh-Nagumo system. The relaxation oscillations, appearing in the oscillatory regime of this system, turn out to exhibit a phase sensitive nonlinear threshold-like response to perturbations, which can be explained by the nonlinear behavior in the vicinity of the canard trajectory. Triggering the phase sensitive excitability of the relaxation oscillations by noise we find a characteristic non-monotone dependence of the mean spiking rate of the relaxation oscillation on the noise level. We explain this non-monotone dependence as a result of an interplay of two competing effects of the increasing noise: the growing efficiency of the excitation and the degradation of the nonlinear response.

*Appeared in*

- Chaos, 28 (2018), pp. 071105/1--071105/6, DOI 10.1063/1.5045179 .

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# Finite elements for scalar convection-dominated equations and incompressible flow problems -- A never ending story?

*Authors*

- John, Volker

ORCID: 0000-0002-2711-4409 - Knobloch, Petr
- Novo, Julia

*2010 Mathematics Subject Classification*

- 65M60 65N30

*Keywords*

- convection-dominated problems, incompressible flow problems, finite element analysis, recent results, open problems

*DOI*

*Abstract*

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed.

*Appeared in*

- Comput. Vis. Sci., 19 (2018), pp. 47--63 .

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# Reproducible research through persistently linked and visualized data

*Authors*

- Drees, Bastian
- Kraft, Angelina
- Koprucki, Thomas

ORCID: 0000-0001-6235-9412

*2010 Mathematics Subject Classification*

- 68P20 68U35 65Yxx

*Keywords*

- Research data, repositories, visualization, persistent identifier

*DOI*

*Abstract*

The demand of reproducible results in the numerical simulation of opto-electronic devices or more general in mathematical modeling and simulation requires the (long-term) accessibility of data and software that were used to generate those results. Moreover, to present those results in a comprehensible manner data visualizations such as videos are useful. Persistent identifier can be used to ensure the permanent connection of these different digital objects thereby preserving all information in the right context. Here we give an overview over the state-of-the art of data preservation, data and software citation and illustrate the benefits and opportunities of enhancing publications with visual simulation data by showing a use case from opto-electronics.

*Appeared in*

- Opt. Quantum Electron., 50 (2018), pp. 59/1--59/10, DOI 10.1007/s11082-018-1327-1 .

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# Hybrid quantum-classical modeling of quantum dot devices

*Authors*

- Kantner, Markus

ORCID: 0000-0003-4576-3135 - Mittnenzweig, Markus
- Koprucki, Thomas

ORCID: 0000-0001-6235-9412

*2010 Mathematics Subject Classification*

- 35Qxx 81V65 81V70, 81V80, 82D37, 82C10, 82C70

*2008 Physics and Astronomy Classification Scheme*

- 05.30.-d 42.50.-p 73.63.Kv 85.30.De 85.35.-p 85.60.Bt.

*Keywords*

- device simulation, quantum dots, Lindblad equation, quantum-classical coupling, single-photon, sources.

*DOI*

*Abstract*

The design of electrically driven quantum dot devices for quantum optical applications asks for modeling approaches combining classical device physics with quantum mechanics. We connect the well-established fields of semi-classical semiconductor transport theory and the theory of open quantum systems to meet this requirement. By coupling the van Roosbroeck system with a quantum master equation in Lindblad form, we obtain a new hybrid quantum-classical modeling approach, which enables a comprehensive description of quantum dot devices on multiple scales: It allows the calculation of quantum optical figures of merit and the spatially resolved simulation of the current flow in realistic semiconductor device geometries in a unified way. We construct the interface between both theories in such a way, that the resulting hybrid system obeys the fundamental axioms of (non-)equilibrium thermodynamics. We show that our approach guarantees the conservation of charge, consistency with the thermodynamic equilibrium and the second law of thermodynamics. The feasibility of the approach is demonstrated by numerical simulations of an electrically driven single-photon source based on a single quantum dot in the stationary and transient operation regime.

*Appeared in*

- Phys. Rev. B., 96 (2017), pp. 205301/1--205301/17, DOI 10.1103/PhysRevB.96.205301 .

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# Symmetry breaking and strong persistent plasma currents via resonant destabilization of atoms

*Authors*

- Brée, Carsten
- Hofmann, Michael
- Babushkin, Ihar
- Demircan, Ayhan
- Morgner, Uwe
- Kosareva, Olga G.
- Savel'ev, Andrei B.
- Husakou, Anton
- Ivanov, Misha

*2010 Mathematics Subject Classification*

- 78A60 35P25

*2008 Physics and Astronomy Classification Scheme*

- 32.80.Rm 42.65.Ky 42.50.Hz

*Keywords*

- THz light generation, photoionization, multiphoton processes

*DOI*

*Abstract*

The ionization rate of an atom in a strong optical field can be resonantly enhanced by the presence of long-living atomic levels (so called Freeman resonances). This process is most prominent in the multiphoton ionization regime meaning that ionization event takes many optical cycles. Nevertheless, here we show that these resonances can lead to fast subcycle-scale plasma buildup at the resonant values of the intensity in the pump pulse. The fast buildup can break the cycle-to-cycle symmetry of the ionization process, resulting in generation of persistent macroscopic plasma currents which remain after the end of the pulse. This, in turn, gibes rise to a broad-band radiation of unusual spectral structure forming a comb from terahertz (THz) to visible. This radiation contains fingerprints of the attosecond electronic dynamics in Rydberg states during ionization.

*Appeared in*

- Phys. Rev. Lett., 119 (2017), pp. 243202/1--243202/5.

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# Stochastic unfolding and homogenization

*Authors*

- Heida, Martin
- Neukamm, Stefan
- Varga, Mario

*2010 Mathematics Subject Classification*

- 49J40 74Nxx 74Qxx 74Q05

*Keywords*

- Stochastic homogenization, unfolding, convex functionals, Allen--Cahn equation

*DOI*

*Abstract*

The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a stochastic homogenization result for an non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method introduced in this paper extends textitdiscrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition.

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# A comparison of delamination models: Modeling, properties, and applications

*Authors*

- Thomas, Marita

*2010 Mathematics Subject Classification*

- 35Q74 74H20 74C05 74C10 49J53 74M15 74R10

*Keywords*

- Adhesive contact, cohesive zone delamination, brittle Griffith-type delamination, coupled rate-independent/rate-dependent system

*DOI*

*Abstract*

This contribution presents recent results in the modeling and the analysis of delamination problems. It addresses adhesive contact, brittle, and cohesive zone models both in a quasistatic and a viscous, dynamic setting for the bulk part. Also different evolution laws for the delaminating surface are discussed.

*Appeared in*

- Mathematical Analysis of Continuum Mechanics and Industrial Applications II, Proceedings of the International Conference CoMFoS16, P. van Meurs, M. Kimura, H. Notsu, eds., vol. 30 of Mathematics for Industry, Springer, Singapore, 2017, pp. 27--38, DOI 10.1007/978-981-10-6283-4_3 .

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# Limiting problems for a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions

*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, asymptotic analyses, long-time behavior

*DOI*

*Abstract*

This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp.105--118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases.

*Appeared in*

- Proceedings of the INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., vol. 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 217--242, DOI 10.1007/978-3-319-75940-1_11 .

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# On a Cahn--Hilliard system with convection and dynamic boundary conditions

*Authors*

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

*2010 Mathematics Subject Classification*

- 35K61 35K25 76R05 80A22

*Keywords*

- Cahn-Hilliard system, convection, dynamic boundary condition, in-tial-boundary value problem, well-posedness, regularity of solutions

*DOI*

*Abstract*

This paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of Cahn--Hilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure Cahn--Hilliard cases are investigated, and a number of results is proven about existence of solutions, uniqueness, regularity, continuous dependence, uniform boundedness of solutions, strict separation property. A complete approximation of the problem, based on the regularization of maximal monotone graphs and the use of a Faedo--Galerkin scheme, is introduced and rigorously discussed.

*Appeared in*

- Ann. Mat. Pura Appl. IV. Ser., 197 (2018), pp. 1445--1475, DOI 10.1007/s10231-018-0732-1 .

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# A multilevel Schur complement preconditioner for complex symmetric matrices

*Authors*

- Schlundt, Rainer

ORCID: 0000-0002-4424-4301

*2010 Mathematics Subject Classification*

- 65F08 65F15 65N22 65Y05

*Keywords*

- Complex symmetric sparse linear system, Schur complement, multilevel preconditioner, domain decomposition, low rank approximation

*DOI*

*Abstract*

This paper describes a multilevel preconditioning technique for solving complex symmetric sparse linear systems. The coefficient matrix is first decoupled by domain decomposition and then an approximate inverse of the original matrix is computed level by level. This approximate inverse is based on low rank approximations of the local Schur complements. For this, a symmetric singular value decomposition of a complex symmetric matix is used. Using the example of Maxwell's equations the generality of the approach is demonstrated.

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# Non-intrusive tensor reconstruction for high dimensional random PDEs

*Authors*

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

*2010 Mathematics Subject Classification*

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

*Keywords*

- non-intrusive, tensor reconstruction, partial differential equations with random coefficients, tensor representation, tensor train, uncertainty quantification, low-rank

*DOI*

*Abstract*

This paper examines a completely non-intrusive, sample-based method for the computation of functional low-rank solutions of high dimensional parametric random PDEs which have become an area of intensive research in Uncertainty Quantification (UQ). In order to obtain a generalized polynomial chaos representation of the approximate stochastic solution, a novel black-box rank-adapted tensor reconstruction procedure is proposed. The performance of the described approach is illustrated with several numerical examples and compared to Monte Carlo sampling.

*Appeared in*

- Comput. Methods Appl. Math., 19 (2019), pp. 39--53 (published online on 25.07.2018), DOI 10.1515/cmam-2018-0028 .

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# Pulses in FitzHugh--Nagumo systems with rapidly oscillating coefficients

*Authors*

- Gurevich, Pavel
- Reichelt, Sina

*2010 Mathematics Subject Classification*

- 35B40 35B40 35K65 37C29 37C75 37N25

*Keywords*

- Traveling waves, pulse solutions, FitzHugh-Nagumo system, two-scale convergence, spectral decomposition, semigroups

*DOI*

*Abstract*

This paper is devoted to pulse solutions in FitzHugh-Nagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. In the limit of vanishing periods, there arises a two-scale FitzHugh-Nagumo system, which qualitatively and quantitatively captures the dynamics of the original system. We prove existence and stability of pulses in the limit system and show their proximity on any finite time interval to pulse-like solutions of the original system.

*Appeared in*

- Multiscale Model. Simul., 16 (2018), pp. 833--856.

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# Consistent operator semigroups and their interpolation

*Authors*

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

*2010 Mathematics Subject Classification*

- 46B70 46M35 47D06

*Keywords*

- Semigroups, interpolation

*DOI*

*Abstract*

Under a mild regularity condition we prove that the generator of the interpolation of two C0-semigroups is the interpolation of the two generators.

*Appeared in*

- J. Operator Theory, 82 (2019), pp. 3--21, DOI 10.7900/jot.2017nov15.2233 .

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# Numerical approach to a model for quasistatic damage with spatial $BV$-regularization

*Authors*

- Bartels, Sören
- Milicevic, Marijo
- Thomas, Marita

*2010 Mathematics Subject Classification*

- 35K85 74R05 49J45 49S05 65M12

*Keywords*

- Partial damage, damage evolution with spatial regularization, functions of bounded variation, semistable energetic solutions, numerical approximation

*DOI*

*Abstract*

We address a model for rate-independent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BV-regularization. Discrete solutions are obtained using an alternate time-discrete scheme and the Variable-ADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove convergence of the method and show that discrete solutions approximate a semistable energetic solution of the rate-independent system. Moreover, we present our numerical results for two benchmark problems.

*Appeared in*

- Proceedings of the INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, eds., vol. 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 179--203, DOI 10.1007/978-3-319-75940-1_9 .

*Download Documents*

# Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi--Dirac and Gauss--Fermi statistics

*Authors*

- Farrell, Patricio

ORCID: 0000-0001-9969-6615 - Patriarca, Matteo
- Fuhrmann, Jürgen

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

ORCID: 0000-0001-6235-9412

*2010 Mathematics Subject Classification*

- 65N08 35K55

*Keywords*

- Scharfetter--Gummel schemes, (organic) semiconductors, nonlinear diffusion, ermodynamic consistency, finite volume scheme, Gauss--Fermi integral, Fermi--Dirac integral

*DOI*

*Abstract*

We compare three thermodynamically consistent Scharfetter--Gummel schemes for different distribution functions for the carrier densities, including the Fermi--Dirac integral of order 1/2 and the Gauss--Fermi integral. The most accurate (but unfortunately also most costly) generalized Scharfetter--Gummel scheme requires the solution of an integral equation. We propose a new method to solve this integral equation numerically based on Gauss quadrature and Newton's method. We discuss the quality of this approximation and plot the resulting currents for Fermi--Dirac and Gauss--Fermi statistics. Finally, by comparing two modified (diffusion-enhanced and inverse activity based) Scharfetter--Gummel schemes with the more accurate generalized scheme, we show that the diffusion-enhanced ansatz leads to considerably lower flux errors, confirming previous results (J. Comp. Phys. 346:497-513, 2017).

*Appeared in*

- Opt. Quantum Electron., 50 (2018), pp. 101/1--101/10, DOI 10.1007/s11082-018-1349-8 .

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# The weighted energy-dissipation principle and evolutionary Gamma-convergence for doubly nonlinear problems

*Authors*

- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Melchionna, Stefano

*2010 Mathematics Subject Classification*

- 58E30 35K55 47J35

*Keywords*

- Doubly nonlinear evolution, weighted-energy-dissipation principle, evolutionary Gamma-convergence, variational principle

*DOI*

*Abstract*

We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizer correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of the associated systems. More precisely, we deal with the limits δ→0, ε→0, as well as δ+ ε→0 either in the sense of Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε→0, in the case of quadratic dissipation potentials and uniformly λ-convex energy functionals. Finally, we discuss a homogenization problem as an example of application.

*Appeared in*

- ESAIM Control Optim. Calc. Var., 25 (2019), pp. 36/1--36/38, DOI 10.1051/cocv/2018023 .

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# Extreme at-the-money skew in a local volatility model

*Authors*

- Pigato, Paolo

*2010 Mathematics Subject Classification*

- 60H08 91G08

*Keywords*

- Implied volatility, local volatility, skew explosion, small-time asymptotics, European option pricing, discontinuous coefficients, regime-switch

*DOI*

*Abstract*

We consider a local volatility model, with volatility taking two possible values, depending on the value of the underlying with respect to a fixed threshold. When the threshold is taken at-the-money, we establish exact pricing formulas and compute short-time asymptotics of the implied volatility surface. We derive an exact formula for the at-the-money implied volatility skew, which explodes as T^{-1/2}, reproducing the empirical "steep short end of the smile". This behavior does not depend on the precise choice of the parameters, but simply follows from the "regime-switch" of the local volatility at-the-money.

*Appeared in*

- Finance Stoch., 23 (2019), pp. 827--859, DOI 10.1007/s00780-019-00406-2 .

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# Random walk on random walks: Low densities

*Authors*

- Blondel, Oriane
- Hilário, Marcelo R.
- Soares dos Santos, Renato
- Sidoravicius, Vladas
- Teixeira, Augusto

*2010 Mathematics Subject Classification*

- 60F15 60K35 82B41 82C22 82C44

*Keywords*

- Random walk, dynamic random environment, law of large numbers, central limit theorem, large deviations, renormalization, regeneration

*DOI*

*Abstract*

We consider a random walker in a dynamic random environment given by a system of independent simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or non-lazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condition.

*Appeared in*

- Ann. Appl. Probab., 30:4 (2020), pp. 1614-1641.

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# A kinetic equation for the distribution of interaction clusters in rarefied gases

*Authors*

- Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857 - Simonella, Sergio
- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 60K35 82B40

*Keywords*

- stochastic particle system, interaction clusters, rarefied gases, kinetic equation, numerical experiments

*DOI*

*Abstract*

We consider a stochastic particle model governed by an arbitrary binary interaction kernel. A kinetic equation for the distribution of interaction clusters is established. Under some additional assumptions a recursive representation of the solution is found. For particular choices of the interaction kernel (including the Boltzmann case) several explicit formulas are obtained. These formulas are confirmed by numerical experiments. The experiments are also used to illustrate various conjectures and open problems.

*Appeared in*

- J. Statist. Phys., 169 (2017), pp. 126--167.

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# Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory

*Authors*

- Renger, D. R. Michiel

ORCID: 0000-0003-3557-3485

*2010 Mathematics Subject Classification*

- 46N55 60F10 60J27 82C35

*2008 Physics and Astronomy Classification Scheme*

- 05.40.-a 05.70.Ln 82.40.Bj 82.20.Fd

*Keywords*

- empirical measure, empirical flux, discrete space, large deviations, Macroscopic Fluctuation Theory

*DOI*

*Abstract*

We consider a system of independent particles on a finite state space, and prove a dynamic large-deviation principle for the empirical measure-empirical flux pair, taking the specific fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a large-deviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finite-space setting.

*Appeared in*

- J. Statist. Phys., 172 (2018), pp. 1291--1326, DOI 10.1007/s10955-018-2083-0 .

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# Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part I: Derivation of the model and survey of the results

*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*

- 82.45Gj, 82.45.Mp, 82.60Lf

*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

*DOI*

*Abstract*

We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a global--in--time weak solution for the full model, allowing for a general structure of the mobility tensor and for chemical reactions with highly non linear rates in the bulk and on the active boundary. We characterise the singular states of the system, showing that the chemical species can vanish only globally in space, and that this phenomenon must be concentrated in a compact set of measure zero in time. With respect to our former study [DDGG16], we also essentially improve the a priori estimates, in particular concerning the relative chemical potentials.

*Appeared in*

- ZAMP Z. Angew. Math. Phys., 71 (2020), pp. 119/1--119/68, DOI 10.1007/s00033-020-01341-5 .

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# Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part II: Approximation and a priori estimates

*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*

- 82.45Gj, 82.45.Mp, 82.60Lf

*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

*DOI*

*Abstract*

We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper, which continues the investigation of [DDGG17a], we derive for thermodynamically consistent approximation schemes the natural uniform estimates associated with the dissipations. Our results essentially improve our former study [DDGG16], in particular the a priori estimates concerning the relative chemical potentials.

*Appeared in*

- ZAMP Z. Angew. Math. Phys., 71 (2020), pp. 119/1--119/68, DOI 10.1007/s00033-020-01341-5 .

*Download Documents*

# Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part III: Compactness and convergence

*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*

- 82.45Gj, 82.45.Mp, 82.60Lf

*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

*DOI*

*Abstract*

We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper, which continues the investigations of [DDGG17a, DDGG17b], we prove the compactness of the solution vector, and existence and convergence for the approximation schemes. We point at simple structural PDE arguments as an adequate substitute to the Aubin--Lions compactness Lemma and its generalisations: These familiar techniques attain their limit in the context of our model in which the relationship between time derivatives (transport) and diffusion gradients is highly non linear.

*Appeared in*

- ZAMP Z. Angew. Math. Phys., 71 (2020), pp. 119/1--119/68, DOI 10.1007/s00033-020-01341-5 .

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# Gibbsian representation for point processes via hyperedge potentials

*Authors*

- Jahnel, Benedikt

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

*2010 Mathematics Subject Classification*

- 82B21 60K35

*Keywords*

- Gibbsian point processes, Koslov theorem, Sullivan theorem, hyperedge potentials, Widom-Rowlinson model

*DOI*

*Abstract*

We consider marked point processes on the *d*-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing uniformly absolutely convergent Hamiltonians in terms of hyperedge potentials in the sense of Georgii [2]. These potentials are natural generalizations of physical multibody potentials which are useful in models of stochastic geometry.

*Appeared in*

- J. Theoret. Probab., published online on 03.11.2019 (2019), pp. 1--27, DOI 10.1007/s10959-019-00960-7 .

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# Global bifurcation analysis of a class of planar systems

*Authors*

- Grin, Alexander
- Schneider, Klaus R.

*2010 Mathematics Subject Classification*

- 34C05 34C23 34D20

*Keywords*

- Global bifurcation limit cycle, planar autonomous system, Dulac--Cherkas function, rotated vector field, singularly perturbed system

*DOI*

*Abstract*

We consider planar autonomous systems dx/dt =P(x,y,λ), dy/dt =Q(x,y,λ) depending on a scalar parameter λ. We present conditions on the functions P and Q which imply that there is a parameter value λ_{0} such that for &lambda > λ_{0} this system has a unique limit cycle which is hyperbolic and stable. Dulac-Cherkas functions, rotated vector fields and singularly perturbed systems play an important role in the proof.

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# Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials

*Authors*

- Biskup, Marek
- Fukushima, Ryoki
- König, Wolfgang

ORCID: 0000-0002-4212-0065

*2010 Mathematics Subject Classification*

- 60H25 82B44 35P20 74Q15 47A75 47H40

*Keywords*

- Random Schrödinger operator, Anderson Hamiltonian, eigenvalue, spectral statistics, homogenization, central limit theorem

*DOI*

*Abstract*

We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schrödinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors' recent work where similar conclusions have been obtained for bounded random potentials. endabstract

*Appeared in*

- Interdiscip. Inform. Sci., 24 (2018), pp. 59--76.

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# Coexistence of Hamiltonian-like and dissipative dynamics in chains of coupled phase oscillators with skew-symmetric coupling

*Authors*

- Burylko, Oleksandr
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Wolfrum, Matthias
- Yanchuk, Serhiy

*2010 Mathematics Subject Classification*

- 34C14 34C15 34C28, 34C30, 37C80, 37L60

*Keywords*

- Phase oscillators, reversible systems, amplitude equations

*DOI*

*Abstract*

We consider rings of coupled phase oscillators with anisotropic coupling. When the coupling is skew-symmetric, i. e. when the anisotropy is balanced in a specific way, the system shows robustly a coexistence of Hamiltonian-like and dissipative regions in the phase space. We relate this phenomenon to the time-reversibility property of the system. The geometry of low-dimensional systems up to five oscillators is described in detail. In particular, we show that the boundary between the dissipative and Hamiltonian-like regions consists of families of heteroclinic connections. For larger chains with skew-symmetric coupling, some sufficient conditions for the coexistence are provided, and in the limit of N → ∞ oscillators, we formally derive an amplitude equation for solutions in the neighborhood of the synchronous solution. It has the form of a nonlinear Schrödinger equation and describes the Hamiltonian-like region existing around the synchronous state similarly to the case of finite rings.

*Appeared in*

- SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 2076--2105, DOI 10.1137/17M1155685 .

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# Short-time near-the-money skew in rough fractional volatility models

*Authors*

- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Friz, Peter

ORCID: 0000-0003-2571-8388 - Gulisashvili, Archil
- Horvath, Blanka
- Stemper, Benjamin

*2010 Mathematics Subject Classification*

- 91G20 60H30 60F10

*DOI*

*Abstract*

We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter H < ½. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order t^{1/2} (recent works of Alòs, León & Vives and Fukasawa) to the wider moderate deviations regime.

*Appeared in*

- Quant. Finance, 19 (2019), pp. 779--798 (published online on 13.11.2018), DOI 10.1080/14697688.2018.1529420 .

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# Duality results and regularization schemes for Prandtl--Reuss perfect plasticity

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Rösel, Simon

*2010 Mathematics Subject Classification*

- 74C05 49M15 49K20 49M29

*Keywords*

- perfect plasticity, Prandtl-Reuss plasticity, small-strain, Fenchel duality, semismooth Newton

*DOI*

*Abstract*

We consider the time-discretized problem of the quasi-static evolution problem in perfect plasticity posed in a non-reflexive Banach space and we derive an equivalent version in a reflexive Banach space. A primal-dual stabilization scheme is shown to be consistent with the initial problem. As a consequence, not only stresses, but also displacement and strains are shown to converge to a solution of the original problem in a suitable topology. This scheme gives rise to a well-defined Fenchel dual problem which is a modification of the usual stress problem in perfect plasticity. The dual problem has a simpler structure and turns out to be well-suited for numerical purposes. For the corresponding subproblems an efficient algorithmic approach in the infinite-dimensional setting based on the semismooth Newton method is proposed.

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# Regularizing aperiodic cycles of resonant radiation in filament light bullets

*Authors*

- Brée, Carsten
- Babushkin, Ihar
- Morgner, Uwe
- Demircan, Ayhan

*2010 Mathematics Subject Classification*

- 76B15

*2008 Physics and Astronomy Classification Scheme*

- 47.35.Bb, 47.27.Sd

*Keywords*

- femtosecond filamentation, nonlinear optical solitions

*DOI*

*Abstract*

We demonstrate an up to now unrecognized and very effective mechanism which prevents filament collapse and allows persistent self-guiding propagation retaining larg portion of the optical energy on-axis over unexpected long distances. The key ingredient is the possibility of leaking continuously energy into the normal dispersion regime via emission of resonant radiation. The frequency of the radiation is determined by the dispersion dynamically modified by photo-generated plasma, thus allowing to excite new frequencies in the spectral ranges which are otherwise difficult to access.

*Appeared in*

- Phys. Rev. Lett., 118 (2017), pp. 163901/1--163901/5, DOI 10.1103/PhysRevLett.118.163901 .

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# Large deviations for the capacity in dynamic spatial relay networks

*Authors*

- Hirsch, Christian
- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065

*2010 Mathematics Subject Classification*

- 60F10 60K35

*Keywords*

- Large deviations, entropy, capacity, relay

*DOI*

*Abstract*

We derive a large deviation principle for the space-time evolution of users in a relay network that are unable to connect due to capacity constraints. The users are distributed according to a Poisson point process with increasing intensity in a bounded domain, whereas the relays are positioned deterministically with given limiting density. The preceding work on capacity for relay networks by the authors describes the highly simplified setting where users can only enter but not leave the system. In the present manuscript we study the more realistic situation where users leave the system after a random transmission time. For this we extend the point process techniques developed in the preceding work thereby showing that they are not limited to settings with strong monotonicity properties.

*Appeared in*

- Markov Process. Related Fields, 25 (2019), pp. 33--73.

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# Three examples concerning the interaction of dry friction and oscillations

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 34C55 47J20 49J40 74N30

*Keywords*

- Rate-independent friction, averaging of friction, play operator with time-dependent thresholds, locomotion, rate-and-state friction, Hopf bifurcation

*DOI*

*Abstract*

We discuss recent work concerning the interaction of dry friction, which is a rate independent effect, and temporal oscillations. First, we consider the temporal averaging of highly oscillatory friction coefficients. Here the effective dry friction is obtained as an infimal convolution. Second, we show that simple models with state-dependent friction may induce a Hopf bifurcation, where constant shear rates give rise to periodic behavior where sticking phases alternate with sliding motion. The essential feature here is the dependence of the friction coefficient on the internal state, which has an internal relaxation time. Finally, we present a simple model for rocking toy animal where walking is made possible by a periodic motion of the body that unloads the legs to be moved.

*Appeared in*

- Proceedings of the INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., vol. 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 159--177, DOI 10.1007/978-3-319-75940-1_8 .

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# Efficient coupling of inhomogeneous current spreading and dynamic electro-optical models for broad-area edge-emitting semiconductor devices

*Authors*

- Radziunas, Mindaugas
- Zeghuzi, Anissa
- Fuhrmann, Jürgen

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

ORCID: 0000-0001-6235-9412 - Wünsche, Hans-Jürgen
- Wenzel, Hans
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 65Z05 78A60 65Y20 78M25 78M12 65N80

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 02.30.Jr 02.60.Lj 02.60.Gf

*Keywords*

- Broad area lasers, modeling, traveling wave, inhomogeneous current spreading, Laplace problem, separation of variables, finite volumes, effective implementation

*DOI*

*Abstract*

We extend a 2 (space) + 1 (time)-dimensional traveling wave model for broad-area edge-emitting semiconductor lasers by a model for inhomogeneous current spreading from the contact to the active zone of the laser. To speedup the performance of the device simulations, we suggest and discuss several approximations of the inhomogeneous current density in the active zone.

*Appeared in*

- Opt. Quantum Electron., 49 (2017), pp. 332/1--332/8, DOI 10.1007/s11082-017-1168-3 .

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# The impact of microcavity wire width on polariton soliton existence and multistability

*Authors*

- Slavcheva, Gabriela
- Koleva, Mirella V.
- Pimenov, Alexander

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a 42.65.Pc 42.65.Tg

*Keywords*

- polaritons, solitons, microcavities, multi-stability, tilted waveguides

*DOI*

*Abstract*

We have developed a model of the nonlinear polariton dynamics in realistic 3D non-planar microcavity wires in the driven-dissipative regime [15]. We find that the typical microcavity optical bistability evolves into multi-stability upon variation of the model parameters. The origin of the multi-stability is discussed in detail. We apply linear perturbation analysis to modulational instabilities, and identify conditions for localisation of composite multi-mode polariton solitons in the triggered parametric oscillator regime. Further, we demonstrate stable polariton soliton propagation in tilted and tapered waveguides, and determine maximum tilt angles for which solitons are still found. Additionally, we study soliton amplitude and velocity dependence on the wire width, with a view towards device applications.

*Appeared in*

- J. Opt., 19 (2017), pp. 065404/1--065404/15.

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# Hybrid finite-volume/finite-element schemes for $p(x)$-Laplace thermistor models

*Authors*

- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Glitzky, Annegret
- Liero, Matthias

ORCID: 0000-0002-0963-2915

*2010 Mathematics Subject Classification*

- 65M08 35J92 35G60 35Q79 80M12 80A20

*Keywords*

- Finite volume scheme, p(x)-Laplace thermistor model, path following

*DOI*

*Abstract*

We introduce an empirical PDE model for the electrothermal description of organic semiconductor devices by means of current and heat flow. The current flow equation is of p(x)-Laplace type, where the piecewise constant exponent p(x) takes the non-Ohmic behavior of the organic layers into account. Moreover, the electrical conductivity contains an Arrhenius-type temperature law. We present a hybrid finite-volume/finite-element discretization scheme for the coupled system, discuss a favorite discretization of the p(x)-Laplacian at hetero interfaces, and explain how path following methods are applied to simulate S-shaped current-voltage relations resulting from the interplay of self-heating and heat flow.

*Appeared in*

- Finite Volumes for Complex Applications VIII -- Hyperbolic, Elliptic and Parabolic Problems -- FVCA 8, Lille, France, June 2017, C. Cancès, P. Omnes, eds., Springer International Publishing, Cham et al., 2017, pp. 397--405.

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# Routeing properties in a Gibbsian model for highly dense multihop networks

*Authors*

- König, Wolfgang

ORCID: 0000-0002-4212-0065 - Tóbiás, András

*2010 Mathematics Subject Classification*

- 60G55 60K30 65K10 82B21 90B15 90B18 91A06

*Keywords*

- Multihop ad-hoc network, signal-to-interference ratio, Gibbs distribution, message routeing, high-density limit, point processes, variational analysis, expected number of hops, expected length of a hop, deviation from the straight line, selfish optimization

*DOI*

*Abstract*

We investigate a probabilistic model for routeing in a multihop ad-hoc communication network, where each user sends a message to the base station. Messages travel in hops via the other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution that favours trajectories with low interference, measured in terms of sum of the signal-to-interference ratios for all the hops, and collections of trajectories with little total congestion, measured in terms of the number of pairs of hops arriving at each relay. This model was introduced in our earlier paper [KT17], where we expressed, in the high-density limit, the distribution of the optimal trajectories as the minimizer of a characteristic variational formula. In the present work, in the special case in which congestion is not penalized, we derive qualitative properties of this minimizer. We encounter and quantify emerging typical pictures in analytic terms in three extreme regimes. We analyze the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line in two regimes, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory turns out to quickly approach a straight line, in regime (1) with equally-sized hops. Surprisingly, in regime (1), the typical length of a hop diverges logarithmically as the distance of the transmitter to the base station diverges. We further analyze the local and global repulsive effect of (3) a densely populated area on the trajectories. Our findings are illustrated by numerical examples. We also discuss a game-theoretic relation of our Gibbsian model with a joint optimization of message trajectories opposite to a selfish optimization, in case congestion is also penalized

*Appeared in*

- IEEE Trans. Inform. Theory, 65 (2019), pp. 6875--6897, DOI 10.1109/TIT.2019.2924187 .

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# A Gibbsian model for message routing in highly dense multi-hop networks

*Authors*

- König, Wolfgang

ORCID: 0000-0002-4212-0065 - Tóbiás, András

*2010 Mathematics Subject Classification*

- 60F10 60G55 60K30 82B21 90B15

*Keywords*

- Gibbs distribution of trajectories, high-density limit, large deviations, empirical measure, variational formula, multihop ad-hoc network, signal-to-interference ratio, message trajectories, congestion

*DOI*

*Abstract*

We investigate a probabilistic model for routing in relay-augmented multihop ad-hoc communication networks, where each user sends one message to the base station. Given the (random) user locations, we weigh the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signal-to-interference ratio) and trajectory families with little congestion (measured by how many pairs of hops use the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of a selfish optimization. We describe the joint routing strategy in terms of the empirical measure of all message trajectories. In the limit of high spatial density of users, we derive the limiting free energy and analyze the optimal strategy, given as the minimizer(s) of a characteristic variational formula. Interestingly, expressing the congestion term requires introducing an additional empirical measure.

*Appeared in*

- ALEA Lat. Am. J. Probab. Math. Stat., 16 (2019), pp. 211--258, DOI DOI: 10.30757/ALEA.v16-08 .

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# Continuum percolation for Cox point processes

*Authors*

- Hirsch, Christian
- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065 - Cali, Elie

*2010 Mathematics Subject Classification*

- 60F10 60K35

*Keywords*

- Cox processes, percolation, stabilization, large deviations

*DOI*

*Abstract*

We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of non-trivial sub- and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in large-radius, high-density and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives.

*Appeared in*

- Stochastic Process. Appl., 366 (2019), pp. 139--172 (published online on 20.11.2018), DOI 10.1016/j.spa.2018.11.002 .

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# Mode-locking in systems of globally coupled phase oscillators

*Authors*

- Eydam, Sebastian
- Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 34C15 37N20

*2008 Physics and Astronomy Classification Scheme*

- 89.75.-k 42.60.Fc 05.45Xt 05.45.-a

*Keywords*

- Phase oscillators, mode-locking

*DOI*

*Abstract*

We investigate the dynamics of a Kuramoto-type system of globally coupled phase oscillators with equidistant natural frequencies and a coupling strength below the synchronization threshold. It turns out that in such cases one can observe a stable regime of sharp pulses in the mean field amplitude with a pulsation frequency given by spacing of the natural frequencies. This resembles a process known as mode-locking in laser and relies on the emergence of a phase relation induced by the nonlinear coupling. We discuss the role of the first and second harmonic in the phase-interaction function for the stability of the pulsations and present various bifurcating dynamical regimes such as periodically and chaotically modulated mode-locking, transitions to phase turbulence and intermittency. Moreover, we study the role of the system size and show that in certain cases one can observe type-II supertransients, where the system reaches the globally stable mode-locking solution only after an exponentially long transient of phase turbulence.

*Appeared in*

- Phys. Rev. E, 96 (2017), pp. 052205/1--052205/8, DOI 10.1103/PhysRevE.96.052205 .

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# Embedding the dynamics of a single delay system into a feed-forward ring

*Authors*

- Klinshov, Vladimir
- Shchapin, Dmitry
- Yanchuk, Serhiy
- Wolfrum, Matthias
- D'Huys, Otti
- Nekorkin, Vladimir

*2010 Mathematics Subject Classification*

- 34K20 34C28

*Keywords*

- Delay systems, coupled oscillators, complex dynamics

*DOI*

*Abstract*

We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where stability of periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example we demonstrate how the complex bifurcation scenario of simultaneously emerging multi-jittering solutions can be transferred from a single oscillator with delayed pulse feedback to multi-jittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.

*Appeared in*

- Phys. Rev. E, 96 (2017) pp. 042217/1--042217/9.

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# Delayed feedback control of self-mobile cavity solitons in a wide-aperture laser with a saturable absorber

*Authors*

- Schemmelmann, Tobias
- Tabbert, Felix
- Pimenov, Alexander
- Vladimirov, Andrei G.
- Gurevich, Svetlana V.

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a, 42.55.Px, 42.60.Fc, 42.65.Pc, 42.65.Tg

*Keywords*

- wide-aperture laser, saturable absorber, cavity solitons, mode-locking, delayed feedback

*DOI*

*Abstract*

We investigate the spatiotemporal dynamics of cavity solitons in a broad area vertical-cavity surface-emitting laser with saturable absorption subjected to time-delayed optical feedback. Using a combination of analytical, numerical and path continuation methods we analyze the bifurcation structure of stationary and moving cavity solitons and identify two different types of traveling localized solutions, corresponding to slow and fast motion. We show that the delay impacts both stationary and moving solutions either causing drifting and wiggling dynamics of initially stationary cavity solitons or leading to stabilization of intrinsically moving solutions. Finally, we demonstrate that the fast cavity solitons can be associated with a lateral mode-locking regime in a broad-area laser with a single longitudinal mode.

*Appeared in*

- Chaos, 27 (2017) pp. 114304/1--114304/9, DOI 10.20347/WIAS.PREPRINT.2400 .

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# New approach to study the van der Pol equation for large damping

*Authors*

- Schneider, Klaus

*2010 Mathematics Subject Classification*

- 34C05 34C26 34D20

*Keywords*

- relaxation oscillation, Dulac-Cherkas function, rotated vector field

*DOI*

*Abstract*

We present a new approach to establish the existence of a unique limit cycle for the van der Pol equation in case of large damping which is hyperbolic and stable. The proof is based on a linear time scaling (instead of the nonlinear Liénard transformation), on a Dulac--Cherkas function and the property of rotating vector fields.

*Appeared in*

- Electron. J. Qual. Theory Differ. Equ., 8 (2018), pp. 1--10, DOI 10.14232/ejqtde/2018.1.8 .

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# A joint model of probabilistic/robust constraints for gas transport management in stationary networks

*Authors*

- González Grandón, Tatiana
- Heitsch, Holger
- Henrion, René

*2010 Mathematics Subject Classification*

- 90B15 90C15

*Keywords*

- chance constraint, robust constraint, uncertainty set, spheric-radial decomposition

*DOI*

*Abstract*

We present a novel mathematical algorithm to assist gas network operators in managing uncertainty, while increasing reliability of transmission and supply. As a result, we solve an optimization problem with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise in the presence of uncertain parameters having partially stochastic and partially non-stochastic character. The application that drives this new approach is a stationary network with uncertain demand (which are stochastic due to the possibility of fitting statistical distributions based on historical measurements) and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of attainable measurements). We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is allowed (shaping a 'maximal' uncertainty set) around nominal roughness coefficients, such that random demands in a stationary gas network can be satisfied at given high probability level for no matter which realization of true roughness coefficients within the uncertainty set. One ends up with a constraint, which is probabilistic with respect to the load of gas and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions. The numerical solution of a corresponding optimization problem is illustrated. The results might assist the network operator with the implementation of cost-intensive roughness measurements.

*Appeared in*

- Comput. Manag. Sci., 14 (2017), pp. 443--460, DOI 10.1007/s10287-017-0284-7 .

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# Anisotropic solid-liquid interface kinetics in silicon: An atomistically informed phase-field model

*Authors*

- Bergmann, Sibylle
- Barragan-Yani, Daniel A.
- Flegel, Elke
- Albe, Karsten
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 74N20 65M70 74E10, 74G65

*2008 Physics and Astronomy Classification Scheme*

- 64.70.D, 02.70.Ns, 34.20.-b

*Keywords*

- phase-field model, molecular dynamics simulation, interface kinetics, silicon recrystallization

*DOI*

*Abstract*

We present an atomistically informed parametrization of a phase-field model for describing the anisotropic mobility of liquid-solid interfaces in silicon. The model is derived from a consistent set of atomistic data and thus allows to directly link molecular dynamics and phase field simulations. Expressions for the free energy density, the interfacial energy and the temperature and orientation dependent interface mobility are systematically fitted to data from molecular dynamics simulations based on the Stillinger-Weber interatomic potential. The temperature-dependent interface velocity follows a Vogel-Fulcher type behavior and allows to properly account for the dynamics in the undercooled melt.

*Appeared in*

- Model. Simul. Mater. Sci. Eng., 25 (2017), pp. 065015/1--065015/20, DOI 10.1088/1361-651X/aa7862 .

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# Local control of globally competing patterns in coupled Swift--Hohenberg equations

*Authors*

- Becker, Maximilian
- Frenzel, Thomas
- Niedermeyer, Thomas
- Reichelt, Sina
- Mielke, Alexander

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

*Keywords*

- Pattern formation, Swift--Hohenberg equation, weakly nonlinear analysis, amplitude equations, control

*DOI*

*Abstract*

We present analytical and numerical investigations of two anti-symmetrically coupled 1D Swift--Hohenberg equations (SHEs) with cubic nonlinearities. The SHE provides a generic formulation for pattern formation at a characteristic length scale. A linear stability analysis of the homogeneous state reveals a wave instability in addition to the usual Turing instability of uncoupled SHEs. We performed weakly nonlinear analysis in the vicinity of the codimension-two point of the Turing-wave instability, resulting in a set of coupled amplitude equations for the Turing pattern as well as left and right traveling waves. In particular, these complex Ginzburg--Landau-type equations predict two major things: there exists a parameter regime where multiple different patterns are stable with respect to each other; and that the amplitudes of different patterns interact by local mutual suppression. In consequence, different patterns can coexist in distinct spatial regions, separated by localized interfaces. We identified specific mechanisms for controlling the position of these interfaces, which distinguish what kinds of patterns the interface connects and thus allow for global pattern selection. Extensive simulations of the original SHEs confirm our results.

*Appeared in*

- Chaos, 28 (2018), pp. 043121/1--043121/11, DOI 10.1063/1.5018139 .

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# Structure formation in thin liquid-liquid films

*Authors*

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

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

*2008 Physics and Astronomy Classification Scheme*

- 68.05.Cf, 02.60.Cb

*Keywords*

- Flowing Matter: Liquids and Complex Fluids, Liquid-liquid interface structure, measurements and simulations

*DOI*

*Abstract*

We revisit the problem of a liquid polymer that dewets from another liquid polymer substrate with the focus on the direct comparison of results from mathematical modeling, rigorous analysis, numerical simulation and experimental investigations of rupture, dewetting dynamics and equilibrium patterns of a thin liquid-liquid system. The experimental system uses as a model system a thin polystyrene (PS) / polymethylmethacrylate (PMMA) bilayer of a few hundred nm. The polymer systems allow for in situ observation of the dewetting process by atomic force microscopy (AFM) and for a precise ex situ imaging of the liquid--liquid interface. In the present study, the molecular chain length of the used polymers is chosen such that the polymers can be considered as Newtonian liquids. However, by increasing the chain length, the rheological properties of the polymers can be also tuned to a viscoelastic flow behavior. The experimental results are compared with the predictions based on the thin film models. The system parameters like contact angle and surface tensions are determined from the experiments and used for a quantitative comparison. We obtain excellent agreement for transient drop shapes on their way towards equilibrium, as well as dewetting rim profiles and dewetting dynamics.

*Appeared in*

- Transport Processes at Fluidic Interfaces, D. Bothe, A. Reusken, eds., Advances in Mathematical Fluid Mechanics, Birkhäuser, Cham, 2017, pp. 531--574, DOI 10.1007/978-3-319-56602-3 .

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# From nonlinear to linear elasticity in a coupled rate-dependent/independent system for brittle delamination

*Authors*

- Rossi, Riccarda
- Thomas, Marita

*2010 Mathematics Subject Classification*

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

*Keywords*

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

*DOI*

*Abstract*

We revisit the weak, energetic-type existence results obtained in [Rossi/Thomas-ESAIM-COCV-21(1):1-59,2015] for a system for rate-independent, brittle delamination between two visco-elastic, physically nonlinear bulk materials and explain how to rigorously extend such results to the case of visco-elastic, linearly elastic bulk materials. Our approximation result is essentially based on deducing the Mosco-convergence of the functionals involved in the energetic formulation of the system. We apply this approximation result in two different situations: Firstly, to pass from a nonlinearly elastic to a linearly elastic, brittle model on the time-continuous level, and secondly, to pass from a time-discrete to a time-continuous model using an adhesive contact approximation of the brittle model, in combination with a vanishing, super-quadratic regularization of the bulk energy. The latter approach is beneficial if the model also accounts for the evolution of temperature.

*Appeared in*

- Proceedings of the INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., vol. 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 127--157, DOI 10.1007/978-3-319-75940-1_7 .

*Download Documents*

# Local well-posedness for thermodynamically motivated quasilinear parabolic systems in divergence form

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35K40 35K51 35K57, 35K59, 35D35, 35B65

*Keywords*

- doubly nonlinear parabolic system, advection--diffusion--reaction equations, a-priori estimates, local--in--time existence and uniqueness, classical solutions

*DOI*

*Abstract*

We show that fully quasilinear parabolic systems are locally well posed in the Hilbert space scala if the coefficients of the differential operator are smooth enough and the spatial domain is sufficiently regular. In the context of diffusion systems driven by entropy, the uniform parabolicity follows from the second law of thermodynamics.

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# Stability of spiral chimera states on a torus

*Authors*

- Omel'chenko, Oleh E.

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

*2010 Mathematics Subject Classification*

- 34C15 37G35 34D06 35B36

*Keywords*

- Coupled oscillators, coherence-incoherence patterns, chimera states, Ott--Antonsen equation, bifurcation analysis

*DOI*

*Abstract*

We study destabilization mechanisms of spiral coherence-incoherence patterns known as spiral chimera states that form on a two-dimensional lattice of nonlocally coupled phase oscillators. For this purpose we employ the linearization of the Ott--Antonsen equation that is valid in the continuum limit and perform a detailed two-parameter stability analysis of a $D_4$-symmetric chimera state, i.e., a four-core spiral state. We identify fold, Hopf and parity-breaking bifurcations as the main mechanisms whereby spiral chimeras can lose stability. Beyond these bifurcations we find new spatio-temporal patterns, in particular, quasiperiodic chimeras, $D_2$-symmetric spiral chimeras as well as drifting states.

*Appeared in*

- SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 97--127, DOI 10.1137/17M1141151 .

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# Tunable semiconductor ring laser with filtered optical feedback: Traveling wave description and experimental validation

*Authors*

- Radziunas, Mindaugas
- Khoder, Mulham
- Tronciu, Vasile
- Danckaert, Jan
- Verschaffelt, Guy

*2010 Mathematics Subject Classification*

- 78A60 35Q60 78-05, 78-04

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px, 42.60.Da, 42.60.Fc, 42.65.Pc, 42.65.Sf, 02.30.Jr, 02.60.Cb

*Keywords*

- Semiconductor ring lasers, optical feedback, wavelength filtering, tunable lasers, traveling wave model, longitudinal modes, mode switching

*DOI*

*Abstract*

We study experimentally and theoretically a semiconductor ring laser with four filtering channels providing filtered delayed optical feedback. To describe and analyze the wavelength selection and tuning in this device, we exploit the traveling-wave model determining the evolution of optical fields and carrier density along the ring cavity and filtering branches. The numerical results agree with the experimental observations: we can reproduce the wavelength tuning, the multiple wavelength emission, and the wavelength switching speed measured in these devices. The traveling-wave model allows us to study in detail the effect of the different laser parameters and can be useful for designing the future devices.

*Appeared in*

- J. Opt. Soc. Amer. B Opt. Phys., 35, pp. 380--390, DOI 10.1364.JOSAB.35.000380 .

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# Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps

*Authors*

- Flegel, Franziska
- Heida, Martin
- Slowik, Martin

*2010 Mathematics Subject Classification*

- 60H25 60K37 35B27 35R60 47B80 47A75

*Keywords*

- Random conductance model, homogenization, Dirichlet eigenvalues, local times, percolation

*DOI*

*Abstract*

We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for thenormalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence

*Appeared in*

- Ann. Inst. H. Poincare Probab. Statist., 55 (2019), pp. 1226--1257, DOI 10.1214/18-AIHP917 .

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# Localized instabilities and spinodal decomposition in driven systems in the presence of elasticity

*Authors*

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

*2008 Physics and Astronomy Classification Scheme*

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

*Keywords*

- Stability Analysis, Interface Dynamics, Asymptotic Analysis, Numerical Methods

*DOI*

*Abstract*

We study numerically and analytically the instabilities associated with phase separation in a solid layer on which an external material flux is imposed. The first instability is localized within a boundary layer at the exposed free surface by a process akin to spinodal decomposition. In the limiting static case, when there is no material flux, the coherent spinodal decomposition is recovered. In the present problem stability analysis of the time-dependent and non-uniform base states as well as numerical simulations of the full governing equations are used to establish the dependence of the wavelength and onset of the instability on parameter settings and its transient nature as the patterns eventually coarsen into a flat moving front. The second instability is related to the Mullins-Sekerka instability in the presence of elasticity and arises at the moving front between the two phases when the flux is reversed. Stability analyses of the full model and the corresponding sharp-interface model are carried out and compared. Our results demonstrate how interface and bulk instabilities can be analysed within the same framework which allows to identify and distinguish each of them clearly. The relevance for a detailed understanding of both instabilities and their interconnections in a realistic setting are demonstrated for a system of equations modelling the lithiation/delithiation processes within the context of Lithium ion batteries.

*Appeared in*

- Phys. Rev. E, 97 (2018), pp. 012801/1--012801/12, DOI 10.1103/PhysRevE.97.012801 .

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# Exit time risk-sensitive stochastic control problems related to systems of cooperative agents

*Authors*

- Dupuis, Paul
- Laschos, Vaios
- Ramanan, Kavita

*2010 Mathematics Subject Classification*

- 93E20

*Keywords*

- Risk-sensitive stochastic control, mean-field interaction, min-max theorems

*DOI*

*Abstract*

We study sequences, parametrized by the number of agents, of exit time stochastic control problems with risk-sensitive costs structures generate by unbounded costs. We identify a fully characterizing assumption, under which, each of them corresponds to a risk-neutral stochastic control problem with additive cost, and also to a risk-neutral stochastic control problem on the simplex, where the specific information about the state of each agent can be discarded. We finally prove that, under some additional assumptions, the sequence of value functions converges to the value function of a deterministic control problem.

*Appeared in*

- Math. Control Signals Systems, 31 (2019), pp. 279-393, changed title: Exit time risk-sensitive control for systems of cooperative agents.

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# Random walk on random walks: Higher dimensions

*Authors*

- Blondel, Oriane
- Hilário, Marcelo R.
- Soares dos Santos, Renato
- Sidoravicius, Vladas
- Teixeira, Augusto

*2010 Mathematics Subject Classification*

- 60F15 60K35 82B41 82C22 82C44

*Keywords*

- Random walk, dynamic random environment, law of large numbers, central limit theorem, large deviations, renormalization, regeneration

*DOI*

*Abstract*

We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].

*Appeared in*

- Electron. J. Probab., 24 (2019), pp. 80/1-80/33.

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# A Hamilton--Jacobi point of view on mean-field Gibbs-non-Gibbs transitions

*Authors*

- Kraaij, Richard
- Redig, Frank
- van Zuijlen, Willem

*2010 Mathematics Subject Classification*

- 49L99 60F10 82C22 82C27

*Keywords*

- Hamiltonian dynamics, Hamilton-Jacobi equation, mean-field models, large deviation principle, Gibbs versus non-Gibbs, dynamical transition, global minimisers of rate functions

*DOI*

*Abstract*

We study the loss, recovery, and preservation of differentiability of time-dependent large deviation rate functions. This study is motivated by mean-field Gibbs-non-Gibbs transitions. The gradient of the rate-function evolves according to a Hamiltonian flow. This Hamiltonian flow is used to analyze the regularity of the time dependent rate function, both for Glauber dynamics for the Curie-Weiss model and Brownian dynamics in a potential. We hereby create a unifying framework for the treatment of mean-field Gibbs-non-Gibbs transitions, based on Hamiltonian dynamics and viscosity solutions of Hamilton-Jacobi equations.

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# On convergences of the squareroot approximation scheme to the Fokker--Planck operator

*Authors*

- Heida, Martin

*2010 Mathematics Subject Classification*

- 35B27 35Q84 49M25 60H25 60K37 80M40 35R60 47B80

*Keywords*

- Finite volumes, Voronoi, discretization, Fokker-Planck, Smoluchowski, Langevin dynamics, square root approximation, stochastic homogenization, G-convergence

*DOI*

*Abstract*

We study the qualitative convergence properties of a finite volume scheme that recently was proposed by Lie, Fackeldey and Weber [SIAM Journal on Matrix Analysis and Applications 2013 (34/2)] in the context of conformation dynamics. The scheme was derived from physical principles and is called the squareroot approximation (SQRA) scheme. We show that solutions to the SQRA equation converge to solutions of the Fokker-Planck equation using a discrete notion of G-convergence. Hence the squareroot approximation turns out to be a usefull approximation scheme to the Fokker-Planck equation in high dimensional spaces. As an example, in the special case of stationary Voronoi tessellations we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property. In particular, the class of tessellations for which the G-convergence result holds is not trivial.

*Appeared in*

- Math. Models Methods Appl. Sci., 28 (2018), pp. 2599--2635, DOI 10.1142/S0218202518500562 .

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# Synchronization of weakly stable oscillators and semiconductor laser arrays

*Authors*

- Kozyreff, Gregory
- Mandel, Paul
- Vladimirov, Andrei

*2010 Mathematics Subject Classification*

- 78A60 34C15 70K50

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Sf 42.55.Px 05.45.Xt

*Keywords*

- semiconductor laser, synchronization, coupled oscillators, bifurcations

*DOI*

*Abstract*

We study the synchronization properties of an array of non identical globally coupled limit cycle oscillators. Above a critical coupling strength, some oscillators undergo a selfpulsing instability. We study analytically the synchronization conditions below and above this instability threshold, thus removing the usual restriction of limit cycle stability. Selfpulsing decreases the order parameter and synchronization degradation can be reduced by delaying the coupling among the oscillators. Semiconductor lasers coupled by an external mirror are used as a convenient realization of that model.

*Appeared in*

- Europhys. Lett., 61 (2003) pp. 613--619.

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# Cancellation of Raman self-frequency shift for compression of optical pulses

*Authors*

- Pickartz, Sabrina
- Brée, Carsten
- Bandelow, Uwe

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

ORCID: 0000-0002-8132-882X

*2010 Mathematics Subject Classification*

- 78A60 78M22 78-06

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k 42.81.Dp 42.65.Dr 05.45.Yv

*Keywords*

- Ultrashort pulses, pulse compression, soliton self-frequency shift, optical event horizons

*DOI*

*Abstract*

We study to which extent a fiber soliton can be manipulated by a specially chosen continuous pump wave. A group velocity matched pump scatters at the soliton, which is compressed due to the energy/momentum transfer. As the pump scattering is very sensitive to the velocity matching condition, soliton compression is quickly destroyed by the soliton self-frequency shift (SSFS). This is especially true for ultrashort pulses: SSFS inevitably impairs the degree of compression. We demonstrate numerically that soliton enhancement can be restored to some extent and the compressed soliton can be stabilized, provided that SSFS is canceled by a second pump wave. Still the available compression degree is considerably smaller than that in the Raman-free nonlinear fibers.

*Appeared in*

- Opt. Quantum Electron., 49 (2017), published online on 21.09.2017, DOI 10.1007/s11082-017-1164-7 .

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# Type II singular perturbation approximation for linear systems with Lévy noise

*Authors*

- Redmann, Martin

ORCID: 0000-0001-5182-9773

*2010 Mathematics Subject Classification*

- 93A15 15A24

*Keywords*

- model order reduction, singular perturbation approximation, Gramians, stochastic systems, Lévy process

*DOI*

*Abstract*

When solving linear stochastic partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is singular perturbation approximation (SPA), a method which has been extensively studied for deterministic systems. As so-called type I SPA it has already been extended to stochastic equations. We provide an alternative generalisation of the deterministic setting to linear systems with Lévy noise which is called type II SPA. It turns out that the ROM from applying type II SPA has better properties than the one of using type I SPA. In this paper, we provide new energy interpretations for stochastic reachability Gramians, show the preservation of mean square stability in the ROM by type II SPA and prove two different error bounds for type II SPA when applied to Lévy driven systems

*Appeared in*

- SIAM J. Control Optim., 56 (2018), pp. 2120--2158, DOI 10.1137/17M113160X .

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# Type II balanced truncation for deterministic bilinear control systems

*Authors*

- Redmann, Martin

ORCID: 0000-0001-5182-9773

*2010 Mathematics Subject Classification*

- 93A15 93B05 93B07

*Keywords*

- Model order reduction, balanced truncation, bilinear systems, Gramians, error bound

*DOI*

*Abstract*

When solving partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT), a method which has been extensively studied for deterministic linear systems. As so-called type I BT it has already been extended to bilinear equations, an important subclass of nonlinear systems. We provide an alternative generalisation of the linear setting to bilinear systems which is called type II BT. The Gramians that we propose in this context contain information about the control. It turns out that the new approach delivers energy bounds which are not just valid in a small neighbourhood of zero. Furthermore, we provide an ℋ∞-error bound which so far is not known when applying type I BT to bilinear systems.

*Appeared in*

- SIAM J. Control Optim., 56 (2018), pp. 2593--2612, DOI 10.1137/17M1147962 .

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# An $H_2$-type error bound for time-limited balanced truncation

*Authors*

- Redmann, Martin

ORCID: 0000-0001-5182-9773 - Kürschner, Patrick

*2010 Mathematics Subject Classification*

- 93A15 93B99 93C05

*Keywords*

- Model reduction, linear systems, time-limited balanced truncation, time-limited Gramians, error bound

*DOI*

*Abstract*

When solving partial differential equations numerically, usually a high order spatial discretization is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT). However, if one aims at finding a good ROM on a certain finite time interval only, time-limited BT (TLBT) can be a more accurate alternative. So far, no error bound on TLBT has been proved. In this paper, we close this gap in the theory by providing an H2 error bound for TLBT with two different representations. The performance of the error bound is then shown in several numerical experiments

*Appeared in*

- Systems Control Lett., 121 (2018), pp. 1--6, DOI 10.1016/j.sysconle.2018.08.004 .

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