# Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations

*Authors*

- Cozzi, Matteo
- Farina, Alberto
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35A15 35J20 35B50

*Keywords*

- Crystal growth, pointwise estimates, rigidity and symmetry results

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# Modeling and simulations of beam stabilization in edge-emitting broad area semiconductor devices

*Authors*

- Radziunas, Mindaugas
- Ciegis, Raimondas

*2010 Mathematics Subject Classification*

- 65M06 65M20 65M99 35Q60 65M12

*Keywords*

- broad area device, traveling wave model numerical scheme, simulations, beam improvement

*Abstract*

A 2+1 dimensional PDE traveling wave model describing spatial-lateral dynamics of edge-emitting broad area semiconductor devices is considered. A numerical scheme based on a split-step Fourier method is presented and implemented on a parallel compute cluster. Simulations of the model equations are used for optimizing of existing devices with respect to the emitted beam quality, as well as for creating and testing of novel device design concepts.

*Appeared in*

- PPAM 2013, Part II, LNCS 8385, R. Wyrzykowski, et al., eds., Springer 2014, 2013, pp. 332--342

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# Influence of slip on the Rayleigh--Plateau rim instability in dewetting viscous films

*Authors*

- Bäumchen, Oliver
- Marquant, Ludovic
- Blossey, Ralf
- Münch, Andreas
- Wagner, Barbara
- Jacobs, Karin

*2010 Mathematics Subject Classification*

- 76A20 76E17 76-05

*2008 Physics and Astronomy Classification Scheme*

- 83.50.Lh

*Keywords*

- thin-film equations, interfacial slip, dewetting liquid polymer, contact-line instability

*Abstract*

A dewetting viscous film develops a characteristic fluid rim at its receding edge due to mass conservation. In the course of the dewetting process the rim becomes unstable via an instability of Rayleigh-Plateau type. An important difference exists between this classic instability of a liquid column and the rim instability in the thin film as the growth of the rim is continuously fueled by the receding film. We explain how the development and macroscopic morphology of the rim instability are controlled by the slip of the film on the substrate. A single thin-film model captures quantitatively the characteristics of the evolution of the rim observed in our experiments.

*Appeared in*

- Physical Review Letters, 113 (2014) pp. 014501/1--014501/4.

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# A coupling of discrete and continuous optimization to solve kinodynamic motion planning problems

*Authors*

- Landry, Chantal
- Welz, Wolfgang
- Gerdts, Matthias

*2010 Mathematics Subject Classification*

- 49J15 49M25 49N90 70E60 90C30 90C35

*Keywords*

- trajectory planning, optimal control problem, collision avoidance, graph search algorithm, initialization, robotics

*Abstract*

A new approach to find the fastest trajectory of a robot avoiding obstacles, is presented. This optimal trajectory is the solution of an optimal control problem with kinematic and dynamics constraints. The approach involves a direct method based on the time discretization of the control variable. We mainly focus on the computation of a good initial trajectory. Our method combines discrete and continuous optimization concepts. First, a graph search algorithm is used to determine a list of via points. Then, an optimal control problem of small size is defined to find the fastest trajectory that passes through the vicinity of the via points. The resulting solution is the initial trajectory. Our approach is applied to a single body mobile robot. The numerical results show the quality of the initial trajectory and its low computational cost.

*Appeared in*

- Optim. Eng. (2015), 1-24, DOI 10.1007/s11081-015-9291-0 .

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# Two-phase flow model for concentrated suspensions

*Authors*

- Ahnert, Tobias
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 35Q35 76T20

*Keywords*

- concentrated suspension, jamming transition, phase space methods, free boundary problem, numerical methods

*Abstract*

A new two-phase model is derived that make use of a constitutive law combining non-Brownian suspension with granular rheology, that was recently proposed by Boyer et al. [PRL, 107(18),188301 (2011)]. It is shown that for the simple channel flow geometry, the stress model naturally exhibits a Bingham type flow property with an unyielded finite-size zone in the center of the channel. As the volume fraction of the solid phase is increased, the various transitions in the flow fields are discussed using phase space methods for a boundary value problem, that is derived from the full model. The predictions of this analysis is then compared to the direct finite-element numerical solutions of the full model.

*Appeared in*

- European J. Appl. Math., 30 (2019), pp. 585--617 (published online on 04.06.2018), DOI 10.1017/S095679251800030X .

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# Regularity of the solution to a nonstandard system of phase field equations

*Authors*

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

*2010 Mathematics Subject Classification*

- 35K61 35D10 35A02

*Keywords*

- nonstandard phase field system, nonlinear differential equations, initial and boundary value problem, regularity of solutions

*Abstract*

A nonstandard systems of differential equations describing two-species phase segregation is considered. This system naturally arises in the asymptotic analysis recently done by Colli, Gilardi, Krejci, and Sprekels as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, a well-posedness result is proved for the limit system. This paper deals with the above limit problem in a less general but still very significant framework and provides a very simple proof of further regularity for the solution. As a byproduct, a simple uniqueness proof is given as well.

*Appeared in*

- Rend. Cl. Sci. Mat. Nat., 147 (2013) pp. 3--19.

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# Simulation of composite materials by a Network FEM with error control

*Authors*

- Eigel, Martin
- Peterseim, Daniel

*2010 Mathematics Subject Classification*

- 65N15 65N30 74Q20

*Keywords*

- a posteriori, error analysis, finite element method, composite material, multiscale, high contrast, generalised Delaunay, network

*Abstract*

A novel Finite Element Method (FEM) for the computational simulation in particle reinforced composite materials with many inclusions is presented. It is based on a specially designed mesh consisting of triangles and channel-like connections between inclusions which form a network structure. The total number of elements and, hence, the number of degrees of freedom are proportional to the number of inclusions. The error of the method is independent of the possibly tiny distances of neighbouring inclusions.

We present algorithmic details for the generation of the problem adapted mesh and derive an efficient residual a posteriori error estimator which enables to compute reliable upper and lower error bounds. Several numerical examples illustrate the performance of the method and the error estimator. In particular, it is demonstrated that the (common) assumption of a lattice structure of inclusions can easily lead to incorrect predictions about material properties.

*Appeared in*

- Comput. Methods Appl. Math., 15 (2015) pp. 21--37.

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# Relating phase field and sharp interface approaches to structural topology optimization

*Authors*

- Blank, Luise
- Farshbaf Shaker, Mohammad Hassan
- Garcke, Harald
- Styles, Vanessa

*2010 Mathematics Subject Classification*

- 49Q10 74P10 49Q20 74P05 65M60

*Keywords*

- Structural topology optimization, linear elasticity, phase-field method, first order conditions, matched asymptotic expansions, shape calculus, numerical simulations

*Abstract*

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for mean compliance problems and a cost involving the least square error to a target displacement.

*Appeared in*

- ESAIM Control Optim. Calc. Var., 20 (2014) pp. 1025--1058.

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# Propagating topological transformations in thin immiscible bilayer films

*Authors*

- Hennessy, Mathew G.
- Burlakov, Victor M.
- Münch, Andreas
- Wagner, Barbara
- Goriely, Alain

*2010 Mathematics Subject Classification*

- 76M45 76E17 35B40 35C20

*2008 Physics and Astronomy Classification Scheme*

- 64.60.an

*Keywords*

- phase separation and segregation in thin films, metastable phases, liquid-liquid interface structure

*Abstract*

A physical mechanism for the topological transformation of a two-layer system confined by two substrates is proposed. Initially the two horizontal layers, A and B, are on top of each other, but upon a sufficiently large disturbance, they can rearrange themselves through a spontaneously propagating sectioning to create a sequence of vertical alternating domains ABABAB. This generic topological transformation could be used to control the morphology of fabricated nanocomposites by first creating metastable layered structures and then triggering their transformation. The generality is underscored by formulating conditions for this topological transformation in terms of the interface energies between phases and substrates. The theoretical estimate for the width of the domains is confirmed by simulations of a phase-field model and its thin-film/sharp-interface approximation.

*Appeared in*

- Europhysics Letters, 105 (2014) pp. 66001/1--66001/6.

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# Stress-driven local-solution approach to quasistatic brittle delamination

*Authors*

- Roubíček, Tomáš
- Thomas, Marita
- Panagiotopoulos, Christos

*2010 Mathematics Subject Classification*

- 35K86 35R35 47J20 49J45 49J40 49S05 65M38 65Z05 74M15 74R10

*Keywords*

- unilateral adhesive contact, brittle limit, rate-independent processes, semi-implicit time discretisation, finite perimeter, property a, (d-1)-thick set, lower density estimate, Hardy's inequality, computational simulations

*Abstract*

A unilateral contact problem between elastic bodies at small strains glued by a brittle adhesive is addressed in the quasistatic rate-independent setting. The delamination process is modelled as governed by stresses rather than by energies. This results in a specific scaling of an approximating elastic adhesive contact problem, discretised by a semi-implicit scheme and regularized by a BV-type gradient term. An analytical zero-dimensional example motivates the model and a specific local-solution concept. Two-dimensional numerical simulations performed on an engineering benchmark problem of debonding a fiber in an elastic matrix further illustrate the validity of the model, convergence, and algorithmical efficiency even for very rigid adhesives with high elastic moduli.

*Appeared in*

- Nonlinear Anal. Real World Appl., 22 (2015) pp. 645--663.

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# Task assignment, sequencing and path-planning in robotic welding cells

*Authors*

- Landry, Chantal
- Welz, Wolfgang
- Henrion, René
- Hömberg, Dietmar
- Skutella, Martin

*2010 Mathematics Subject Classification*

- 49N90 65D18 90C27 90C35 90C90

*Keywords*

- discrete optimization, vehicle routing problem, optimal control problem, collision detection, motion planning, cooperative robots

*Abstract*

A workcell composed of a workpiece and several welding robots is considered. We are interested in minimizing the makespan in the workcell. Hence, one needs i) to assign tasks between the robots, ii) to do the sequencing of the tasks for each robot and iii) to compute the fastest collision-free paths between the tasks. Up to now, task assignment and path-planning were always handled separately, the former being a typical Vehicle Routing Problem whereas the later is modelled using an optimal control problem. In this paper, we present a complete algorithm which combines discrete optimization techniques with collision detection and optimal control problems efficiently.

*Appeared in*

- Methods and Models in Automation and Robotics (MMAR), 2013 -- 18th International Conference on, Miedzyzdroje, Poland, August 26 - 29, 2013, IEEE, 2013, pp. 252--257.

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# On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698

*2010 Mathematics Subject Classification*

- 65N30 65N12 35Q30

*Keywords*

- mixed finite elements, incompressible Navier-Stokes equations, poor mass conservation, variational crime, stability

*Abstract*

According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier-Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of L2-orthogonality between discretely divergence-free velocities and irrotational vector fields. In order to cure this, a new variational crime using divergence-free velocity reconstructions is proposed. Applying lowest order Raviart-Thomas velocity reconstructions to the nonconforming Crouzeix-Raviart element allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a-priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 268 (2014) pp. 782--800.

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# Adaptive smoothing as inference strategy: More specificity for unequally sized or neighboring regions

*Authors*

- Welvaert, Marijke
- Tabelow, Karsten

ORCID: 0000-0003-1274-9951 - Seurinck, Ruth
- Rosseel, Yves

*2010 Mathematics Subject Classification*

- 62P10 62G10 62G05

*Keywords*

- false positive rate, fmri, Gaussian smoothing, power, structural adaptive segmentation

*Abstract*

Although spatial smoothing of fMRI data can serve multiple purposes, increasing the sensitivity of activation detection is probably its greatest benefit. However, this increased detection power comes with a loss of specificity when non-adaptive smoothing (i.e. the standard in most software packages) is used. Simulation studies and analysis of experimental data was performed using the R packages neuRosim and fmri. In these studies, we systematically investigated the effect of spatial smoothing on the power and number of false positives in two particular cases that are often encountered in fMRI research: (1) Single condition activation detection for regions that differ in size, and (2) multiple condition activation detection for neighbouring regions. Our results demonstrate that adaptive smoothing is superior in both cases because less false positives are introduced by the spatial smoothing process compared to standard Gaussian smoothing or FDR inference of unsmoothed data.

*Appeared in*

- Neuroinformatics, 11 (2013) pp. 435--445.

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# An adaptive SUPG method for evolutionary convection-diffusion equations

*Authors*

- de Frutos, Javier
- Garc'ıa-Archilla, Bosco
- John, Volker

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

*2010 Mathematics Subject Classification*

- 65N30

*Keywords*

- Evolutionary convection-diffusion-reaction equations, streamline upwind Petrov--Galerkin (SUPG) method, residual-based a posteriori error estimators, effectivity index, adaptive grid generation

*Abstract*

An adaptive algorithm for the numerical simulation of time-dependent convection-diffusion-reaction equations will be proposed and studied. The algorithm allows the use of the natural extension of any error estimator for the steady-state problem for controlling local refinement and coarsening. The main idea consists in considering the SUPG solution of the evolutionary problem as the SUPG solution of a particular steady-state convection-diffusion problem with data depending on the computed solution. The application of the error estimator is based on a heuristic argument by considering a certain term to be of higher order. This argument is supported in the one-dimensional case by numerical analysis. In the numerical studies, particularly the residual-based error estimator from [18] will be applied, which has proved to be robust in the SUPG norm. The effectivity of this error estimator will be studied and the numerical results (accuracy of the solution, fineness of the meshes) will be compared with results obtained by utilizing the adaptive algorithm proposed in [6].

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 273 (2014) pp. 219--237.

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# Collision detection between robots moving along specified trajectories

*Authors*

- Feyeux, Nelson
- Landry, Chantal

*2010 Mathematics Subject Classification*

- 51M20 51K99 52B10 68T40

*Keywords*

- collision detection, distance computation, motion planning, robotics

*Abstract*

An algorithm to detect collisions between robots moving along given trajectories is presented. The method is a combination of the adaptive dynamic collision checking developed by Schwarzer et al. and Lin and Canny's algorithm, which computes efficiently the distance between two polyhedra. The resulting algorithm is part of a global model that computes the optimal task assignment, sequencing and kinodynamic motion planning in a robotic work-cell.

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# Transient Cherenkov radiation from an inhomogeneous string excited by an ultrashort laser pulse at superluminal velocity

*Authors*

- Arkhipov, Rostislav M.
- Babushkin, Ihar
- Lebedev, Mikhail K.
- Tolmachev, Yurii A.
- Arkhipov, Mikhail V.

*2010 Mathematics Subject Classification*

- 78A60 78A10 78A40 78A45

*2008 Physics and Astronomy Classification Scheme*

- 41.60.Bq 42.25.Fx 42.25.Hz 42.65.Ky

*Keywords*

- Cherenkov radiation, superluminal pulse propagation, ultrashort laser pulse, difraction of ultrashort laser pulses

*Abstract*

An optical response of one-dimensional string made of dipoles with a periodically varying density excited by a spot of light moving along the string at the superluminal (subluminal) velocity is studied. We consider in details the spectral and temporal dynamics of the Cherenkov radiation, which occurs in such system in the transient regime. We point out the resonance character of radiation and the appearance of a new Doppler-like frequency in the spectrum of the transient Cherenkov radiation. Possible applications of the effect as well as different string topologies are discussed

*Appeared in*

- Phys. Rev. A, 89 (2014) pp. 043811/1--043811/10.

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# Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems

*Authors*

- Mielke, Alexander

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

*2010 Mathematics Subject Classification*

- 49Q20 58E99

*Keywords*

- Doubly nonlinear equations, generalized gradient flows, rate-independent systems, vanishing-viscosity limit, variational Gamma convergence, energy-dissipation balance, arclength parameterized solutions

*Abstract*

Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent ows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for innite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their dierentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on rened lower semicontinuity-compactness arguments, and on new BVestimates that are of independent interest.

*Appeared in*

- J. Eur. Math. Soc. (JEMS), 18 (2016), pp. 2107--2165.

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# On the construction of a class of generalized Kukles systems having at most one limit cycle

*Authors*

- Schneider, Klaus R.
- Grin, Alexander

*2010 Mathematics Subject Classification*

- 34C05 34C23

*Keywords*

- Kukles system, Duclac-Cherkas function, limit cycle

*Abstract*

Consider the class of planar systems $$fracdxdt = y, quad fracdydt = -x + mu sum_j=0^3 h_j(x,mu) y^j$$ depending on the real parameter $mu$. We are concerned with the inverse problem: How to construct the functions $h_j$ such that the system has not more than a given number of limit cycles for $mu$ belonging to some (global) interval. Our approach to treat this problem is based on the construction of suitable Dulac-Cherkas functions $Psi(x,y,mu)$ and exploiting the fact that in a simply connected region the number of limit cycles is not greater than the number of ovals contained in the set defined by $Psi(x,y,mu)=0.$

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# On some random forests with determinantal roots

*Authors*

- Avena, Luca
- Gaudillière, Alexandre

*2010 Mathematics Subject Classification*

- 05C81 60J20 15A15 4215A18 05C85

*Keywords*

- finite networks, spanning forests, determinantal processes, random sets, hitting times, local equilibria, Wilson's algorithm, random partitions, coalescence and fragmentation

*Abstract*

Consider a finite weighted oriented graph. We study a probability measure on the set of spanning rooted oriented forests on the graph. We prove that the set of roots sampled from this measure is a determinantal process, characterized by a possibly non-symmetric kernel with complex eigenvalues. We then derive several results relating this measure to the Markov process associated with the starting graph, to the spectrum of its generator and to hitting times of subsets of the graph. In particular, the mean hitting time of the set of roots turns out to be independent of the starting point, conditioning or not to a given number of roots. Wilson's algorithm provides a way to sample this measure and, in absence of complex eigenvalues of the generator, we explain how to get samples with a number of roots approximating a prescribed integer. We also exploit the properties of this measure to give some probabilistic insight into the proof of an algebraic result due to Micchelli and Willoughby [13]. Further, we present two different related coalescence and fragmentation processes.

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# Direct and inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves

*Authors*

- Hu, Guanghui
- Kirsch, Andreas

*2010 Mathematics Subject Classification*

- 74F10 35R30 78A45 35B27

*Keywords*

- fluid-solid interactions, bi-periodic structures, inverse scattering, factorization method

*Abstract*

Consider a time-harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with homogeneous compressible inviscid fluid with a constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by the Lamé constants. This paper is concerned with direct (or forward) and inverse fluid-solid interaction (FSI) problems with unbounded bi-periodic interfaces between acoustic and elastic waves. We present a variational approach to the forward interaction problem with Lipschitz interfaces. Existence of quasi-periodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. Concerning the inverse problem, we show that the factorization method by Kirsch (1998) is applicable to the FSI problem in periodic structures. A computational criterion and a uniqueness result are justified for precisely characterizing the elastic body by utilizing the scattered acoustic near field measured in the fluid.

*Appeared in*

- Inverse Probl. Imaging, 10 pp. 103--129.

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# Delayed feedback control of self-mobile cavity solitons

*Authors*

- Pimenov, Alexander
- Vladimirov, Andrei G.
- Gurevich, Svetlana V.
- Panajotov, Krassimir
- Huyet, Guillaume
- Tlidi, Mustapha

*2010 Mathematics Subject Classification*

- 35B32 35B36 37K40 78A60

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a 87.23.Cc 42.65.Hw 42.65.Pc

*Keywords*

- cavity solitons, motion with constant velocity, optical feedback, VCSEL, drift bifurcation, asymptotic analysis

*Abstract*

Control of the motion of cavity solitons is one the central problems in nonlinear optical pattern formation. We report on the impact of the phase of the time-delayed optical feedback and carrier lifetime on the self-mobility of localized structures of light in broad area semiconductor cavities. We show both analytically and numerically that the feedback phase strongly affects the drift instability threshold as well as the velocity of cavity soliton motion above this threshold. In addition we demonstrate that non-instantaneous carrier response in the semiconductor medium is responsible for the increase in critical feedback rate corresponding to the drift instability.

*Appeared in*

- Phys. Rev. A, 88 (2013) pp. 053830/1--053830/11.

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# Sharp limit of the viscous Cahn--Hilliard equation and thermodynamic consistency

*Authors*

- Dreyer, Wolfgang
- Guhlke, Clemens

*2010 Mathematics Subject Classification*

- 35K25 35C20 82C26

*Keywords*

- Cahn-Hilliard equation, thermodynamics, phase transitions, asymptotic expansions

*Abstract*

Diffuse and sharp interface models represent two alternatives to describe phase transitions with an interface between two coexisting phases. The two model classes can be independently formulated. Thus there arises the problem whether the sharp limit of the diffuse model fits into the setting of a corresponding sharp interface model. We call a diffuse model admissible if its sharp limit produces interfacial jump conditions that are consistent with the balance equations and the 2nd law of thermodynamics for sharp interfaces. We use special cases of the viscous Cahn-Hilliard equation to show that there are admissible as well as non-admissible diffuse interface models.

*Appeared in*

- Contin. Mech. Thermodyn., 29 (2017) pp. 913--934.

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# From rough path estimates to multilevel Monte Carlo

*Authors*

- Bayer, Christian

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

ORCID: 0000-0003-2571-8388 - Riedel, Sebastian
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 60H35 65C05 65C30

*Keywords*

- Rough paths, Fractional Brownian motion, Euler scheme, Multilevel Monte Carlo

*Abstract*

Discrete approximations to solutions of stochastic differential equations are well-known to converge with strong rate 1/2. Such rates have played a key-role in Giles' multilevel Monte Carlo method [Giles, Oper. Res. 2008] which gives a substantial reduction of the computational effort necessary for the evaluation of diffusion functionals. In the present article similar results are established for large classes of rough differential equations driven by Gaussian processes (including fractional Brownian motion with H>1/4 as special case).

*Appeared in*

- SIAM J. Numer. Anal., 54 (2016) pp. 1449--1483.

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# Feel the heat: Nonlinear electrothermal feedback in organic LEDs

*Authors*

- Fischer, Axel
- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Gärtner, Klaus
- Tietze, Max L.
- Brückner, Jacqueline
- Lüssem, Björn
- Leo, Karl
- Glitzky, Annegret
- Scholz, Reinhard

*2010 Mathematics Subject Classification*

- 82D37 80A20

*2008 Physics and Astronomy Classification Scheme*

- 81.05.Fb 85.60.Jb 72.80.Le

*Keywords*

- Organic light emitting diodes, Joule self-heating, negative differential resistance, device temperature, luminance

*Abstract*

For lighting applications, Organic light-emitting diodes (OLED) need much higher brightness than for displays, leading to self-heating. Due to the temperature-activated transport in organic semiconductors, this can result in brightness inhomogeneities and catastrophic failure. Here, we show that due to the strong electrothermal feedback of OLEDs, the common spatial current and voltage distribution is completely changed, requiring advanced device modeling and operation concepts. Our study clearly demonstrates the effect of negative differential resistance (NDR) in OLEDs induced by self-heating. As a consequence, for increasing voltage, regions with declining voltages are propagating through the device, and even more interestingly, a part of these regions show even decreasing currents, leading to strong local variation in luminance. The expected breakthrough of OLED lighting technology will require an improved price performance ratio, and the realization of modules with very high brightness but untainted appearance is considered to be an essential step into this direction. Thus, a deeper understanding of the control of electrothermal feedback will help to make OLEDs in lighting more competitive.

*Appeared in*

- Appeared in: Adv. Funct. Mater., 24 (2014) pp. 3367--3374

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# Perturbation determinants for singular perturbations

*Authors*

- Malamud, Mark
- Neidhardt, Hagen

*2010 Mathematics Subject Classification*

- 47B25 34B20 34B24

*Keywords*

- symmetric operators, boundary triplets, almost solvable extensions, perturbation determinants, second order elliptic operators

*Abstract*

For proper extensions of a densely defined closed symmetric operator with trace class resolvent difference the perturbation determinant is studied in the framework of boundary triplet approach to extension theory.

*Appeared in*

- Russ. J. Math. Phys., 21 (2014) pp. 55--98.

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# On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions

*Authors*

- Klein, Olaf

ORCID: 0000-0002-4142-3603

*2010 Mathematics Subject Classification*

- 47J40 53C35

*Keywords*

- hysteresis operators, vectorial hysteresis, string representation, discontinuous inputs

*Abstract*

In Brokate-Sprekels 1996, it is shown that hysteresis operators acting on scalar-valued, continuous, piecewise monotone input functions can be represented by functionals acting on alternating strings. In a number of recent papers, this representation result is extended to hysteresis operators dealing with input functions in a general topological vector space. The input functions have to be continuous and piecewise monotaffine, i.e., being piecewise the composition of two functions such that the output of a monotone increasing function is used as input for an affine function. In the current paper, a representation result is formulated for hysteresis operators dealing with input functions being left-continuous and piecewise monotaffine and continuous. The operators are generated by functions acting on an admissible subset of the set of all strings of pairs of elements of the vector space. of the set of all strings of pairs of elements of the vector space.

*Appeared in*

- Discrete Contin. Dyn. Syst., 35 (2015) pp. 2591--2614.

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# A tangential regularization method for backflow stabilization in hemodynamics

*Authors*

- Bertoglio, Cristóbal
- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645

*2010 Mathematics Subject Classification*

- 62P10, 76D05, 76M10, 76Z05

*2008 Physics and Astronomy Classification Scheme*

- 47.63.-b, 47.11.-j

*Keywords*

- Navier-Stokes equations, backflow stabilization, blood flow, finite element method

*Abstract*

In computational simulations of fluid flows, instabilities at the Neumann boundaries may appear during backflow regime. It is widely accepted that this is due to the incoming energy at the boundary, coming from the convection term, which cannot be controlled when the velocity field is unknown. We propose a stabilized formulation based on a local regularization of the fluid velocity along the tangential directions on the Neumann boundaries. The stabilization term is proportional to the amount of backflow, and does not require any further assumption on the velocity profile. The perfomance of the method is assessed on a two- and three-dimensional Womersley flows, as well as considering a hemodynamic physiological regime in a patient-specific aortic geometry.

*Appeared in*

- J. Comput. Phys., 261 (2014) pp. 162--171.

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# The fundamental solution of unidirectional pulse propagation equation

*Authors*

- Babushkin, Ihar

*2010 Mathematics Subject Classification*

- 78A60 35Q61 35Q60

*Keywords*

- fundamental solution, unidirectional pulse propagating equation

*Abstract*

In the article the fundamental solution of a variant of wave equation known as ``unidirectional pulse propagation equation'' (UPPE) and its paraxial approximation is obtained. It is shown that the fundamental solution can be presented as a projection of a fundamental solution of the wave equation to some functional subspace. We discuss the degree of equivalence of UPPE and wave equation in this respect. In particular, we show that UPPE, in contrast to the widespread belief, describes the wave propagation in both directions simultaneously, and remark non-causal character of its solutions.

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# Fast cubature of volume potentials over rectangular domains

*Authors*

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

*2010 Mathematics Subject Classification*

- 65D32 65-05 41A30 41A63

*Keywords*

- Multi-dimensional convolution, advection-diffusion potential, tensor product representation, higher dimensions

*Abstract*

In the present paper we study high-order cubature formulas for the computation of advection-diffusion potentials over boxes. By using the basis functions introduced in the theory of approximate approximations, the cubature of a potential is reduced to the quadrature of one dimensional integrals. For densities with separated approximation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures in very high dimensions. Numerical tests show that these formulas are accurate and provide approximation of order *O*(h^{6}) up to dimension 10^{8}.

*Appeared in*

- Appl. Comput. Harmon. Anal., 36 (2014) pp. 167--182 with modified title: Fast cubature of volume potentials over rectangular domains by approximate approximations

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# Extremal aging for trap models

*Authors*

- Gün, Onur

*2010 Mathematics Subject Classification*

- 82C44 82D30 60G70

*Keywords*

- random walk, random environment, REM, dynamics of spin glasses, aging, extremal processes

*Abstract*

In the seminal work [5], Ben Arous and Cerný give a general characterization of aging for trap models in terms of α-stable subordinators with α ∈ (0,1). Some of the important examples that fall into this universality class are Random Hopping Time (RHT) dynamics of Random Energy Model (REM) and *p*-spin models observed on exponential time scales. In this paper, we explain a different aging mechanism in terms of *extremal processes* that can be seen as the extension of α-stable aging to the case α=0. We apply this mechanism to the RHT dynamics of the REM for a wide range of temperature and time scales. The other examples that exhibit extremal aging include the Sherrington Kirkpatrick (SK) model and *p*-spin models [6, 9], and biased random walk on critical Galton-Watson trees conditioned to survive [11].

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# A curvature estimate for open surfaces subject to a general mean curvature operator and natural contact conditions at their boundary

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35J93, 35B65, 58J99

*Keywords*

- Mean curvature equation, contact-angle boundary conditions, regularity theory, K-K' quasi-conformal Gaussian map

*Abstract*

In the seventies, L. Simon showed that the main curvatures of two-dimensional hypersurfaces obeying a general equation of mean curvature type are a priori bounded by the Hölder norm of the coefficients of the surface differential operator. This was an essentially interior estimate. In this paper, we provide a complement to the theory, proving a global curvature estimate for open surfaces that satisfy natural contact conditions at the intersection with a given boundary.

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# Singular limiting induced from continuum solutions and the problem of dynamic cavitation

*Authors*

- Giesselmann, Jan
- Tzavaras, Athanasios E.

*2010 Mathematics Subject Classification*

- 35L15

*Keywords*

- nonlinear elasticity, singular solutions, cavitation

*Abstract*

In the works of K.A. Pericak-Spector and S. Spector (1988,1995) a class of self-similar solutions are constructed for the equations of radial isotropic elastodynamics that describe cavitating solutions. Cavitating solutions decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (for polyconvex energies) due to point-singularities at the cavity. To resolve this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution), according to which a discontinuous motion is a slic-solution if its averages form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for creating the cavity, which is captured by the notion of slic-solution but neglected by the usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the cavitating solution is in fact larger than that of the homogeneously deformed state. We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture, and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan.

*Appeared in*

- Arch. Ration. Mech. Anal., 212 (2014) pp. 241--281

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# Adaptive smoothing of multi-shell diffusion-weighted magnetic resonance data by msPOAS

*Authors*

- Becker, Saskia
- Tabelow, Karsten

ORCID: 0000-0003-1274-9951 - Mohammadi, Siawoosh
- Weiskopf, Nikolaus
- Polzehl, Jörg

ORCID: 0000-0001-7471-2658

*2010 Mathematics Subject Classification*

- 62P10 62G05

*Keywords*

- Diffusion weighted magnetic resonance imaging, POAS, Structural adaptive smoothing, Parameter choice, Multi-shell

*Abstract*

In this article we present a noise reduction method (msPOAS) for multi-shell diffusion-weighted magnetic resonance data. To our knowledge, this is the first smoothing method which allows simultaneous smoothing of all q-shells. It is applied directly to the diffusion weighted data and consequently allows subsequent analysis by any model. Due to its adaptivity, the procedure avoids blurring of the inherent structures and preserves discontinuities. MsPOAS extends the recently developed position-orientation adaptive smoothing (POAS) procedure to multi-shell experiments. At the same time it considerably simplifies and accelerates the calculations. The behavior of the algorithm msPOAS is evaluated on diffusion-weighted data measured on a single shell and on multiple shells.

*Appeared in*

- NeuroImage, 95 (2014) pp. 90--105.

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# Optimal control of a cooling line for production of hot rolled dual phase steel

*Authors*

- Bleck, Wolfgang
- Hömberg, Dietmar
- Prahl, Ulrich
- Suwanpinij, Piyada
- Togobytska, Nataliya

*2010 Mathematics Subject Classification*

- 35K05 49N90

*Keywords*

- dual phase steel, hot rolling, optimal control, phase transformation

*Abstract*

In this article, the optimal control of a cooling line for production of dual phase steel in a hot rolling process is discussed. In order to achieve a desired dual phase steel microstructure an optimal cooling strategy has to be found. The cooling strategy should be such that a desired final distribution of ferrite in the steel slab is reached most accurately. This problem has been solved by means of mathematical control theory. The results of the optimal control of the cooling line have been verified in hot rolling experiments at the pilot hot rolling mill at the Institute for Metal Forming (IMF), TU Bergakademie Freiberg.

*Appeared in*

- Steel Res. Int., 85 (2014) pp. 1328--1333.

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# Conditioning of linear-quadratic two-stage stochastic optimization problems

*Authors*

- Emich, Konstantin
- Henrion, René
- Römisch, Werner

*2010 Mathematics Subject Classification*

- 90C15 90C31 49K40

*Keywords*

- Stochastic optimization, two-stage linear-quadratic problems, conditioning, coderivative calculus, simple recourse

*Abstract*

In this paper a condition number for linear-quadratic two-stage stochastic optimization problems is introduced as the Lipschitz modulus of the multifunction assigning to a (discrete) probability distribution the solution set of the problem. Being the outer norm of the Mordukhovich coderivative of this multifunction, the condition number can be estimated from above explicitly in terms of the problem data by applying appropriate calculus rules. Here, a chain rule for the extended partial second-order subdifferential recently proved by Mordukhovich and Rockafellar plays a crucial role. The obtained results are illustrated for the example of two-stage stochastic optimization problems with simple recourse.

*Appeared in*

- Math. Program., 148 (2014) pp. 201--221.

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# Controlled topological transitions in thin film phase separation

*Authors*

- Hennessy, Mathew G.
- Burlakov, Victor M.
- Münch, Andreas
- Wagner, Barbara
- Goriely, Alain

*2010 Mathematics Subject Classification*

- 76M45 76E17 35B40 35C20

*2008 Physics and Astronomy Classification Scheme*

- 64.60.an

*Keywords*

- phase separation and segregation in thin films, metastable phases

*Abstract*

In this paper the evolution of a binary mixture in a thin-film geometry with a wall at the top and bottom is considered. Bringing the mixture into its miscibility gap so that no spinodal decomposition occurs in the bulk, a slight energetic bias of the walls towards each one of the constituents ensures the nucleation of thin boundary layers that grow until the constituents have moved into one of the two layers. These layers are separated by an interfacial region, where the composition changes rapidly. Conditions that ensure the separation into two layers with a thin interfacial region are investigated based on a phase-field model and using matched asymptotic expansions a corresponding sharp-interface problem for the location of the interface is established. It is then argued that a thus created two-layer system is not at its energetic minimum but destabilizes into a controlled self-replicating pattern of trapezoidal vertical stripes by minimizing the interfacial energy between the phases while conserving their area. A quantitative analysis of this mechanism is carried out via a new thin-film model for the free interfaces, which is derived asymptotically from the sharp-interface model.

*Appeared in*

- SIAM J. Appl. Math., 75 (2015) pp. 38--60.

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# Optimal control of elastic vector-valued Allen--Cahn variational inequalities

*Authors*

- Farshbaf Shaker, Mohammad Hassan
- Hecht, Claudia

*2010 Mathematics Subject Classification*

- 35K86 49K20 49K21 49J20 35R35

*Keywords*

- Vector-valued Allen--Cahn system, parabolic obstacle problems, linear elasticity, MPCCs, mathematical programs with complementarity constraints, optimality conditions

*Abstract*

In this paper we consider a elastic vector-valued Allen--Cahn MPCC (Mathematical Programs with Complementarity Constraints) problem. We use a regularization approach to get the optimality system for the subproblems. By passing to the limit in the optimality conditions for the regularized subproblems, we derive certain generalized first-order necessary optimality conditions for the original problem.

*Appeared in*

- SIAM J. Control Optim., 54 (2016) pp. 129--152.

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# A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces

*Authors*

- Disser, Karoline
- Meyries, Martin
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35K20 35M13 35R05 35K65 35R01

*Keywords*

- Parabolic equations, mixed boundary conditions, dynamical boundary conditions, Lipschitz domain, degenerate diffusion, surface diffusion, power weights, maximal parabolic Lp-regularity

*Abstract*

In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem.

*Appeared in*

- J. Math. Anal. Appl., 430 (2015) pp. 1102--1123.

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# Simulation of conditional diffusions via forward-reverse stochastic representations

*Authors*

- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 65C05 65C30

*Keywords*

- Forward-reverse representations, pinned diffusions, conditional diffusions, Monte Carlo simulation

*Abstract*

In this paper we derive stochastic representations for the finite dimensional distributions of a multidimensional diffusion on a fixed time interval, conditioned on the terminal state. The conditioning can be with respect to a fixed point or more generally with respect to some subset. The representations rely on a reverse process connected with the given (forward) diffusion as introduced in Milstein et al. [Bernoulli, 10(2):281-312, 2004] in the context of a forward-reverse transition density estimator. The corresponding Monte Carlo estimators have essentially root-N accuracy, hence they do not suffer from the curse of dimensionality. We provide a detailed convergence analysis and give a numerical example involving the realized variance in a stochastic volatility asset model conditioned on a fixed terminal value of the asset.

*Appeared in*

- Ann. Appl. Probab., 24 (2014) pp. 1994--2032

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# Analysis of a time discretization scheme for a nonstandard viscous Cahn--Hilliard system

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Krejčí, Pavel
- Podio-Guidugli, Paolo
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35A40 35K55 35Q70 65M12 65M15

*Keywords*

- Cahn--Hilliard equation, phase field model, time discretization, convergence, error estimates

*Abstract*

In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, for the difference of discrete and continuous solutions we prove an error estimate of order one with respect to the time step.

*Appeared in*

- ESAIM Math. Model. Numer. Anal., 48 (2014) pp. 1061--1087.

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# Nonisothermal nematic liquid crystal flows with the Ball--Majumdar free energy

*Authors*

- Feireisl, Eduard
- Rocca, Elisabetta
- Schimperna, Giulio
- Zarnescu, Arghir

*2010 Mathematics Subject Classification*

- 76A15 74G25 35D30

*Keywords*

- nematic liquid crystal, Ball-Majumdar free energy, nonisothermal model, existence theorem

*Abstract*

In this paper we prove the existence of global in time weak solutions for an evolutionary PDE system modelling nonisothermal Landau-de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier-Stokes system for the macroscopic velocity $vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Q-tensor $QQ$, namely a tensor-valued variable representing the normalized second order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature $vt$ are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential $f$ introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Q-tensor equation, a term which is at the same time singular in $QQ$ and degenerate in $vt$. To treat it a careful analysis of the properties of $f$, particularly of its blow-up rate, is carried out.

*Appeared in*

- Ann. Mat. Pura Appl. IV. Ser., 194 (2015) pp. 1269--1299.

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# Homogenization of elliptic systems with non-periodic, state dependent coefficients

*Authors*

- Hanke, Hauke
- Knees, Dorothee

*2010 Mathematics Subject Classification*

- 74Q15 35B27 35R05 74A45

*Keywords*

- Two-scale convergence, folding and unfolding operator, Γ-convergence, discrete gradient, state dependent coefficient

*Abstract*

In this paper, a homogenization problem for an elliptic system with non-periodic, state dependent coefficients representing microstructure is investigated. The state functions defining the tensor of coefficients are assumed to have an intrinsic length scale denoted by ε > 0. The aim is the derivation of an effective model by investigating the limit process ε → 0 of the state functions rigorously. The effective model is independent of the parameter ε > 0 but preserves the microscopic structure of the state functions (ε > 0), meaning that the effective tensor is given by a unit cell problem prescribed by a suitable microscopic tensor. Due to the non-periodic structure of the state functions and the corresponding microstructure, the effective tensor turns out to vary from point to point (in contrast to a periodic microscopic model). In a forthcoming paper, these states will be solutions of an additional evolution law describing changes of the microstructure. Such changes could be the consequences of temperature changes, phase separation or damage progression, for instance. Here, in addition to the above and as a preparation for an application to time-dependent damage models (discussed in a future paper), we provide a Γ-convergence result of sequences of functionals being related to the previous microscopic models with state dependent coefficients. This requires a penalization term for piecewise constant state functions that allows us to extract from bounded sequences those sequences converging to a Sobolev function in some sense. The construction of the penalization term is inspired by techniques for Discontinuous Galerkin methods and is of own interest. A compactness and a density result are provided.

*Appeared in*

- Asymptot. Anal., 92 (2015) pp. 203--234.

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# Inequalities for Markov operators, majorization and the direction of time

*Authors*

- Stephan, Holger

*2010 Mathematics Subject Classification*

- 54C45 60J05 15B51 82C03

*Keywords*

- function-measure-duality, general classical physical system, Jensen's inequality, rearrangements of functions, stochastic matrix, Robin-Hood method

*Abstract*

In this paper, we connect the following partial orders: majorization of vectors in linear algebra, majorization of functions in integration theory and the order of states of a physical system due to their temporal-causal connection.

Each of these partial orders is based on two general inequalities for Markov operators and their adjoints. The first inequality compares pairs composed of a continuous function (observables) and a probability measure (statistical states), the second inequality compares pairs of probability measure. We propose two new definitions of majorization, related to these two inequalities. We derive several identities and inequalities illustrating these new definitions. They can be useful for the comparison of two measures if the Radon-Nikodym Theorem is not applicable.

The problem is considered in a general setting, where probability measures are defined as convex combinations of the images of the points of a topological space (the physical state space) under the canonical embedding into its bidual. This approach allows to limit the necessary assumptions to functions and measures.

In two appendices, the finite dimensional non-uniform distributed case is described, in detail. Here, majorization is connected with the comparison of general piecewise affine convex functions. Moreover, the existence of a Markov matrix, connecting two given majorizing pairs, is shown.

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# A deep quench approach to the optimal control of an Allen--Cahn equation with dynamic boundary conditions and double obstacles

*Authors*

- Colli, Pierluigi
- Farshbaf Shaker, Mohammad Hassan
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 74M15 49K20 35K61

*Keywords*

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

*Abstract*

In this paper, we investigate optimal control problems for Allen-Cahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy is the following: we use the results that were recently established by two of the authors 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 (non-differentiable) double obstacle potentials.

*Appeared in*

- Appl. Math. Optim., 71 (2015) pp. 1--24.

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# On uniform decay of the entropy for reaction-diffusion systems

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Haskovec, Jan
- Markowich, Peter A.

*2010 Mathematics Subject Classification*

- 35K57 35B40 92E20

*Keywords*

- Reaction-diffusion, mass-action law, log-Sobolev inequality, exponential decay of relative entropy

*Abstract*

In this work we derive entropy decay estimates for a class of nonlinear reaction-diffusion systems modeling reversible chemical reactions under the assumption of detailed balance. In particular, we provide explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the log-Sobolev inequality and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: we allow for vanishing diffusion constants in some chemical components, and we consider different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions.

*Appeared in*

- J. Dynam. Differential Equations, 27 (2015) pp. 897--928.

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# A mixture theory of electrolytes containing solvation effects

*Authors*

- Dreyer, Wolfgang
- Guhlke, Clemens
- Landstorfer, Manuel

*2010 Mathematics Subject Classification*

- 78A57 35Q99 34B15

*2008 Physics and Astronomy Classification Scheme*

- 82.45.Gj 82.45.Mp 82.60.Lf

*Keywords*

- Double Layer, Poisson-Boltzmann, Solvation, Mixture theory, Gouy-Chapman-Stern Model

*Abstract*

In this work we present a new mixture theory of a liquid solvent containing completely dissociated ions to study the space charge layer of electrolytes in contact with some inert metal. We incorporate solvation shell effects (i) in our derivation of the mixing entropy and (ii) in the pressure model. Chemical potentials of ions and solvent molecules in the incompressible limit are then derived from a free energy function. For the thermodynamic equilibrium the coupled equation system of mass and momentum balance, the incompressibility constraint and the Poisson equation are summarized. With that we study the space charge layer of the electrolytic solution for an applied half cell potential and compare our results to historic and recent interpretations of the double layer in liquid electrolytes. The novelties of the new model are: (i) coupling of momentum- and mass-balance equations, (ii) calculation of entropic contributions due to solvated ions and (iii) the potential and pressure dependence of the free charge density in equilibrium.

*Appeared in*

- Electrochem. Commun., 43 (2014) pp. 75--78.

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# Modeling and analysis of a phase field system for damage and phase separation processes in solids

*Authors*

- Bonetti, Elena
- Heinemann, Christian
- Kraus, Christiane
- Segatti, Antonio

*2010 Mathematics Subject Classification*

- 35K55 74G25 74A45

*Keywords*

- Cahn-Hilliard system, phase separation, elliptic-parabolic systems, doubly nonlinear differential inclusions, complete damage, existence results, energetic solutions, weak solutions, linear elasticity, rate-dependent systems

*Abstract*

In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced system, we start with a regularized version. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.

*Appeared in*

- J. Partial Differ. Equ., 258 (2015) pp. 3928--3959.

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# A study of isogeometric analysis for scalar convection-diffusion equations

*Authors*

- John, Volker

ORCID: 0000-0002-2711-4409 - Schumacher, Liesel

*2010 Mathematics Subject Classification*

- 65N30

*Keywords*

- Isogeometric analysis, streamline upwind, Petrov--Galerkin stabilization, Hemker problem, over- and undershoots, sharpness of layers

*Abstract*

Isogeometric Analysis (IGA), in combination with the Streamline Upwind Petrov--Galerkin (SUPG) stabilization, is studied for the discretization of steady-state convection-diffusion equations. Numerical results obtained for the Hemker problem are compared with results computed with the SUPG finite element method of the same order. Using an appropriate parameterization for IGA, the computed solutions are much more accurate than those obtained with the finite element method, both in terms of the size of spurious oscillations and of the sharpness of layers.

*Appeared in*

- Appl. Math. Lett., 27 (2014) pp. 43--48.

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# On iterative subdomain methods for the Stokes--Darcy problem

*Authors*

- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - John, Volker

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

*2010 Mathematics Subject Classification*

- 65N30 65N55

*Keywords*

- Stokes--Darcy problem, iterative subdomain methods, Robin boundary conditions, finite element methods, continuous and discontinuous, updating strategy, applications from geosciences

*Abstract*

Iterative subdomain methods for the Stokes--Darcy problem that use Robin boundary conditions on the interface are reviewed. Their common underlying structure and their main differences are identified. In particular, it is clarified that there are different updating strategies for the interface conditions. For small values of fluid viscosity and hydraulic permeability, which are relevant in applications from geosciences, it is shown in numerical studies that only one of these updating strategies leads to an efficient numerical method, if this strategy is used in combination with appropriate parameters in the Robin boundary conditions. In particular, it is observed that the values of appropriate parameters are larger than those proposed so far. Not only the size but also the ratio of appropriate Robin parameters depends on the coefficients of the problem.

*Appeared in*

- Comput. Geosci., 18 (2014) pp. 711--728.

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# A new model for quantum dot light emitting-absorbing devices

*Authors*

- Neidhardt, Hagen
- Wilhelm, Lukas
- Zagrebnov, Valentin A.

*2010 Mathematics Subject Classification*

- 47A40 47A55 81Q37 81V80

*Keywords*

- Landauer-Büttiker formula, Jaynes-Cummings model, coupling to leads, light emission, solar cells

*Abstract*

Motivated by the Jaynes-Cummings (JC) model, we consider here a quantum dot coupled simultaneously to a reservoir of photons and to two electric leads (free-fermion reservoirs). This Jaynes-Cummings-Leads (JCL) model makes possible that the fermion current through the dot creates a photon flux, which describes a light-emitting device. The same model is also describe a transformation of the photon flux into current of fermions, i.e. a quantum dot light-absorbing device. The key tool to obtain these results is an abstract Landauer-Büttiker formula.

*Appeared in*

- J. Mathematical Physics Analysis Geometry (MAG), 10 (2014) pp. 1--37.

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# On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Peletier, Mark A.
- Renger, D. R. Michiel

ORCID: 0000-0003-3557-3485

*2010 Mathematics Subject Classification*

- 35Q82 35Q84 49S05 60F10 60J25 60J27

*Keywords*

- Generalized gradient flows, large deviations, convex analysis, particle systems

*Abstract*

Motivated by the occurence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions ℒ that induce a flow, given by ℒ(z_{t},ż_{t})=0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when ℒ is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.

*Appeared in*

- Potential Anal., 41 (2014) pp. 1293--1325.

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# Multi-material phase field approach to structural topology optimization

*Authors*

- Blank, Luise
- Farshbaf Shaker, Mohammad Hassan
- Garcke, Harald
- Rupprecht, Christoph
- Styles, Vanessa

*2010 Mathematics Subject Classification*

- 49Q10 74P05 74P15 90C52 65K15

*Keywords*

- shape and topology optimization, phase field approach, shape sensitivity analysis, gradient projection method

*Abstract*

Multi-material structural topology and shape optimization problems are formulated within a phase field approach. First-order conditions are stated and the relation of the necessary conditions to classical shape derivatives are discussed. An efficient numerical method based on an *H*^{1}-gradient projection method is introduced and finally several numerical results demonstrate the applicability of the approach.

*Appeared in*

- Trends in PDE Constrained Optimization, G. Leugering, P. Benner et al., eds., vol. 165 of International Series of Numerical Mathematics, Birkhäuser, Basel et al., 2014, pp. 231--246

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# Analysis of the PSPG stabilization for the continuous-in-time discretization of the evolutionary Stokes equations

*Authors*

- John, Volker

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

*2010 Mathematics Subject Classification*

- 665M12 65M60

*Keywords*

- Evolutionary Stokes equations, PSPG stabilization, equal order pairs of finite element spaces, optimal error estimates, small time step instability

*Abstract*

Optimal error estimates for the pressure stabilized Petrov--Galerkin (PSPG) method for the continuous-in-time discretization of the evolutionary Stokes equations are proved in the case of regular solutions. The main result is applicable to higher order finite elements. The error bounds for the pressure depend on the error of the pressure at the initial time. An approach is suggested for choosing the discrete initial velocity in such a way that this error is bounded. The ``instability of the discrete pressure for small time steps'', which is reported in the literature, is discussed on the basis of the analytical results. Numerical studies confirm the theoretical results, showing in particular that this instability does not occur for the proposed initial condition.

*Appeared in*

- SIAM J. Numer. Anal., 53 (2015) pp. 1005--1031.

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# Optimal control of Allen--Cahn systems

*Authors*

- Blank, Luise
- Farshbaf Shaker, Mohammad Hassan
- Hecht, Claudia
- Michl, Josef
- Rupprecht, Christoph

*2010 Mathematics Subject Classification*

- 49J40 49K20 49J20 49M15 74P99

*Keywords*

- Allen-Cahn system, parabolic obstacle problems, linear elasticity, mathematical programs with complementarity constraints, optimality conditions, Trust-Region-Newton method

*Abstract*

Optimization problems governed by Allen-Cahn systems including elastic effects are formulated and first-order necessary optimality conditions are presented. Smooth as well as obstacle potentials are considered, where the latter leads to an MPEC. Numerically, for smooth potential the problem is solved efficiently by the Trust-Region-Newton-Steihaug-cg method. In case of an obstacle potential first numerical results are presented.

*Appeared in*

- Trends in PDE Constrained Optimization, G. Leugering, P. Benner et al., eds., vol. 165 of International Series of Numerical Mathematics, Birkhäuser, Basel et al., 2014, pp. 11--26

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# Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions

*Authors*

- van Ackooij, Wim
- Henrion, René

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- Stochastic optimization, probabilistic constraints, chance constraints, gradients of probability functions

*Abstract*

Probabilistic constraints represent a major model of stochastic optimization. A possible approach for solving probabilistically constrained optimization problems consists in applying nonlinear programming methods. In order to do so, one has to provide sufficiently precise approximations for values and gradients of probability functions. For linear probabilistic constraints under Gaussian distribution this can be successfully done by analytically reducing these values and gradients to values of Gaussian distribution functions and computing the latter, for instance, by Genz' code. For nonlinear models one may fall back on the spherical-radial decomposition of Gaussian random vectors and apply, for instance, Deák's sampling scheme for the uniform distribution on the sphere in order to compute values of corresponding probability functions. The present paper demonstrates how the same sampling scheme can be used in order to simultaneously compute gradients of these probability functions. More precisely, we prove a formula representing these gradients in the Gaussian case as a certain integral over the sphere again. Later, the result is extended to alternative distributions with an emphasis on the multivariate Student (or T-) distribution.

*Appeared in*

- SIAM J. Optim., 24 (2014) pp. 1864--1889.

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# Efficient linear solvers for incompressible flow simulations using Scott--Vogelius finite elements

*Authors*

- Cousins, Benjamin
- Le Borne, Sabine
- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Rebholz, Leo G.
- Wang, Zhen

*2010 Mathematics Subject Classification*

- 76D07 65F05 65F08

*Keywords*

- Scott-Vogelius elements, linear solvers, static condensation, augmented Lagrangian preconditioning, H-LU

*Abstract*

Recent research has shown that in some practically relevant situations like multiphysics flows (Galvin et al., Comput Methods Appl Mech Eng, 2012) divergence-free mixed finite elements may have a significantly smaller discretization error than standard nondivergence-free mixed finite elements. To judge the overall performance of divergence-free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in Scott-Vogelius finite element implementations of the incompressible Navier-Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott-Vogelius (SV) elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements such as Taylor-Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes and discuss extensions. For iterative methods, we test augmented Lagrangian and H -LU preconditioners with GMRES, on both full and statically condensed systems. Several numerical experiments are provided that show these classes of solvers are well suited for use with SV elements and could deliver an interesting overall performance in several applications.

*Appeared in*

- Numer. Methods Partial Differential Equations, 29 (2013) pp. 1217--1237.

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# Rogue wave formation by accelerated solitons at an optical event horizon

*Authors*

- Demircan, Ayhan
- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Brée, Carsten
- Mahnke, Christoph
- Mitschke, Fedor
- Steinmeyer, Günter

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Re 42.65.Ky 42.65.Tg 42.81.Dp

*Keywords*

- Rogue wave, Optical soliton, Ultrashort pulse, Optical event horizon

*Abstract*

Rogue waves, by definition, are rare events of extreme amplitude, but at the same time they are frequent in the sense that they can exist in a wide range of physical contexts. While many mechanisms have been demonstrated to explain the appearance of rogue waves in various specific systems, there is no known generic mechanism or general set of criteria shown to rule their appearance. Presupposing only the existence of a nonlinear Schrödinger-type equation together with a concave dispersion profile around a zero dispersion wavelength we demonstrate that solitons may experience acceleration and strong reshaping due to the interaction with continuum radiation, giving rise to extreme-value phenomena. The mechanism is independent of the optical Raman effect. A strong increase of the peak power is accompanied by a mild increase of the pulse energy and carrier frequency, whereas the photon number of the soliton remains practically constant. This reshaping mechanism is particularly robust and is naturally given in optics in the supercontinuum generation process.

*Appeared in*

- Appl. Phys. B, 115 (2013) pp. 343--354.

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# Computational haemodynamics in stenotic internal jugular veins

*Authors*

- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Montecinos, Gino
- Müller, Lucas O.
- Haacke, E. Mark
- Toro, Eleuterio F.

*2008 Physics and Astronomy Classification Scheme*

- 07.05.Tp, 47.11.-j, 47.63.Cb, 87.19.uj

*Keywords*

- Computational haemodynamics, stenosis, internal jugular veins, anomalous haemodynamics, CCSVI

*Abstract*

Stenosis in internal jugular veins (IJVs) are frequently associated to pathological venous circulation and insufficient cerebral blood drainage. In this work, we set up a computational framework to assess the relevance of IJV stenoses through numerical simulation, combining medical imaging, patient-specific data and a mathematical model for venous occlusions. Coupling a three-dimensional (3D) description of blood flow in IJVs with a reduced one-dimesional model (1D) for major intracranial veins, we are able to model different anatomical configurations, an aspect of importance to understand the impact of IJV stenosis in intracranial venous haemodynamics. We investigate several stenotic configurations in a physiologic patient-specific regime, quantifying the effect of the stenosis in terms of venous pressure increase and wall shear stress patterns. Simulation results are in qualitative agreement with reported pressure anomalies in pathological cases. Moreover, they demonstrate the potential of the proposed multiscale framework for individual-based studies and computer-aided diagnosis.

*Appeared in*

- J. Math. Biol., 70 (2015) pp. 745--772.

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# Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator

*Authors*

- Pimenov, Alexander
- Rachinskii, Dmitrii

*2010 Mathematics Subject Classification*

- 47J40 92D25 37L15

*Keywords*

- robust homoclinic orbit, Preisach operator, operator-differential equations, predator-prey model, hysteresis operators, rate-independent systems

*Abstract*

Systems of operator-differential equations which hysteresis operators can have unstable equilibrium points with an open basin of attraction. In this paper, a numerical example of a robust homoclinic loop is presented for the first time in a population dynamics model with hysteretic response of prey to variations of predator. A mechanism creating this homoclinic trajectory is discussed.

*Appeared in*

- , 139 (2014) pp. 285--298.

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# TetGen, towards a quality tetrahedral mesh generator

*Authors*

- Si, Hang

*2010 Mathematics Subject Classification*

- 65M50 65N50 65D18 68U05 68N99

*Keywords*

- tetrahedral mesh generation, Delaunay tetrahedralization, constrained Delaunay, boundary recovery, mesh refinement, mesh quality, flips, edge removal

*Abstract*

TetGen is a C++ program for generating quality tetrahedral meshes aimed to support numerical methods and scientific computing. It is also a research project for studying the underlying mathematical problems and evaluating algorithms. This paper presents the essential meshing components developed in TetGen for robust and efficient software implementation. And it highlights the state-of-the-art algorithms and technologies currently implemented and developed in TetGen for automatic quality tetrahedral mesh generation.

*Appeared in*

- ACM Trans. Math. Software, 41 (2015) pp. 11:1--11:36.

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# Path-wise approximation of the Cox--Ingersoll--Ross process

*Authors*

- Milstein, Grigori N.
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 65C30 60H35

*Keywords*

- Cox-Ingersoll-Ross process, Heston model, Doss-Sussmann formalism, exact simulation

*Abstract*

The Doss-Sussmann (DS) approach is used for simulating the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows for expressing trajectories of the CIR process by solutions of some ordinary differential equation (ODE) that depend on realizations of the Wiener process involved. Via simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving an ODE, we approximately construct the trajectories of the CIR process. From a conceptual point of view the proposed method may be considered as an exact simulation approach.

*Appeared in*

- Adv. Appl. Probab., 1132--1156 (2015) pp. .

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# A new perspective on the Propagation-Separation Approach: Taking advantage of the propagation condition

*Authors*

- Becker, Saskia
- Mathé, Peter

ORCID: 0000-0002-1208-1421

*2010 Mathematics Subject Classification*

- 62G05

*Keywords*

- Structural adaptive smoothing, Propagation, Separation, Local likelihood, Exponential families

*Abstract*

The Propagation-Separation approach is an iterative procedure for pointwise estimation of local constant and local polynomial functions. The estimator is defined as a weighted mean of the observations with data-driven weights. Within homogeneous regions it ensures a similar behavior as non-adaptive smoothing (propagation), while avoiding smoothing among distinct regions (separation). In order to enable a proof of stability of estimates, the authors of the original study introduced an additional memory step aggregating the estimators of the successive iteration steps. Here, we study theoretical properties of the simplified algorithm, where the memory step is omitted. In particular, we introduce a new strategy for the choice of the adaptation parameter yielding propagation and stability for local constant functions with sharp discontinuities.

*Appeared in*

- Electron. J. Stat., 7 (2013) pp. 2702--2736.

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# Direct discretizations of bi-variate population balance systems with finite difference schemes of different order

*Authors*

- John, Volker

ORCID: 0000-0002-2711-4409 - Suciu, Carina

*2010 Mathematics Subject Classification*

- 76T20

*Keywords*

- bi-variate population balance systems, direct discretizations, finite difference methods, accuracy of numerical results

*Abstract*

The accurate and efficient simulation of bi-variate population balance systems is nowadays a great challenge since the domain spanned by the external and internal coordinates is five-dimensional. This report considers direct discretizations of this equation in tensor-product domains. In this situation, finite difference methods can be applied. The studied model includes the transport of dissolved potassium dihydrogen phosphate (KDP) and of energy (temperature) in a laminar flow field as well as the nucleation and growth of KDP particles. Two discretizations of the coupled model will be considered which differ only in the discretization of the population balance equation: a first order monotone upwind scheme and a third order essentially non-oscillatory (ENO) scheme. The Dirac term on the right-hand side of this equation is discretized with a finite volume method. The numerical results show that much different results are obtained even in the class of direct discretizations.

*Appeared in*

- Chem. Engng. Sci., 106 (2014) pp. 39--52.

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# Uniform Poincaré--Sobolev and relative isoperimetric inequalities for classes of domains

*Authors*

- Thomas, Marita

*2010 Mathematics Subject Classification*

- 46E35 26D10 52A38

*Keywords*

- Poincaré-Sobolev inequality, relative isoperimetric inequality, uniform cone property

*Abstract*

The aim of this paper is to prove an isoperimetric inequality relative to a d-dimensional, bounded, convex domain &Omega intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius r>0 and the position y∈cl(&Omega) of the center of the ball. For this, uniform Sobolev, Poincaré and Poincaré-Sobolev inequalities are deduced for classes of (not necessarily convex) domains that satisfy a uniform cone property. It is shown that the constants in all of these inequalities solely depend on the dimensions of the cone, space dimension d, the diameter of the domain and the integrability exponent p∈[1,d).

*Appeared in*

- Discrete Contin. Dyn. Syst., 35 (2015) pp. 2741--2761.

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# Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains

*Authors*

- Heinemann, Christian
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 35L20 35L51 35K85 35K55 49J40 49S05 74A45 74G25 34A12 35K92 35K35

*Keywords*

- hyperbolic-parabolic systems, doubly nonlinear differential inclusions, existence results, energetic solutions, weak solutions, linear elasticity, rate-dependent systems

*Abstract*

The aim of this paper is to prove existence of weak solutions of hyperbolic-parabolic evolution inclusions defined on Lipschitz domains with mixed boundary conditions describing, for instance, damage processes and elasticity with inertia terms. To this end, a suitable weak formulation to deal with such evolution inclusions in a non-smooth setting is presented. Then, existence of weak solutions is proven by utilizing time-discretization, regularization of the displacement variable and variational techniques from [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321-359] to recover the subgradients after the limit passages.

*Appeared in*

- SIAM J. Math. Anal., 47 (2015) pp. 2044--2073.

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# The Propagation-Separation Approach: Consequences of model misspecification

*Authors*

- Becker, Saskia

*2010 Mathematics Subject Classification*

- 62G05

*Keywords*

- structural adaptive smoothing, propagation, separation, local likelihood, exponential families, model misspecification

*Abstract*

The article presents new results on the Propagation-Separation Approach by Polzehl and Spokoiny (2006). This iterative procedure provides a unified approach for nonparametric estimation, supposing a local parametric model. The adaptivity of the estimator ensures sensitivity to structural changes. Originally, an additional memory step was included into the algorithm, where most of the theoretical properties were based on. However, in practice, a simplified version of the algorithm is used, where the memory step is omitted. Hence, we aim to justify this simplified procedure by means of a theoretical study and numerical simulations. In our previous study, we analyzed the simplified Propagation-Separation Approach, supposing piecewise constant parameter functions with sharp discontinuities. Here, we consider the case of a misspecified model.

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# Quantum transport in cylindrical semiconductor nanowires with constrictions

*Authors*

- Racec, Paul N.

*2010 Mathematics Subject Classification*

- 47A40 35Q40 35P25

*2008 Physics and Astronomy Classification Scheme*

- 62.23.Hj 73.63.-b 72.20.Dp

*Keywords*

- Quantum point contact, cylindrical nanowire, Schrödinger operator, R-matrix formalism, quasi-bound states, evanescent channels

*Abstract*

The energy dependence of the tunneling coefficient for a cylindrical semiconductor nanowire, i.e. a one-dimensional electron gas, with one or two constrictions is studied. Using the R-matrix formalism the localization probabilities at the resonant energies can be computed. They give decisive information about the physical meaning of the resonant peaks and dips that appear. The nanowire with two constrictions yields a well-defined system for the experimental evidence of the quasi-bound states of the evanescent channels. Clearly marked dips due to them should appear in the linear conductance at low temperatures.

*Appeared in*

- Phys. Status Solidi B, 251 (2014) pp. 195--200 under the title ``Cylindrical semiconductor nanowires with constrictions''

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# A study on the conditioning of finite element equations with arbitrary anisotropic meshes via a density function approach

*Authors*

- Kamenski, Lennard
- Huang, Weizhang

*2010 Mathematics Subject Classification*

- 65N30 65N50 65F35 65F15

*Keywords*

- conditioning, finite element, anisotropic diffusion, anisotropic mesh, stiffness matrix, extreme eigenvalue, Jacobi preconditioning, diagonal scaling

*Abstract*

The linear finite element approximation of a general linear diffusion problem with arbitrary anisotropic meshes is considered. The conditioning of the resultant stiffness matrix and the Jacobi preconditioned stiffness matrix is investigated using a density function approach proposed by Fried in 1973. It is shown that the approach can be made mathematically rigorous for general domains and used to develop bounds on the smallest eigenvalue and the condition number that are sharper than existing estimates in one and two dimensions and comparable in three and higher dimensions. The new results reveal that the mesh concentration near the boundary has less influence on the condition number than the mesh concentration in the interior of the domain. This is especially true for the Jacobi preconditioned system where the former has little or almost no influence on the condition number. Numerical examples are presented.

*Appeared in*

- J. Math. Study, 47 (2014) pp. 151--172.

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# A Widder's type theorem for the heat equation with nonlocal diffusion

*Authors*

- Barrios, Begoña
- Peral, Ireneo
- Soria, Fernando
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35K05 35K15 35C15 35B30 35B99

*Keywords*

- Heat equation, fractional Laplacian, trace of strong solutions, uniqueness of non-negative solutions

*Abstract*

The main goal of this work is to prove that every non-negative strong solution of the fractional heat equation can be written as a kernel convolution with its initial datum. This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by D. V. Widder to the nonlocal diffusion framework.

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# Droplets on liquids and their long way into equilibrium

*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*

- Liquid-liquid interface structure: measurements and simulations, Numerical simulation, solution of equations

*Abstract*

The morphological paths towards equilibrium droplets during the late stages of the dewetting process of a liquid film from a liquid substrate is investigated experimentally and theoretically. As liquids, short chained polystyrene (PS) and polymethyl-methacrylate (PMMA) are used, which can be considered as Newontian liquids well above their glass transition temperatures. Careful imaging of the PS/air interface of the droplets during equilibration by in situ scanning force microscopy and the PS/PMMA interface after removal of the PS droplets reveal a surprisingly deep penetration of the PS droplets into the PMMA layer. Droplets of sufficiently small volumes develop the typical lens shape and were used to extract the ratio of the PS/air and PS/PMMA surface tensions and the contact angles by comparison to theoretical exact equilibrium solutions of the liquid/liquid system. Using these results in our dynamical thin-film model we find that before the droplets reach their equilibrium they undergo several intermediate stages each with a well-defined signature in shape. Moreover, the intermediate droplet shapes are independent of the details of the initial configuration, while the time scale they are reached depend strongly on the droplet volume. This is shown by the numerical solutions of the thin-film model and demonstrated by quantitative comparison to experimental results.

*Appeared in*

- Eur. Phys. J. E -- Soft Matter, 36 (2013) pp. 87--99.

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# Identification, simulation and optimal control of heat transfer in cooling line of hot strip rolling mill

*Authors*

- Graf, Marcel
- Hömberg, Dietmar
- Kawalla, Rudolf
- Togobytska, Nataliya
- Weiss, Wolf

*Keywords*

- Simulation, optimal control, heat transfer, hot rolling process, cooling line

*Abstract*

The numerical simulation of mechanical properties of hot-rolled products has a major significance for material characterisation as well as material development. The basis for these is the knowledge about the material-specific phase transformations in combination with the initial microstructure from the deformation steps before entering into the cooling line. Additionally, the technological conditions in the run-out table (ROT) are essentially for transformation kinetics. In order to simulate these processes, the plant-specific heat transfer coefficient must be measured. Therefore, steel samples with thermocouples inside are transported with defined velocities through the cooling line of the continuous pilot plant at the Institute of Metal Forming in Freiberg. Furthermore, the material and its movement must be taken into account as characteristics of the ROT (e.g. amount and distribution of the cooling medium, the streaming situation in several segments, the nozzle geometry and, as a consequence, the water jet shape, and the impact pressure of the cooling medium on the surface of the rolled material) as influencing parameters. This paper describes the possibilities for determining and simulating the heat transfer in the cooling line with industrial conditions. Moreover, this paper discusses the optimal control of the cooling line to achieve the desired temperature and phase distribution on the run-out table. The resulting information contributes to new technology and material developments at the pilot plant, as well as for the transfer of results into the industry.

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# A random cloud model for the Schrödinger equation

*Authors*

- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 35Q41 60J25 81Q05

*Keywords*

- Schrödinger equation, probabilistic representation, stochastic particle model, Markov jump process

*Abstract*

The paper is concerned with the construction of a stochastic model for the spatially discretized time-dependent Schrödinger equation. The model is based on a particle system with a Markov jump evolution. The particles are characterized by a sign (plus or minus), a position (discrete grid) and a type (real or imaginary). The jumps are determined by the creation of offsprings. The main result is the construction of a family of complex-valued random variables such that their expected values coincide with the solution of the Schrödinger equation.

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# Cell size error in stochastic particle methods for coagulation equations with advection

*Authors*

- Patterson, Robert I. A.

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

*2010 Mathematics Subject Classification*

- 65C35 65M12 65M75

*Keywords*

- Stochastic particle methods, spatial discretization error, coagulation equation, advection

*Abstract*

The paper studies the approximation error in stochastic particle methods for spatially inhomogeneous population balance equations. The model includes advection, coagulation and inception. Sufficient conditions for second order approximation with respect to the spatial discretization parameter (cell size) are provided. Examples are given, where only first order approximation is observed.

*Appeared in*

- SIAM J. Numer. Anal., 52 (2014) pp. 424--442.

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# Some properties of the kinetic equation for electron transport in semiconductors

*Authors*

- Wagner, Wolfgang

*2010 Mathematics Subject Classification*

- 82D37, 65C05

*Keywords*

- Electron transport equation, semiconductors, heat generation, steady state distribution, Monte Carlo algorithm

*Abstract*

The paper studies the kinetic equation for electron transport in semiconductors. New formulas for the heat generation rate are derived by analyzing the basic scattering mechanisms. In addition, properties of the steady state distribution are discussed and possible extensions of the deviational particle Monte Carlo method to the area of electron transport are proposed.

*Appeared in*

- Kinet. Relat. Models, 6 (2013) pp. 955--967.

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# A continuous dependence result for a nonstandard system of phase field equations

*Authors*

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

*2010 Mathematics Subject Classification*

- 35K61 35A05 35B30

*Keywords*

- nonstandard phase field system, nonlinear differential equations, uniqueness, continuous dependence

*Abstract*

The present note deals with a nonstandard systems of differential equations describing a two-species phase segregation. This system naturally arises in the asymptotic analysis carried out recently by the same authors, as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, an existence result has been proved for the limit system in a very general framework. On the contrary, uniqueness was shown by assuming a constant mobility coefficient. Here, we generalize this result and prove a continuous dependence property in the case that the mobility coefficient suitably depends on the chemical potential.

*Appeared in*

- Math. Methods Appl. Sci., 37 (2014) pp. 1318--1324.

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# On the probability density function of baskets

*Authors*

- Bayer, Christian

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

ORCID: 0000-0003-2571-8388 - Laurence, Peter

*2010 Mathematics Subject Classification*

- 91G20 60F10 65C50

*Keywords*

- Sums of lognormals, Focality, Pricing of butterfly spreads on baskets

*Abstract*

The state price density of a basket, even under uncorrelated Black-Scholes dynamics, does not allow for a closed from density. (This may be rephrased as statement on the sum of lognormals and is especially annoying for such are used most frequently in Financial and Actuarial Mathematics.) In this note we discuss short time and small volatility expansions, respectively. The method works for general multi-factor models with correlations and leads to the analysis of a system of ordinary (Hamiltonian) differential equations. Surprisingly perhaps, even in two asset Black-Scholes situation (with its flat geometry), the expansion can degenerate at a critical (basket) strike level; a phenomena which seems to have gone unnoticed in the literature to date. Explicit computations relate this to a phase transition from a unique to more than one "most-likely" paths (along which the diffusion, if suitably conditioned, concentrates in the afore-mentioned regimes). This also provides a (quantifiable) understanding of how precisely a presently out-of-money basket option may still end up in-the-money.

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# Spectrum and amplitude equations for scalar delay-differential equations with large delay

*Authors*

- Yanchuk, Serhiy
- Lücken, Leonhard
- Wolfrum, Matthias
- Mielke, Alexander

ORCID: 0000-0002-4583-3888

*2010 Mathematics Subject Classification*

- 34K08 35Q56

*Keywords*

- amplitude equations, delay-differential equations, large delay

*Abstract*

The subject of the paper are scalar delay-differential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-differential equations close to the destabilization threshold.

*Appeared in*

- Discrete Contin. Dyn. Syst., 35 (2015) pp. 537--553.

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# Domain expression of the shape derivative and application to electrical impedance tomography

*Authors*

- Laurain, Antoine
- Sturm, Kevin

*2010 Mathematics Subject Classification*

- 49Q10 35Q93 35R30 35R05

*Keywords*

- Shape optimization, distributed shape gradient, electrical impedance tomography, Lagrangian method, level set method

*Abstract*

The well-known structure theorem of Hadamard-Zolesio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. However a volume representation (distributed shape derivative) is more general than the boundary form and allows to work with shapes having a lower regularity. It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithm. In this paper we describe the numerous advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We give several examples of numerical applications such as the inverse conductivity problem and the level set method.

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# Multiscale modeling of weakly compressible elastic materials in harmonic regime and application to microscale structure estimation

*Authors*

- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Mura, Joaquin

*2010 Mathematics Subject Classification*

- 65Z05 74Q05 74Q15

*Keywords*

- Homogenization, linear elasticity, weakly compressible materials, inverse problem, elastography

*Abstract*

This article is devoted to the modeling of elastic materials composed by an incompressible elastic matrix and small compressible gaseous inclusions, under a time harmonic excitation. In a biomedical context, this model describes the dynamics of a biological tissue (e.g. lung or liver) when wave analysis methods (such as Magnetic Resonance Elastography) are used to estimate tissue properties. Due to the multiscale nature of the problem, direct numerical simulations are prohibitive. We extend the homogenized model introduced in [Baffico, Grandmont, Maday, Osses, SIAM J. Mult. Mod. Sim., 7(1), 2008] to a time harmonic regime to describe the solid-gas mixture from a macroscopic point of view in terms of an effective elasticity tensor. Furthermore, we derive and validate numerically analytical approximations for the effective elastic coefficients in terms of macroscopic parameters. This simplified description is used to to set up an efficient variational approach for the estimation of the tissue porosity, using the mechanical response to external harmonic excitations.

*Appeared in*

- Multiscale Model. Simul., 12 (2014) pp. 514--537.

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# A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

*Authors*

- Knees, Dorothee
- Rossi, Riccarda
- Zanini, Chiara

*2010 Mathematics Subject Classification*

- 74R05 74C05 35D40 35K86 49J40

*Keywords*

- rate-independent damage evolution, vanishing viscosity method, arclength reparametrization, time discretization, regularity estimates

*Abstract*

This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps. Therefore, we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling and approximate the model by its viscous regularization. In fact, the analysis of the latter PDE system presents remarkable difficulties, due to its highly nonlinear character. We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with careful regularity estimates tailored to this specific system relying on a q-Laplacian type gradient regularization of the damage variable. Hence, for the viscous problem we conclude the existence of weak solutions satisfying a suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis. The latter leads to the notion of (weak) parameterized solution to our rate-independent system, which encompasses the influence of viscosity in the description of the jump regime.

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# The factorization method for inverse elastic scattering from periodic structures

*Authors*

- Hu, Guanghui
- Lu, Yulong
- Zhang, Bo

*2010 Mathematics Subject Classification*

- 35R30 74B05 78A46 35Q93

*Keywords*

- Inverse elastic scattering, factorization method, Dirichlet boundary condition, Navier equation, uniqueness

*Abstract*

This paper is concerned with the inverse scattering of time-harmonic elastic waves from rigid periodic structures. We establish the factorization method to identify an unknown grating surface from knowledge of the scattered compressional or shear waves measured on a line above the scattering surface. Near-field operators are factorized by selecting appropriate incident waves derived from quasi-periodic half-space Green's tensor to the Navier equation. The factorization method gives rise to a uniqueness result for the inverse scattering problem by utilizing only the compressional or shear components of the scattered field corresponding to all quasi-periodic incident plane waves with a common phase-shift. A number of computational examples are provided to show the accuracy of the inversion algorithms, with an emphasis placed on comparing reconstructions from the scattered near-field and those from its compressional and shear components.

*Appeared in*

- Inverse Problems, 29 (2013) pp. 115005/1--115005/25.

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# Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures

*Authors*

- Bugert, Beatrice
- Schmidt, Gunther

*2010 Mathematics Subject Classification*

- 31B10 35Q60 35Q61 45A05 78A45

*Keywords*

- biperiodic scattering problems, Maxwell's equations, boundary integral equations, Lipschitz domains, Gårding inequalities

*Abstract*

This paper is concerned with the study of a new integral equation formulation for electromagnetic scattering by a 2π-biperiodic polyhedral Lipschitz profile. Using a combined potential ansatz, we derive a singular integral equation with Fredholm operator of index zero from time-harmonic Maxwell's equations and prove its equivalence to the electromagnetic scattering problem. Moreover, under certain assumptions on the electric permittivity and the magnetic permeability, we obtain existence and uniqueness results in the special case that the grating is smooth and, under more restrictive assumptions, in the case that the grating is of polyhedral Lipschitz regularity.

*Appeared in*

- Discrete Contin. Dyn. Syst. Ser. S, 8 (2015) pp. 435--473.

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# Elastic scattering by unbounded rough surfaces: Solvability in weighted Sobolev spaces

*Authors*

- Elschner, Johannes
- Hu, Guanghui

*2010 Mathematics Subject Classification*

- 74B05 35J05 35J20 35J25 42B10 78A45 74J20 35J57 35Q74

*Keywords*

- non-smooth rough surface, linear elasticity, radiation condition, variational formulation, weighted Sobolev spaces, Navier equation

*Abstract*

This paper is concerned with the variational approach in weighted Sobolev spaces to time-harmonic elastic scattering by two-dimensional unbounded rough surfaces. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the total elastic displacement satisfies either the Dirichlet or impedance boundary condition. We establish uniqueness and existence results for both elastic plane and point source (spherical) wave incidence, following the recently developed variational approach in [SIAM J. Math. Anal., 42: 6 (2010), pp. 2554-2580] for the Helmholtz equation. This paper extends our previous solvability results [SIAM J. Math. Anal., 44: 6 (2012), pp. 4101-4127] in the standard Sobolev space to the weighted Sobolev spaces.

*Appeared in*

- Appl. Anal., 94 (2015) pp. 251--278.

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# Critical dimension in profile semiparametric estimation

*Authors*

- Andresen, Andreas
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 62F10 62J12 62F25 62H12

*Keywords*

- maximum likelihood, local quadratic bracketing, spread, local concentration

*Abstract*

This paper revisits the classical inference results for profile quasi maximum likelihood estimators (profile MLE) in the semiparametric estimation problem. We mainly focus on two prominent theorems: the Wilks phenomenon and Fisher expansion for the profile MLE are stated in a new fashion allowing finite samples and model misspecification. The method of study is also essentially different from the usual analysis of the semiparametric problem based on the notion of the hardest parametric submodel. Instead we apply the local bracketing and the upper function devices from Spokoiny (2011). This novel approach particularly allows to address the important issue of the effective target and nuisance dimension and it does not involve any pilot estimator of the target parameter. The obtained nonasymptotic results are surprisingly sharp and yield the classical asymptotic statements including the asymptotic normality and efficiency of the profile MLE. The general results are specified to the important special cases of an i.i.d. sample.

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# Geometric error of finite volume schemes for conservation laws on evolving surfaces

*Authors*

- Giesselmann, Jan
- Müller, Thomas

*2010 Mathematics Subject Classification*

- 65M08 35L65 58J45

*Keywords*

- hyperbolic conservation laws, finite volume schemes, curved surfaces, error bound

*Abstract*

This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of $mathbbR^3$. We compare theoretical schemes assuming knowledge of all geometric quantities to (practical) schemes defined on moving polyhedra approximating the surface. For the former schemes error estimates have already been proven, but the implementation of such schemes is not feasible for complex geometries. The latter schemes, in contrast, only require (easily) computable geometric quantities and are thus more useful for actual computations. We prove that the difference between approximate solutions defined by the respective families of schemes is of the order of the mesh width. In particular, the practical scheme converges to the entropy solution with the same rate as the theoretical one. Numerical experiments show that the proven order of convergence is optimal.

*Appeared in*

- Numerische Mathematik 128 (2014), pp. 489--516.

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# Lagrange method in shape optimization for non-linear partial differential equations: A material derivative free approach

*Authors*

- Sturm, Kevin

*2010 Mathematics Subject Classification*

- 49Q10 49Q12

*Keywords*

- Lagrange approach, shape derivative, non-linear partial differential equations, material derivative

*Abstract*

This paper studies the relationship between the material derivative method, the shape derivative method, the min-max formulation of Correa and Seeger, and the Lagrange method introduced by Céa. A theorem is formulated which allows a rigorous proof of the shape differentiability without the usage of material derivative; the domain expression is automatically obtained and the boundary expression is easy to derive. Furthermore, the theorem is applied to a cost function which depends on a quasi-linear transmission problem. Using a Gagliardo penalization the existence of optimal shapes is established.

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# A numerical investigation of velocity-pressure reduced order models for incompressible flows

*Authors*

- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Iliescu, Traian
- John, Volker

ORCID: 0000-0002-2711-4409 - Schyschlowa, Swetlana

*2010 Mathematics Subject Classification*

- 76D05

*Keywords*

- Navier--Stokes equations, proper orthogonal decomposition, velocity-pressure reduced order models, snapshot accuracy

*Abstract*

This report has two main goals. First, it numerically investigates three velocity-pressure reduced order models (ROMs) for incompressible flows. The proper orthogonal decomposition (POD) is used to generate the modes. One method computes the ROM pressure solely based on the velocity POD modes, whereas the other two ROMs use pressure modes as well. To the best of the authors' knowledge, one of the latter methods is novel. The second goal is to numerically investigate the impact of the snapshot accuracy on the ROMs accuracy. Numerical studies are performed on a two-dimensional laminar flow past a circular obstacle. It turns out that, both in terms of accuracy and efficiency, the two ROMs that utilize pressure modes are clearly superior to the ROM that uses only velocity modes. The numerical results also show a strong correlation of the accuracy of the snap shots with the accuracy of the ROMs.

*Appeared in*

- J. Comput. Phys., 259 (2014) pp. 598--616.

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# Equilibrium shapes of poly-crystalline silicon nanodots

*Authors*

- Korzec, Maciek D.
- Roczen, Maurizio
- Schade, Martin
- Wagner, Barbara
- Rech, Bernd

*2010 Mathematics Subject Classification*

- 37D35 74E10 49Q10

*2008 Physics and Astronomy Classification Scheme*

- 61.50.Ah

*Keywords*

- equilibrium shapes, anisotropic surface energy, adhesion energy, constrained optimization, Wulff construction, silicon nanodots, TEM

*Abstract*

This study is concerned with the topography of nanostructures consisting of arrays of poly-crystalline nanodots. Guided by transmission electron microscopy (TEM) measurements of crystalline Si (c-Si) nanodots that evolved from a 'dewetting' process of an amorphous Si (a-Si) layer from a SiO$_2$ coated substrate, we investigate appropriate formulations for the surface energy density and transitions of energy density states at grain boundaries. We introduce a new numerical minimization formulation that allows to account for adhesion energy from an underlying substrate. We demonstrate our approach first for the free standing case, where the solutions can be compared to well-known Wulff constructions, before we treat the general case for interfacial energy settings that support 'partial wetting'. We then use our method to predict the morphologies of poly-crystalline silicon nanodots.

*Appeared in*

- Journal of Applied Physics, 115 (2014) pp. 074304/1--074304/12.

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# Electronic states in a quantum well -- nanobridge -- quantum dot structure

*Authors*

- Racec, Paul N.
- Goray, Leonid I.

*2010 Mathematics Subject Classification*

- 65N30 65Z05 35P99

*2008 Physics and Astronomy Classification Scheme*

- 73.22.-f 71.15.-m 78.67.-n

*Keywords*

- Nanobridge, hybrid bound states, optical matrix elements, cylindrical nanowire

*Abstract*

Using the finite volume method we compute within effective mass approximation the single-particle eigenstates for electrons and holes in a InGaAs/GaAs quantum well -- nanobridge -- quantum dot structure. It is shown that hybrid states appear in this complex system. The interaction between the eigenvalues may be an explanation for the additional photoluminescence peak measured for inverted structures with smaller nanobridge lengths.

*Appeared in*

- Proceeding of the International Conference ``Days of Diffraction 2014'', O.V. Motygin, A.P. Kiselev, L.I. Goray, A.Y. Kazakov, A.S. Kirpichnikova, eds., IEEE, Danvers (USA), 2014, pp. 89--95

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# A compressible mixture model with phase transition

*Authors*

- Dreyer, Wolfgang
- Giesselmann, Jan
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 35C20 35R35 76T10 76T30 35Q30 35Q35 76D45 76N10 76T99 80A22 82B26

*Keywords*

- Multi-component flow, phase transition, asymptotic analysis, sharp interface limit, free boundary problems, Allen-Cahn equation, Euler system

*Abstract*

We introduce a new thermodynamically consistent diffuse interface model of Allen-Cahn/Navier-Stokes type for multi-component flows with phase transitions and chemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized Allen-Cahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a Young-Laplace and a Stefan type law.

*Appeared in*

- Phys. D, 273-274 (2014) pp. 1--13.

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# Stability analysis of non-constant base states in thin film equations

*Authors*

- Dziwnik, Marion
- Korzec, Maciek D.
- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 35B40 35C20 76M45

*2008 Physics and Astronomy Classification Scheme*

- 81.07.-b

*Keywords*

- multiple-scale methods, stability analysis, rim instability, free boundaries, dewetting film

*Abstract*

We address the linear stability of non-constant base states within the class of mass conserving free boundary problems for degenerate and non-degenerate thin film equations. Well-known examples are the finger-instabilities of growing rims that appear in retracting thin solid and liquid films. Since the base states are time dependent and do not have a simple travelling wave or self-similar form, a classical eigenvalue analysis fails to provide the dominant wavelength of the instability. However, the initial fronts evolve on a slower time-scale than the typical perturbations. We exploit this time-scale separation and develop a multiple-scale approach for this class of stability problems. We show that the value of the dominant wavelength is rapidly attained once the base state has entered an approximately self-similar scaling. We note that this value is different from the one obtained by the linear stability analysis with "frozen modes", frequently found in the literature. Furthermore we show that for the present class of stability problems the dispersion relation behaves linear for large wavelengths, which is in contrast to many other instability problems in thin film flows.

*Appeared in*

- with the title : ``Stability analysis of unsteady, nonuniform base states in thin film equations'', Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 12 (2014) pp. 755--780.

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# A sequence of order relations, encoding heteroclinic connections in scalar parabolic PDE

*Authors*

- Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 35K57 37L30 35B41 34C37

*Keywords*

- calar semilinear parabolic PDE, order structures; attractors, heteroclinic connections, meandric permutations, nodal properties

*Abstract*

We address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations

u_{t} = u_{xx} + ƒ (x, u, u_{x}), 0 < x < 1

on a bounded interval with Neumann conditions. Introducing a sequence of order relations, we prove a new and simple criterion for the existence of heteroclinic connections, using only information about nodal properties of solutions to the stationary ODE problem. This result allows also for a complete classiffication of possible attractors in terms of the permutation of the equilibria, given by their order at the two boundaries of the interval.

*Appeared in*

- J. Differential Equations, 183, (2002) pp. 56-78

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# A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes

*Authors*

- Eigel, Martin
- Gittelson, Claude Jeffrey
- Schwab, Christoph
- Zander, Elmar

*2010 Mathematics Subject Classification*

- 65N30

*Keywords*

- generalized polynomial chaos, adaptive Finite Element Methods, contraction property, residual a-posteriori error estimation, uncertainty quantification

*Abstract*

We analyze a-posteriori error estimation and adaptive refinement algorithms for stochastic Galerkin Finite Element methods for countably-parametric elliptic boundary value problems. A residual error estimator which separates the effects of gpc-Galerkin discretization in parameter space and of the Finite Element discretization in physical space in energy norm is established. It is proved that the adaptive algorithm converges, and to this end we establish a contraction property satisfied by its iterates. It is shown that the sequences of triangulations which are produced by the algorithm in the FE discretization of the active gpc coefficients are asymptotically optimal. Numerical experiments illustrate the theoretical results.

*Appeared in*

- ESAIM Math. Model. Numer. Anal., 49 (2015) pp. 1367--1398.

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# Bifurcations in the Sakaguchi--Kuramoto model

*Authors*

- Omel'chenko, Oleh

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

*2010 Mathematics Subject Classification*

- 34C15 37N20 37N25

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt 89.79.Kd

*Keywords*

- synchronization, coupled oscillators, Sakaguchi-Kuramoto model, Ott-Antonsen reduction

*Abstract*

We analyze the Sakaguchi-Kuramoto model of coupled phase oscillators in a continuum limit given by a frequency dependent version of the Ott-Antonsen system. Based on a self-consistency equation, we provide a detailed analysis of partially synchronized states, their bifurcation from the completely incoherent state and their stability properties. We use this method to analyze the bifurcations for various types of frequency distributions and explain the appearance of non-universal synchronization transitions.

*Appeared in*

- Phys. D, 263 (2013) pp. 74--85.

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# Probing nonlinear adiabatic paths with a universal integrator

*Authors*

- Hofmann, Michael
- Schaller, Gernot

*2008 Physics and Astronomy Classification Scheme*

- 03.67.Ac 75.10.Nr 75.10.Dg 02.60.-x

*Keywords*

- Quantum algorithms and protocols, Spin Hamiltonians, Numerical methods

*Abstract*

We apply a flexible numerical integrator to the simulation of adiabatic quantum computation with nonlinear paths. We find that a nonlinear path may significantly improve the performance of adiabatic algorithms versus the conventional straight-line interpolations. The employed integrator is suitable for solving the time-dependent Schrödinger equation for any qubit Hamiltonian. Its flexible storage format significantly reduces cost for storage and matrix-vector multiplication in comparison to common sparse matrix schemes.

*Appeared in*

- Phys. Rev. A, 89 (2014) pp. 032308/1--032308/8.

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# Direct and inverse acoustic scattering by a collection of extended and point-like scatterers

*Authors*

- Hu, Guanghui
- Mantile, Andrea
- Sini, Mourad

*2010 Mathematics Subject Classification*

- 35R30 78A45 35J05 35J25 35J57 35L05

*Keywords*

- Inverse scattering, point interaction, two-scale problem, factorization method

*Abstract*

We are concerned with the acoustic scattering by an extended obstacle surrounded by point-like obstacles. The extended obstacle is supposed to be rigid while the point-like obstacles are modeled by point perturbations of the exterior Laplacian. In the first part, we consider the forward problem. Following two equivalent approaches (the Foldy formal method and the Krein resolvent method), we show that the scattered field is a sum of two contributions: one is due to the diffusion by the extended obstacle and the other arises from the linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one. In the second part, we deal with the inverse problem. It consists in reconstructing both the extended and point-like scatterers from the corresponding far-field pattern. To solve this problem, we describe and justify the factorization method of Kirsch. Using this method, we provide several numerical results and discuss the multiple scattering effect concerning both the interactions between the point-like obstacles and between these obstacles and the extended one.

*Appeared in*

- Multiscale Model. Simul., 12 (2014) pp. 996--1027.

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# Influence of the carrier reservoir dimensionality on electron-electron scattering in quantum dot materials

*Authors*

- Wilms, Alexander
- Mathé, Peter

ORCID: 0000-0002-1208-1421 - Schulze, Franz
- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Knorr, Andreas
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2010 Mathematics Subject Classification*

- 81V65 65C05

*2008 Physics and Astronomy Classification Scheme*

- 73.21.La 73.63.Kv.

*Keywords*

- quantum dots, Coulomb scattering, quasi-Monte Carlo

*Abstract*

We calculated Coulomb scattering rates from quantum dots (QDs) coupled to a 2D carrier reservoir and QDs coupled to a 3D reservoir. For this purpose, we used a microscopic theory in the limit of Born-Markov approximation, in which the numerical evaluation of high dimensional integrals is done via a quasi-Monte Carlo method. Via a comparison of the so determined scattering rates, we investigated the question whether scattering from 2D is generally more efficient than scattering from 3D. In agreement with experimental findings, we did not observe a significant reduction of the scattering efficiency of a QD directly coupled to a 3D reservoir. In turn, we found that 3D scattering benefits from it?s additional degree of freedom in the momentum space.

*Appeared in*

- Phys. Rev. B., 88 (2013) pp. 235421/1--235421/11.

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# Homogenization of the nonlinear bending theory for plates

*Authors*

- Neukamm, Stefan
- Olbermann, Heiner

*2010 Mathematics Subject Classification*

- 74B20 74Q05 49Q10 74K20

*Keywords*

- homogenization, Kirchhoff plate theory, two-scale convergence, nonlinear differential constraint

*Abstract*

We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Gamma-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a "cylinder" or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions, using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.

*Appeared in*

- Calc. Var. Partial Differ. Equ., (published online on Sept. 14, 2014) pp. , DOI 10.1007/s00526-014-0765-2 .

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# Existence and asymptotic stability of a periodic solution with boundary layers of reaction-diffusion equations with singularly perturbed Neumann boundary conditions

*Authors*

- Butuzov, Valentin F.
- Nefedov, Nikolai N.
- Recke, Lutz
- Schneider, Klaus

*2010 Mathematics Subject Classification*

- 35B25 35B35 35B12 35K57

*Keywords*

- singularly perturbed first order ordinary differential equation, initial value problem, boundary layer, double root of degenerate equation, asymptotic expansion

*Abstract*

We consider singularly perturbed reaction-diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution $u(x,t,ve)$ with boundary layers and derive conditions for their asymptotic stability The boundary layer part of $u(x,t,ve)$ is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order $ve$. Another peculiarity of our problem is that - in contrast to the case of Dirichlet boundary conditions - it may have several asymptotically stable time-periodic solutions, where these solutions differ only in the desribtion of the boundary layers. Our approach is based on the construction of sufficiently precise lower and upper solutions

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# Asymptotics for at the money local vol basket options

*Authors*

- Bayer, Christian

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

*2010 Mathematics Subject Classification*

- 91G60 91G20 58J90

*Keywords*

- Basket options, spread options, implied volatility, asymptotic formulas, heat kernel expansion

*Abstract*

We consider a basket or spread option on based on a multi-dimensional local volatility model. Bayer and Laurence [Comm. Pure. Appl. Math., to appear] derived highly accurate analytic formulas for prices and implied volatilities of such options when the options are not at the money. We now extend these results to the ATM case. Moreover, we also derive similar formulas for the local volatility of the basket.

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# Minimization of a fractional perimeter-Dirichlet integral functional

*Authors*

- Caffarelli, Luis
- Savin, Ovidiu
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35B65 49Q05 31A05 49Q15

*Keywords*

- regularity results, free boundary problems, monotonicity formulas

*Abstract*

We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set. We obtain regularity results for the minimizers and for their free boundaries using blow-up analysis, density estimates, monotonicity formulas, Euler-Lagrange equations and extension problems.

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# Partially coherent twisted states in arrays of coupled phase oscillators

*Authors*

- Omel'chenko, Oleh

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

*2010 Mathematics Subject Classification*

- 34C15 37N20 37N25

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt 89.75.Kd

*Keywords*

- coupled oscillators, Kuramoto model, twisted states, Ott/Antonsen, Eckhaus bifurcation

*Abstract*

We consider a one-dimensional array of phase oscillators with non-local coupling and a Lorentzian distribution of natural frequencies. The primary objects of interest are partially coherent states that are uniformly "twisted" in space. To analyze these we take the continuum limit, perform an Ott/Antonsen reduction, integrate over the natural frequencies and study the resulting spatio-temporal system on an unbounded domain. We show that these twisted states and their stability can be calculated explicitly. We find that stable twisted states with different wave numbers appear for increasing coupling strength in the well-known Eckhaus scenario. Simulations of finite arrays of oscillators show good agreement with results of the analysis of the infinite system.

*Appeared in*

- Chaos, 24 (2014) pp. 023102/1--023102/9.

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# Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators

*Authors*

- Omel'chenko, Oleh

ORCID: 0000-0003-0526-1878

*2010 Mathematics Subject Classification*

- 34C15 35B36 5B32 35B42 35Q83

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Xt, 89.75.Kd

*Keywords*

- coupled phase oscillators, non-local coupling, coherence-incoherence patterns, chimera states, bifurcation analysis, pattern formation

*Abstract*

We consider a paradigmatic spatially extended model of non-locally coupled phase oscillators which are uniformly distributed within a one-dimensional interval and interact depending on the distance between their sites modulo periodic boundary conditions. This model can display peculiar spatio-temporal patterns consisting of alternating patches with synchronized (coherent) or irregular (incoherent) oscillator dynamics, hence the name coherence-incoherence pattern, or chimera state. For such patterns we formulate a general bifurcation analysis scheme based on a hierarchy of continuum limit equations. This gives us possibility to classify known coherence-incoherence patterns and to suggest directions for searching new ones.

*Appeared in*

- Nonlinearity, 26 (2013) pp. 2469--2498.

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# Strongly nonlocal dislocation dynamics in crystals

*Authors*

- Dipierro, Serena
- Figalli, Alessio
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 49N60 35B05 35Q99 35B40 35J25 35D30 35G25

*Keywords*

- nonlocal Peierls-Nabarro model, dislocation dynamics, fractional Laplacian, oscillation and regularity results

*Abstract*

We consider an equation motivated by crystal dynamics and driven by a strongly nonlocal elliptic operator of fractional type. We study the evolution of the dislocation function for macroscopic space and time scales, by showing that the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. We also prove that the motion of these dislocation points is governed by an interior repulsive potential that is superposed to an elastic reaction to the external stress.

*Appeared in*

- Comm. Partial Differential Equations, 39 (2014) pp. 2351--2387.

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# Dislocation dynamics in crystals: A macroscopic theory in a fractional Laplace setting

*Authors*

- Dipierro, Serena
- Palatucci, Giampiero
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35Q99 35B40 35J25 35D30 35G25 70F99

*Keywords*

- nonlinear problems, nonlocal Allen-Cahn equation, reaction-diffusion, Peierls--Nabarro model, dislocation dynamics, particle systems, fractional Laplacian, fractional Sobolev spaces

*Abstract*

We consider an evolution equation arising in the Peierls--Nabarro model for crystal dislocation. we study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. these dislocation points evolve according to the external stress and an interior repulsive potential

*Appeared in*

- Comm. Math. Phys., 333 (2015) pp. 1061--1105.

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# Closed-loop optimal experiment design: Solution via moment extension

*Authors*

- Hildebrand, Roland
- Gevers, Michel
- Solari, Gabriel

*2010 Mathematics Subject Classification*

- 93E12

*Keywords*

- Optimal experiment design, Closed-loop identification, Convex programming, Power spectral density, Moment method

*Abstract*

We consider optimal experiment design for parametric prediction error system identification of linear time-invariant multiple-input multiple-output (MIMO) systems in closed-loop when the true system is in the model set. The optimization is performed jointly over the controller and the spectrum of the external excitation, which can be reparametrized as a joint spectral density matrix. We have shown in [18] that the optimal solution consists of first computing a finite set of generalized moments of this spectrum as the solution of a semi-definite program. A second step then consists of constructing a spectrum that matches this finite set of optimal moments and satisfies some constraints due to the particular closed-loop nature of the optimization problem. This problem can be seen as a moment extension problem under constraints. Here we first show that the so-called central extension always satisfies these constraints, leading to a constructive procedure for the optimal controller and excitation spectrum.We then show that, using this central extension, one can construct a broader set of parametrized optimal solutions that also satisfy the constraints; the additional degrees of freedom can then be used to achieve additional objectives. Finally, our new solution method for the MIMO case allows us to considerably simplify the proofs given in [18] for the single-input single-output case.

*Appeared in*

- IEEE Trans. Autom. Control, 60 (2015) pp. 1731--1744.

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# Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails

*Authors*

- Biskup, Marek
- König, Wolfgang

*2010 Mathematics Subject Classification*

- 60F05 60G55 70H40 70B80

*Keywords*

- parabolic Anderson model, random Schrödinger operator, eigenvalue order statistics, Poisson point process convergence, Anderson localisation

*Abstract*

We consider random Schrödinger operators of the form Δ + ξ, where $Delta; is the lattice Laplacian on *Z*^{d} and ξ is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of *Z*^{d}. We show that for ξ with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where ξ takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem.

*Appeared in*

- Comm. Math. Phys., 341 (2016) pp. 179--218.

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# Uniqueness and nondegeneracy of positive solutions of $(-Delta)^s u+u = u^p$ in $R^N$ when $s$ is close to 1

*Authors*

- Fall, Mouhamed Moustapha
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 26A33 35A15 35B40

*Keywords*

- Fractional Laplacian, uniqueness results, nondegeneracy of minimizers, asymptotic methods

*Abstract*

We consider the equation (-Δ)^{s} *u+u = u ^{p}* with

*s*∈ (0,1) in the subcritical range of

*p*. We prove that if

*s*is sufficiently close to 1 the equation possesses a unique minimizer, which is nondegenerate.

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# Pathwise stability of likelihood estimators for diffusions via rough paths

*Authors*

- Diehl, Joscha
- Friz, Peter

ORCID: 0000-0003-2571-8388 - Mai, Hilmar

*2010 Mathematics Subject Classification*

- 62F35 62M99

*Keywords*

- rough paths, MLE for diffusions, pathwise stability, model misspecification

*Abstract*

We consider the estimation problem of an unknown drift parameter within classes of non-degenerate diffusion processes. The Maximum Likelihood Estimator (MLE) is analyzed with regard to its pathwise stability properties and robustness towards misspecification in volatility and even the very nature of noise. We construct a version of the estimator based on rough integrals (in the sense of T. Lyons) and present strong evidence that this construction resolves a number of stability issues inherent to the standard MLEs.

*Appeared in*

- The Annals of Probalility, 26(2016) pp. 2169--2192

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# Ultrashort optical solitons in transparent nonlinear media with arbitrary dispersion

*Authors*

- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Bandelow, Uwe

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

*2010 Mathematics Subject Classification*

- 78A60 35Q51 35Q55

*2008 Physics and Astronomy Classification Scheme*

- 42.81.Dp 42.65.Tg 05.45.Yv 42.65.Re

*Keywords*

- Ultrashort pulses, Nonlinear Schrödinger Equation, Solitons

*Abstract*

We consider the propagation of ultrashort optical pulses in nonlinear fibers and suggest a new theoretical framework for the description of pulse dynamics and exact characterization of solitary solutions. Our approach deals with a proper complex generalization of the nonlinear Maxwell equations and completely avoids the use of the slowly varying envelope approximation. The only essential restriction is that fiber dispersion does not favor both the so-called Cherenkov radiation, as well as the resonant generation of the third harmonics, as these effects destroy ultrashort solitons. Assuming that it is not the case, we derive a continuous family of solitary solutions connecting fundamental solitons to nearly single-cycle ultrashort ones for arbitrary anomalous dispersion and cubic nonlinearity.

*Appeared in*

- Opt. Quantum Electron., (2013) pp. .

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# On non-asymptotic optimal stopping criteria in Monte Carlo simulations

*Authors*

- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Hoel, Håkon
- von Schwerin, Erik
- Tempone, Raúl

*2010 Mathematics Subject Classification*

- 65C05 62L12 62L15

*Keywords*

- Monte Carlo methods, optimal stopping, sequential stopping rules, non-asymptotic

*Abstract*

We consider the setting of estimating the mean of a random variable by a sequential stopping rule Monte Carlo (MC) method. The performance of a typical second moment based sequential stopping rule MC method is shown to be unreliable in such settings both by numerical examples and through analysis. By analysis and approximations, we construct a higher moment based stopping rule which is shown in numerical examples to perform more reliably and only slightly less efficiently than the second moment based stopping rule.

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# Continuum thermodynamics of chemically reacting fluid mixtures

*Authors*

- Bothe, Dieter
- Dreyer, Wolfgang

*2010 Mathematics Subject Classification*

- 76T10 35Q35 35Q80 76A02 80A17 80A32 92E20

*Keywords*

- partial balances, entropy principle, parity and time reversal, Onsager reciprocity relations, reactive flows, multicomponent diffusion, Maxwell-Stefan equations, generalized driving forces, incompressible mixture, constitutive theory, mixture free energy

*Abstract*

We consider viscous and heat conducting mixtures of molecularly miscible chemical species forming a fluid in which the constituents can undergo chemical reactions. Assuming a common temperature for all components, a first main aim is the derivation of a closed system of partial mass and partial momentum balances plus a common balance of internal energy. This is achieved by careful exploitation of the entropy principle which, in particular, requires appropriate definitions of absolute temperature and chemical potentials based on an adequate definition of thermal energy that excludes diffusive contributions. The latter is crucial in order to obtain a closure framework for the interaction forces between the different species. The interaction forces split into a thermo-mechanical and a chemical part, where the former turns out to be symmetric if binary interactions are assumed. In the non-reactive case, this leads to a system of Navier-Stokes type sub-systems, coupled by interspecies friction forces. For chemically reacting systems and as a new result, the chemical interaction force is identified as a contribution which is non-symmetric, unless chemical equilibrium holds. The theory also provides a rigorous derivation of the so-called generalized thermodynamic driving forces, avoiding the use of approximate solutions to the Boltzmann equations which is common in the engineering literature. Moreover, starting with a continuum thermodynamic field theory right away, local versions of fundamental relations known from thermodynamics of homogeneous systems, like the Gibbs-Duhem equation, are derived. Furthermore, using an appropriately extended version of the entropy principle and introducing cross-effects already before closure as entropy invariant couplings between principal dissipative mechanisms, the Onsager symmetry relations are a strict consequence. With a classification of the factors forming the binary products in the entropy production according to their parity instead of the classical distinction between so-called fluxes and driving forces, the apparent anti-symmetry of certain couplings is thereby also revealed. If the diffusion velocities are small compared to the speed of sound, the well-known Maxwell-Stefan equations together with the so-called generalized thermodynamic driving forces follow in the special case without chemical reactions, thereby neglecting wave phenomena in the diffusive motion. This results in a reduced model having only the constituents' mass balances individually. In the reactive case, this approximation via a scale separation argument is no longer possible. Instead, we first employ the partial mass and mixture internal energy balances, common to both model classes, to identify all constitutive quantities. Combined with the concept of entropy invariant model reduction, leaving the entropy production unchanged under the reduction from partial momentum balances to a single common mixture momentum balance, the chemical interactions yield an additional contribution to the transport coefficients, leading to an extension of the Maxwell-Stefan equations to chemically active mixtures. Within the considered model class for reactive fluid mixtures the new results are achieved for arbitrary free energy functions.

*Appeared in*

- Acta Mech., 226 (2015) pp. 1757--1805.

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# A simple formula for the second-order subdifferential of maximum functions

*Authors*

- Emich, Konstantin
- Henrion, René

*2010 Mathematics Subject Classification*

- 49J52 49J53

*Keywords*

- second-order subdifferential, extended partial second-order subdifferential, maximum function, calculus rules

*Abstract*

We derive a simple formula for the second-order subdifferential of the maximum of coordinates which allows us to construct this set immediately from its argument and the direction to which it is applied. This formula can be combined with a chain rule recently proved by Mordukhovich and Rockafellar [9] in order to derive a similarly simple formula for the extended partial second-order subdifferential of finite maxima of smooth functions. Analogous formulae can be derived immediately for the full and conventional partial second-order subdifferentials.

*Appeared in*

- Vietnam J. Math., 42 (2014) pp. 467--478.

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# Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Reichelt, Sina
- Thomas, Marita

*2010 Mathematics Subject Classification*

- 35B25 35A01 35K57 35K65 35M10

*Keywords*

- Two-scale convergence, folding and unfolding, coupled reaction-diffusion equations, nonlinear reaction, degenerating diffusion, Gronwall estimate

*Abstract*

We derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit.

*Appeared in*

- Networks Heterogeneous Media, 9 (2014) pp. 353--382.

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# Large deviations for the local times of a random walk among random conductances in a growing box

*Authors*

- König, Wolfgang
- Wolff, Tilman

*2010 Mathematics Subject Classification*

- 60K37 60J65 60J55 60F10

*Keywords*

- random conductances, random walk, randomised Laplace operator, local times, large deviations, Donsker-Varadhan-Gärtner theory, spectral homogenisation, Lifshitz tails

*Abstract*

We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in *Z ^{d}*. We work in the interesting case that the conductances are positive, but may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small conductance values and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution as well as the time-dependent size of the box.

An interesting phase transition occurs if the thickness parameter of the conductance tails exceeds a certain threshold: for thicker tails, the random walk spreads out over the entire growing box, for thinner tails it stays confined to some bounded region. In fact, in the first case, the rate function turns out to be equal to the

*p*-th power of the

*p*-norm of the gradient of the square root for some

*2d/(d+2) < p < 2*. This extends the Donsker-Varadhan-Gärtner rate function for the local times of Brownian motion (with deterministic environment) from

*p=2*to these values.

As corollaries of our LDP, we derive the logarithmic asymptotics of the non-exit probability of the RWRC from the growing box, and the Lifshitz tails of the generator of the RWRC, the randomised Laplace operator. To contrast with the annealed, not uniformly elliptic case, we also provide an LDP in the quenched setting for conductances that are bounded and bounded away from zero. The main tool here is a spectral homogenisation result, based on a quenched invariance principle for the RWRC.

*Appeared in*

- Markov Process. Related Fields, 21 (2015) pp. 591--638.

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# Uncertainty quantification for the family-wise error rate in multivariate copula models

*Authors*

- Stange, Jens
- Bodnar, Taras
- Dickhaus, Thorsten

*2010 Mathematics Subject Classification*

- 62J15 62F05 62F03

*Keywords*

- Delta method, Gumbel-Hougaard copula, multiple testing, simultaneous test procedure, subset pivotality

*Abstract*

We derive confidence regions for the realized family-wise error rate (FWER) of certain multiple tests which are empirically calibrated at a given (global) level of significance. To this end, we regard the FWER as a derived parameter of a multivariate parametric copula model. It turns out that the resulting confidence regions are typically very much concentrated around the target FWER level, while generic multiple tests with fixed thresholds are in general not FWER-exhausting. Since FWER level exhaustion and optimization of power are equivalent for the classes of multiple test problems studied in this paper, the aforementioned findings militate strongly in favour of estimating the dependency structure (i. e., copula) and incorporating it in a multivariate multiple test procedure. We illustrate our theoretical results by considering two particular classes of multiple test problems of practical relevance in detail, namely, multiple tests for components of a mean vector and multiple support tests.

*Appeared in*

- AStA Advances in Statistical Analysis,,issue ( ) pp. --

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# Rational modeling of electrochemical double-layers and derivation of Butler--Volmer equations

*Authors*

- Dreyer, Wolfgang
- Guhlke, Clemens
- Müller, Rüdiger

ORCID: 0000-0003-2643-722X

*2010 Mathematics Subject Classification*

- 35Q35 76T30 35C20

*2008 Physics and Astronomy Classification Scheme*

- 82.45.Gj, 82.45.Fk

*Keywords*

- electrolyte, double-layer, Butler-Volmer, thermodynamics, asymptotic analysis

*Abstract*

We derive the boundary conditions for the contact between an electrolyte and a solid electrode. At first we revisit the thermodynamic consistent complete model that resolves the actual electrode--electrolyte interface and its adjacent boundary layers. The width of these layers is controlled by the Debye length that is typically very small, leading to strongly different length scales in the system. We apply the method of asymptotic analysis to derive a simpler reduced model that does not resolve the boundary layers but instead incorporates the electrochemical properties of the layers into a set of new boundary conditions. This approach fully determines the relation of bulk quantities to the boundary conditions of the reduced model. In particular, the Butler-Volmer equations for electrochemical reactions, which are still under discussion in the literature, are rational consequences of our approach. For illustration and to compare with the literature, we consider a simple generic reaction.

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# Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model

*Authors*

- Giesselmann, Jan
- Pryer, Tristan

*2010 Mathematics Subject Classification*

- 65M12 65M60 76T99 76D45

*Keywords*

- Quasi-incompressibility, Allen--Cahn, Cahn--Hilliard, Navier--Stokes--Korteweg, phase transition, energy consistent/mimetic, discontinuous Galerkin finite element method

*Abstract*

We design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of Allen--Cahn/Cahn--Hilliard/Navier--Stokes--Korteweg type which allows for phase transitions. We show that the scheme is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to discrete equivalents of those effects already causing dissipation on the continuous level, that is, there is no artificial numerical dissipation added into the scheme. In this sense the methods are consistent with the energy dissipation of the continuous PDE system.

*Appeared in*

- Mathematical Modeling and Numerical Analysis M2AN, 49(1) (2015), pp. 275--301.

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# Forward-reverse EM algorithm for Markov chains

*Authors*

- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Mai, Hilmar
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 65C05 60J20

*Keywords*

- forward-reverse representations, EM algorithm, Monte Carlo simulation, maximum likelihood estimation, Markov chain estimation

*Abstract*

We develop an EM algorithm for estimating parameters that determine the dynamics of a discrete time Markov chain evolving through a certain measurable state space. As a key tool for the construction of the EM method we also develop forward-reverse representations for Markov chains conditioned on a certain terminal state. These representations may be considered as an extension of the earlier work of Bayer and Schoenmakers (2013) on conditional diffusions. We present several experiments and consider the convergence of the new EM algorithm.

*Appeared in*

- Adv. Appl. Probab., 2 (2018), pp. 621--644 , under the new title: Forward-reverse expectation-maximization algorithm for Markov chains: Convergence and numerical analysis, DOI 10.1017/apr.2018.27 .

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# Saturation of the all-optical Kerr effect in solids

*Authors*

- Borchers, Bastian
- Brée, Carsten
- Birkholz, Simon
- Demircan, Ayhan
- Steinmeyer, Günter

*2010 Mathematics Subject Classification*

- 78A60

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k 42.65.An 42.65.Jx 42.65.Hw

*Keywords*

- Nonlinear Optics, All-optical Kerr effect, Pulse compression

*Abstract*

We discuss the influence of the higher-order Kerr effect (HOKE) in wide band gap solids at extreme intensities below the onset of optically induced damage. Using different theo- retical models, we employ multiphoton absorption rates to compute the nonlinear refractive index by a Kramers-Kronig transform. Within this theoretical framework we provide an esti- mate for the appearance of significant deviations from the standard optical Kerr effect pre- dicting a linear index change with intensity. We discuss the role of the observed saturation behavior in practically relevant situations, including Kerr lens mode-locking and supercon- tinuum generation in photonic crystal fibers. Furthermore we present experimental data from a multi-wave mixing experiment in BaF2 which can be explained by the appearance of the HOKE.

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# The Cayley transform applied to non-interacting quantum transport

*Authors*

- Cornean, Horia
- Neidhardt, Hagen
- Wilhelm, Lukas
- Zagrebnov, Valentin

*2010 Mathematics Subject Classification*

- 47A40 47A55 81Q37 81V80

*Keywords*

- Landauer-Büttiker formula, dissipative Schrödinger operators, self-adjoint dilations, Dirac operators

*Abstract*

We extend the Landauer-Büttiker formalism in order to accommodate both unitary and self-adjoint operators which are not bounded from below. We also prove that the pure point and singular continuous subspaces of the decoupled Hamiltonian do not contribute to the steady current. One of the physical applications is a stationary charge current formula for a system with four pseudo-relativistic semi-infinite leads and with an inner sample which is described by a Schrödinger operator defined on a bounded interval with dissipative boundary conditions. Another application is a current formula for electrons described by a one dimensional Dirac operator; here the system consists of two semi-infinite leads coupled through a point interaction at zero.

*Appeared in*

- J. Funct. Anal., 266 (2014) pp. 1421--1475.

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# An approach to nonlinear viscoelasticity via metric gradient flows

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Ortner, Christoph
- Şengül, Yasemin

*2010 Mathematics Subject Classification*

- 74D10 35A15 35Q74 37L05 53C22

*Keywords*

- Nonlinear viscoelasticity, gradient flow, dissipative distance, generalized geodesics

*Abstract*

We formulate quasistatic nonlinear finite-strain viscoelasticity of rate-type as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure time-dependent frame-indifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the time-incremental problem. Because of the missing compactness the limit of vanishing timesteps can only be obtained by proving some kind of strong convergence. We show that this is possible in the one-dimensional case by using a suitably generalized convexity in the sense of geodesic convexity of gradient flows. For a general class of distances we derive discrete evolutionary variational inequalities and are able to pass to the time-continuous in some case in a specific case.

*Appeared in*

- SIAM J. Math. Anal., 46 (2014) pp. 1317--1347.

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# A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints

*Authors*

- Arnold, Thomas
- Henrion, René
- Möller, Andris
- Vigerske, Stefan

*2010 Mathematics Subject Classification*

- 90B05 90C15 90C10

*Keywords*

- stochastic optimization, probabilistic constraints, mixed-integer nonlinear programming, power management

*Abstract*

We illustrate the solution of a mixed-integer stochastic nonlinear optimization problem in an application of power management. In this application, a coupled system consisting of a hydro power station and a wind farm is considered. The objective is to satisfy the local energy demand and sell any surplus energy on a spot market for a short time horizon. Generation of wind energy is assumed to be random, so that demand satisfaction is modeled by a joint probabilistic constraint taking into account the multivariate distribution. The turbine is forced to either operate between given positive limits or to be shut down. This introduces additional binary decisions. The numerical solution procedure is presented and results are illustrated.

*Appeared in*

- Pac. J. Optim., 10 (2014) pp. 5--20.

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# On a reduced sparsity stabilization of grad-div type for incompressible flow problems

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Rebholz, Leo

*2010 Mathematics Subject Classification*

- 76D07 65N30

*Keywords*

- grad-div stabilization, mass conservation, Navier-Stokes equations, modified pressure

*Abstract*

We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator's matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar.

*Appeared in*

- Comput. Methods Appl. Mech. Engrg., 261--262 (2013) pp. 142--153.

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# Beam shaping mechanism in spatially modulated edge emitting broad area semiconductor amplifiers

*Authors*

- Radziunas, Mindaugas
- Botey, Muriel
- Herrero, Ramon
- Staliunas, Kestutis

*2010 Mathematics Subject Classification*

- 35Q60 35B27 37M05 78A60 78A45

*Keywords*

- broad area semiconductor amplifiers, periodic modulation, spatial modulation, angular filtering, beam shaping, far field, quality improvement, traveling wave model

*Abstract*

We investigate beam shaping in broad area semiconductor amplifiers induced by a periodic modulation of the pump on a scale of several microns. The study is performed by solving numerically a (2+1)-dimensional model for the semiconductor amplifier. We show that, under realistic conditions, the anisotropic gain induced by the pump periodicity can show narrow angular profile of enhanced gain of less than one degree, providing an intrinsic filtering mechanism and eventually improving the spatial beam quality.

*Appeared in*

- Appl. Phys. Lett., 103 (2013) pp. 132101/1--132101/4.

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# Regularity of second derivatives in elliptic transmission problems near an interior regular multiple line of contact

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35B65 35J25

*Keywords*

- Elliptic transmission problems, second order generalized derivatives, interior multiple contact line, best Hölder exponent

*Abstract*

We investigate the regularity of the weak solution to elliptic transmission problems that involve several materials intersecting at a closed interior line of contact. We prove that local weak solutions possess second order generalized derivatives up to the contact line, mainly exploiting their higher regularity in the direction tangential to the line. Moreover we are thus able to characterize the higher regularity of the gradient and the Hoelder exponent by means of explicit estimates known in the literature for two dimensional problems. They show that strong regularity properties, for instance the integrability of the gradient to a power larger than the space dimension d =3, are to expect if the oscillations of the diffusion coefficient are moderate (that is for far larger a range than what a theory of small perturbations would allow), or if the number of involved materials does not exceed three.

*Appeared in*

- Math. Methods Appl. Sci., 41 (2018), pp. 6457--6479, DOI 10.1002/mma.5170 .

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# Convergence of an implicit Voronoi finite volume method for reaction-diffusion problems

*Authors*

- Fiebach, André
- Glitzky, Annegret
- Linke, Alexander

ORCID: 0000-0002-0165-2698

*2010 Mathematics Subject Classification*

- 35K57 35R05 65M08, 65M12, 80A30

*Keywords*

- reaction-diffusion systems, heterostructures, finite volume method, convergence, long-term simulation

*Abstract*

We investigate the convergence of an implicit Voronoi finite volume method for reaction- diffusion problems including nonlinear diffusion in two space dimensions. The model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. The numerical scheme uses boundary conforming Delaunay meshes and preserves positivity and the dissipative property of the continuous system. Starting from a result on the global stability of the scheme (uniform, mesh-independent global upper and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. In order to illustrate the preservation of qualitative properties by the numerical scheme, we present a long-term simulation of the Michaelis-Menten-Henri system. Especially, we investigate the decay properties of the relative free energy and the evolution of the dissipation rate over several magnitudes of time, and obtain experimental orders of convergence for these quantities.

*Appeared in*

- Numer. Methods Partial Differential Equations, 32 (2016), pp. 141--174.

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# Directional reversals and multimode dynamics in semiconductor ring lasers

*Authors*

- Pérez-Serrano, Antonio
- Javaloyes, Julien
- Balle, Salvador

*2010 Mathematics Subject Classification*

- 78A60 65P30 65M99 58J45 35B32

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.55.Sa 42.60.Mi 42.65.Pc 42.65.Sf

*Keywords*

- Semiconductor lasers, Travelling wave model (TWM), Multimode dynamics, Instabilities

*Abstract*

We investigate the dynamics of longitudinal modes in quantum-well semiconductor ring lasers by means of a spatio-temporal travelling wave model. We report the existence of a novel multimode instability in such a system that provokes a periodic deterministic directional reversal involving jumps between consecutive longitudinal modes. The switching sequence follows the modal frequencies from blue to red, and every modal jump is accompanied by a reversal of the direction of emission. We characterize and analyze such instability via the bifurcation analysis of the full travelling wave model as well as by performing the linear stability analysis of the monochromatic solutions.

*Appeared in*

- Phys. Rev. A, 89 (2014) pp. 023818/1--023818/14.

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# Controlling unstable chaos: Stabilizing chimera states by feedback

*Authors*

- Sieber, Jan
- Omel'chenko, Oleh

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

*2010 Mathematics Subject Classification*

- 34H10 34C15

*2008 Physics and Astronomy Classification Scheme*

- 05.45.Gg, 05.45.Xt, 89.75.Kd

*Keywords*

- chaos control, chimera state

*Abstract*

We present a control scheme that is able to find and stabilize a chaotic saddle in a system with a large number of interacting particles. This allows us to track a high dimensional chaotic attractor through a bifurcation where it loses its attractivity. Similar to a classical delayed feedback control, the scheme is non-invasive, however, only in an appropriately relaxed sense considering the chaotic regime as a statistical equilibrium displaying random fluctuations as a finite size effects. We demonstrate the control scheme for so called chimera states, which are coherence-incoherence patterns in coupled oscillator systems. The control makes chimera states observable close to coherence, for small numbers of oscillators, and for random initial conditions.

*Appeared in*

- Phys. Rev. Lett., 112 (2014) pp. 054102/1--054102/5.

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# A curvature-adapted anisotropic surface remeshing method

*Authors*

- Dassi, Franco
- Si, Hang

*2010 Mathematics Subject Classification*

- 65M50 65N50 65D18 68U05 68N99

*Keywords*

- surface mesh generation, surface remeshing, curvature-adapted, anisotropic, mesh optimization, edge flip

*Abstract*

We present a new method for remeshing surfaces that respect the intrinsic anisotropy of the surfaces. In particular, we use the normal informations of the surfaces, and embed the surfaces into a higher dimensional space (here we use 6d). This allow us to form an isotropic mesh optimization problem in this embedded space. Starting from an initial mesh of a surface, we optimize the mesh by improving the mesh quality measured in the embedded space. The mesh is optimized by combining common local modifications operations, i.e., edge flip, edge contraction, vertex smoothing, and vertex insertion. All operations are applied directly on the 3d surface mesh. This method results a curvature-adapted mesh of the surface. This method can be easily adapted to mesh multi-patches surfaces, i.e., containing corner singularities and sharp features. We present examples of remeshed surfaces from implicit functions and CAD models.

*Appeared in*

- New Challenges in Grid Generation and Adaptivity for Scientific Computing, S. Perotto, L. Formaggia, eds., vol. 5 of SEMA SIMAI Springer Series, Springer International Publishing, Cham, 2015, pp. 19--41

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# Low Mach asymptotic preserving scheme for the Euler--Korteweg model

*Authors*

- Giesselmann, Jan

*2010 Mathematics Subject Classification*

- 65M06 65M12 76T10

*Keywords*

- Multi-phase flows, phase transition, all-speed scheme, asymptotic preserving, low Mach number flows, finite difference scheme

*Abstract*

We present an all speed scheme for the Euler-Korteweg model. We study a semi-implicit time-discretisation which treats the terms, which are stiff for low Mach numbers, implicitly and thereby avoids a dependence of the timestep restriction on the Mach number. Based on this we present a fully discrete finite difference scheme. In particular, the scheme is asymptotic preserving, i.e., it converges to a stable discretisation of the incompressible limit of the Euler-Korteweg model when the Mach number tends to zero.

*Appeared in*

- IMA J. Numer. Anal. 35 (2) (2015) , pp. 802--833.

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# POAS4SPM --- A toolbox for SPM to denoise diffusion MRI data

*Authors*

- Tabelow, Karsten

ORCID: 0000-0003-1274-9951 - Mohammadi, Siawoosh
- Weiskopf, Nikolaus
- Polzehl, Jörg

ORCID: 0000-0001-7471-2658

*2010 Mathematics Subject Classification*

- 62P10 62G05

*Keywords*

- dMRI, Noise reduction, POAS, msPOAS, SPM8

*Abstract*

We present an implementation of a recently developed noise reduction algorithm for dMRI data, called multi-shell position orientation adaptive smoothing (msPOAS), as a toolbox for SPM. The method intrinsically adapts to the structures of different size and shape in dMRI and hence avoids blurring typically observed in non-adaptive smoothing. We give examples for the usage of the toolbox and explain the determination of experiment-dependent parameters for an optimal performance of msPOAS.

*Appeared in*

- Neuroinformatics, 13 (2015) pp. 19--29.

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# Numerical methods for generalized nonlinear Schrödinger equations

*Authors*

- Čiegis, Raimondas
- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Radziunas, Mindaugas

*2010 Mathematics Subject Classification*

- 35Q55 65M70 65M06

*2008 Physics and Astronomy Classification Scheme*

- 02.70.Hm 02.70.Bf 02.60.Jh 42.81.Dp

*Keywords*

- Nonlinear Schrödinger Equation, Splitting algorithm, Pseudo-spectral scheme, Finite-difference scheme, Numerical experiments

*Abstract*

We present and analyze different splitting algorithms for numerical solution of the both classical and generalized nonlinear Schrödinger equations describing propagation of wave packets with special emphasis on applications to nonlinear fiber-optics. The considered generalizations take into account the higher-order corrections of the linear differential dispersion operator as well as the saturation of nonlinearity and the self-steepening of the field envelope function. For stabilization of the pseudo-spectral splitting schemes for generalized Schrödinger equations a regularization based on the approximation of the derivatives by the low number of Fourier modes is proposed. To illustrate the theoretically predicted performance of these schemes several numerical experiments have been done.

*Appeared in*

- Kinet. Relat. Models, 8 (2015) pp. 215--234.

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# Solitons on a background, rogue waves and classical soliton solutions of Sasa--Satsuma equation

*Authors*

- Bandelow, Uwe

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

*2010 Mathematics Subject Classification*

- 35Q55 35Q60 37K40

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Tg 05.45.Yv 42.81.Dp

*Keywords*

- Generalized nonlinear Schrödinger equations, Sasa-Satsuma equation, soliton, rogue waves

*Abstract*

We present the most general multi-parameter family of a soliton on a background solutions to the Sasa-Satsuma equation. The solution contains a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits nontrivial limiting cases, such as rogue waves and classical solitons, that are considered in detail.

*Appeared in*

- J. Opt., 15 (2013) pp. 064006/1--064006/10.

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# Gevrey regularity for integro-differential operators

*Authors*

- Albanese, Guglielmo
- Fiscella, Alessio
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 35R11 60G22 35B65

*Keywords*

- Integro-differential equations, Gevrey class, Fractional Laplacian

*Abstract*

We prove a regularity theory in the Gevrey class for a family of nonlocal differential equations of elliptic type.

*Appeared in*

- J. Math. Anal. Appl., 428 (2015) pp. 1225--1238.

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# Improving efficiency of coupled schemes for Navier--Stokes equations by a connection to grad-div stabilized projection methods

*Authors*

- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Neilan, Michael
- Rebholz, Leo G.
- Wilson, Nicholas

*2010 Mathematics Subject Classification*

- 76D05 65M60

*Keywords*

- incompressible Navier-Stokes equations, projection method, grad-div stabilization, divergence-free finite elements

*Abstract*

We prove that in finite element settings where the divergence-free subspace of the velocity space has optimal approximation properties, the solution of Chorin/Temam projection methods for Navier-Stokes equations equipped with grad-div stabilization with parameter γ, converge to the associated coupled method solution with rate γ^{-1} as γ → ∞. We prove this first for backward Euler schemes, and then extend the results to BDF2 schemes, and finally to schemes with outflow boundary conditions. Several numerical experiments are given which verify the convergence rate, and show how using projection methods in this setting with large grad-div stabilization parameters can dramatically improve accuracy.

*Appeared in*

- J. Numer. Math., 25 (2017), pp. 229--248, DOI 10.1515/jnma-2016-1024 ; changed title: A connection between coupled and penalty projection timestepping schemes with FE spatial discretization for the Navier--Stokes equations.

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# Regularity and Bernstein-type results for nonlocal minimal surfaces

*Authors*

- Figalli, Alessio
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 49Q05 35B65 35R11 28A75

*Keywords*

- s-minimal surfaces, regularity theory, Bernstein's Theorem

*Abstract*

We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi stating that the validity of Bernstein's theorem as a consequence of the nonexistence of singular minimal cones in one dimension less.

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# Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35D10 35J55 35Q60

*Keywords*

- Low-frequency Maxwell equations, transmission conditions, regularity theory, Div-Curl inequality, Div-Curl Lemma

*Abstract*

We show that Lp vector fields over a Lipschitz domain are integrable to higher exponents if their generalized divergence and rotation can be identified with bounded linear operators acting on standard Sobolev spaces. A Div-Curl Lemma-type argument provides compact embedding results for such vector fields. We investigate the regularity of the solution fields for the low-frequency approximation of the Maxwell equations in time-harmonic regime. We focus on the weak formulation 'in H' of the problem, in a reference geometrical setting allowing for material heterogeneities.

*Appeared in*

- Discrete Contin. Dyn. Syst., 8 (2015) pp. 479--496.

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# A critical Kirchhoff type problem involving a non-local operator

*Authors*

- Fiscella, Alessio
- Valdinoci, Enrico

ORCID: 0000-0001-6222-2272

*2010 Mathematics Subject Classification*

- 45K05 47G20 26A33 74K05

*Keywords*

- Vibrating strings, nonlocal equations, fractional Laplacia

*Abstract*

We show the existence of non-negative solutions for a Kirchhoff type problem driven by a non-local integrodifferential operator.

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# Moment asymptotics for multitype branching random walks in random environment

*Authors*

- Gün, Onur
- König, Wolfgang
- Sekulović, Ozren

*2010 Mathematics Subject Classification*

- 60J80 60J55 60F10 60K37 60J10

*Keywords*

- multitype branching random walk, Feynman-Kac-type formula, variational analysis, annealed moments, large deviations

*Abstract*

We study a discrete time multitype branching random walk on a finite space with finite set of types. Particles follow a Markov chain on the spatial space whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman-Kac formula. We choose Weibull-type distributions with parameter 1/ρ_{ij} for the upper tail of the mean number of *j* type particles produced by an *i* type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.

*Appeared in*

- J. Theoret. Probab., 28 (2015) pp. 1726--1742.

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# Shape optimization for a sharp interface model of distortion compensation

*Authors*

- Sturm, Kevin
- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Hömberg, Dietmar

*2010 Mathematics Subject Classification*

- 49Q10 35Q74 74N15

*Keywords*

- Distortion, phase transition, shape optimisation, speed method, transmission problem

*Abstract*

We study a mechanical equilibrium problem for a material consisting of two components with different densities, which allows to change the outer shape by changing the interface between the subdomains. We formulate the shape design problem of compensating unwanted workpiece changes by controlling the interface, employ regularity results for transmission problems for a rigorous derivation of optimality conditions based on the speed method, and conclude with some numerical results based on a spline approximation of the interface.

*Appeared in*

- Comput. Optim. Appl., 64 (2016) pp. 557--588 under the title ``Distortion compensation as a shape optimisation problem for a sharp interface model''

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# Analysis and simulations of multifrequency induction hardening

*Authors*

- Hömberg, Dietmar
- Petzold, Thomas
- Rocca, Elisabetta

*2010 Mathematics Subject Classification*

- 35Q61 35K40 78M10

*Keywords*

- multifrequency induction hardening, heat equation, Maxwell's equations, well-posedness of the initial boundary value problem, stability estimates, finite element simulations

*Abstract*

We study a model for induction hardening of steel. The related differential system consists of a time domain vector potential formulation of the Maxwell's equations coupled with an internal energy balance and an ODE for the volume fraction of austenite, the high temperature phase in steel. We first solve the initial boundary value problem associated by means of a Schauder fixed point argument coupled with suitable a-priori estimates and regularity results. Moreover, we prove a stability estimate entailing, in particular, uniqueness of solutions for our Cauchy problem. We conclude with some finite element simulations for the coupled system.

*Appeared in*

- Nonlinear Anal. Real World Appl., 22 (2015) pp. 84--97.

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# Robust equilibration a posteriori error estimation for convection-diffusion-reaction problems

*Authors*

- Eigel, Martin
- Merdon, Christian

*2010 Mathematics Subject Classification*

- 65N30 65N15 65J15 65N22 65J10

*Keywords*

- a posteriori, error analysis, finite element method, equilibrated, convection dominated, adaptivity, inhomogeneous Dirichlet, augmented norm

*Abstract*

We study a posteriori error estimates for convection-diffusion-reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation (SDM), we derive robust and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of H (div, Ω). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of some part of the differential operator, complete independence from the coefficients of the problem is achieved.

Numerical benchmarks illustrate the very good performance of the error estimators in the convection dominated and the singularly perturbed cases.

*Appeared in*

- J. Sci. Comput., 67 (2016) pp. 747--768 under the title ``Equilibration a posteriori error estimation for convection-diffusion-reaction problems''

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# Connection times in large ad hoc mobile networks

*Authors*

- Döring, Hanna
- Faraud, Gabriel
- König, Wolfgang

*2010 Mathematics Subject Classification*

- 60K35 82C21 60J20 60F10

*Keywords*

- ad-hoc networks, connectivity, random waypoint model, dynamic continuum percolation, large deviations

*Abstract*

We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, i.e., they are iteratedly forwarded from participant to participant over distances $leq 2R$, with $2R$ the communication radius, until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random, and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model (RWP). Here we describe the decay rate, in the limit of large time horizons, of the probability that the portion of the connection time is less than the expectation.

*Appeared in*

- Bernoulli, 22 (2016) pp. 2143--2176.

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# Gradient flow perspective of thin-film bilayer flows

*Authors*

- Huth, Robert
- Jachalski, Sebastian
- Kitavtsev, Georgy
- Peschka, Dirk

ORCID: 0000-0002-3047-1140

*2010 Mathematics Subject Classification*

- 76A20 35R37 37D35 35K65

*Keywords*

- thin-film, gradient flow, moving boundary problem, bilayer

*Abstract*

We study gradient flow formulations of thin-film bilayer flows with triple-junctions between liquid/liquid/air. First we highlight the gradient structure in the Stokes free-boundary flow and identify its solutions with the well known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding thin-film model and formally identify solutions with those of the corresponding free-boundary problem. A robust numerical algorithm for the thin-film gradient flow structure is then provided. Using this algorithm we compare the sharp triple-junction model with precursor models. For their stationary solutions a rigorous connection is established using Gamma-convergence. For time-dependent solutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zero-contact angle case, where we compare numerical solutions with asymptotically exact source-type solutions.

*Appeared in*

- J. Engrg. Math., 94 (2015) pp. 43--61.

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# Parameter identification in non-isothermal nucleation and growth processes

*Authors*

- Hömberg, Dietmar
- Lu, Shuai
- Sakamoto, Kenichi
- Yamamoto, Masahiro

*2010 Mathematics Subject Classification*

- 80A22 49K20 49N45

*Keywords*

- Phase transition, nucleation and growth, inverse problem

*Abstract*

We study non-isothermal nucleation and growth phase transformations, which are described by a generalized Avrami model for the phase transition coupled with an energy balance to account for recalescence effects. The main novelty of our work is the identification of temperature dependent nucleation rates. We prove that such rates can be uniquely identified from measurements in a subdomain and apply an optimal control approach to develop a numerical strategy for its computation.

*Appeared in*

- Inverse Problems, 30 (2014) pp. 035003/1--035003/24.

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# Quantification of ergodicity in stochastic homogenization: Optimal bounds via spectral gap on Glauber dynamics

*Authors*

- Gloria, Antoine
- Neukamm, Stefan
- Otto, Felix

*2010 Mathematics Subject Classification*

- 35B27 39A70 60H25 60F99

*Keywords*

- stochastic homogenization, homogenization error, corrector equation, quantitative results

*Abstract*

We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice Zd with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w. r. t. a Glauber dynamics on coefficient fields - as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i. e. of the generator of the process called "random environment as seen from the particle"). As a corollary we recover existence of stationary correctors (in dimensions d > 2) and prove new optimal estimates for regularized versions of the corrector (in dimensions d >= 2). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green's function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.

*Appeared in*

- Invent. math. 199 (2015) pp. 455--515.

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# On gradient structures for Markov chains and the passage to Wasserstein gradient flows

*Authors*

- Disser, Karoline
- Liero, Matthias

ORCID: 0000-0002-0963-2915

*2010 Mathematics Subject Classification*

- 35K10 35K20 37L05 49M25 60F99 65M08 60J27 70G75

*Keywords*

- Wasserstein gradient flow, relative entropy, finite-volume scheme, entropy/entropy-dissipation formulation, gradient structures, Markov chains

*Abstract*

We study the approximation of Wasserstein gradient structures by their finite-dimensional analog. We show that simple finite-volume discretizations of the linear Fokker-Planck equation exhibit the recently established entropic gradient-flow structure for reversible Markov chains. Then, we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradient-flow structures. In particular, we make no use of the linearity of the equations nor of the fact that the Fokker-Planck equation is of second order.

*Appeared in*

- Netw. Heterog. Media, 10 (2015) pp. 233-253.

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# Moment bounds for the corrector in stochastic homogenization of a percolation model

*Authors*

- Lamacz, Agnes
- Neukamm, Stefan
- Otto, Felix

*2010 Mathematics Subject Classification*

- 35B27 60K37 60F99 60H25 39A70

*Keywords*

- stochastic homogenization, percolation, corrector equation, quantitative results

*Abstract*

We study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on Z^d, d > 2. The model is obtained from the classical Bernoulli bond percolation by conditioning all bonds parallel to the first coordinate direction to be open. As a main result we prove (in fact for a slightly more general model) that stationary correctors exist and that all finite moments of the corrector are bounded. This extends a previous result by Gloria and the third author, where uniformly elliptic conductances are treated, to the degenerate case. Our argument is based on estimates on the gradient of the elliptic Green's function.

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# Uniform asymptotic expansions for the infinite harmonic chain

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Patz, Carsten

*2010 Mathematics Subject Classification*

- 37K60 41A60 42B20

*Keywords*

- Asymptotic analysis, oscillatory integrals, Fermi-Pasta-Ulam chain, Airy function, dispersive decay, method of stationary phase

*Abstract*

We study the dispersive behavior of waves in linear oscillator chains. We show that for general general dispersions it is possible to construct an expansion such that the remainder can be estimated by $1/t$ uniformly in space. In particalur we give precise asymptotics for the transition from the $1/t^1/2$ decay of nondegenerate wave numbers to the generate $1/t^1/3$ decay of generate wave numbers. This involves a careful description of the oscillatory integral involving the Airy function.

*Appeared in*

- Z. Anal. Anwendungen, 36 (2017), pp. 437--475, DOI 10.4171/ZAA/1596 .

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# Stability of explicit Runge--Kutta methods for finite element approximation of linear parabolic equations on anisotropic meshes

*Authors*

- Huang, Weizhang
- Kamenski, Lennard
- Lang, Jens

*2010 Mathematics Subject Classification*

- 65M60 65M50 65F15

*Keywords*

- finite element method, anisotropic mesh, stability condition, parabolic equation, explicit Runge--Kutta method

*Abstract*

We study the stability of explicit Runge-Kutta integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of 2 (d + 1), where d is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first one depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The other factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.

*Appeared in*

- SIAM J. Numer. Anal., 54 (2016) pp. 1612--1634.

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# Stability of explicit Runge--Kutta methods for high order finite element approximation of linear parabolic equations

*Authors*

- Huang, Weizhang
- Kamenski, Lennard
- Lang, Jens

*2010 Mathematics Subject Classification*

- 65M60 65M50 65F15

*Keywords*

- finite element method, anisotropic mesh, stability condition, parabolic equation

*Abstract*

We study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order finite element approximation of linear parabolic equations on general nonuniform meshes.

*Appeared in*

- Proceedings of Numerical Mathematics and Advanced Applications, vol. 103 of Lecture Notes in Computational Science and Engineering, Springer International Publishing, Switzerland, 2015, pp. 165--173

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# The total mass of super-Brownian motion upon exiting balls and Sheu's compact support condition

*Authors*

- Hesse, Marion
- Kyprianou, Andreas

*2010 Mathematics Subject Classification*

- 60J68 60J80

*Keywords*

- super-Brownian motion, exit measures, time-dependent continuous state branching processes, compact support condition

*Abstract*

We study the total mass of a

Our results identify the compact support criterion in Sheu (1994) as Grey's condition (1974) for the aforementioned limiting branching mechanism.

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# Aspects of guaranteed error control in CPDEs

*Authors*

- Carstensen, Carsten
- Merdon, Christian
- Neumann, Johannes

*2010 Mathematics Subject Classification*

- 65N30 65N15

*Keywords*

- guaranteed error control, equilibration error estimators, Poisson model problem, conforming finite element methods, Crouzeix-Raviart nonconforming finite element methods, curved boundaries, guaranteed goal-oriented error control

*Abstract*

Whenever numerical algorithms are employed for a reliable computational forecast, they need to allow for an error control in the final quantity of interest. The discretisation error control is of some particular importance in computational PDEs (CPDEs) where guaranteed upper error bounds (GUB) are of vital relevance. After a quick overview over energy norm error control in second-order elliptic PDEs, this paper focuses on three particular aspects. First, the variational crimes from a nonconforming finite element discretisation and guaranteed error bounds in the discrete norm with improved postprocessing of the GUB. Second, the reliable approximation of the discretisation error on curved boundaries and, finally, the reliable bounds of the error with respect to some goal-functional, namely, the error in the approximation of the directional derivative at a given point.

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