### On the Darwin--Howie--Whelan equations for the scattering of fast electrons described by the Schrödinger equation

Thomas Koprucki, Anieza Maltsi & Alexander Mielke
The Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrödinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to...

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

Eleonora Cinti, Joaquim Serra & Enrico Valdinoci
We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the $s$-fractional perimeter as a particular case. On the one hand, we establish universal $BV$-estimates in every dimension $nge 2$ for stable sets. Namely, we prove that any stable set in $B_1$ has finite classical perimeter in $B_1/2$, with a universal bound. This nonlocal result is new even in the case of $s$-perimeters and its local counterpart (for classical stable minimal...

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

Tao Yin, Guanghui Hu, Liwei Xu & Bo Zhang
Consider a time-harmonic acoustic point source incident on a bounded isotropic linearly elastic body immersed in a homogeneous compressible inviscid fluid. This paper is concerned with the inverse fluid-solid interaction (FSI) problem of recovering the elastic body from near-field data generated by infinitely many incident point source waves at a fixed energy. The incident point sources and the receivers for recording scattered signals are both located on a non-spherical closed surface, on which an outgoing-to-incoming...

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

Karoline Disser, Matthias Liero & Jonathan Zinsl
We investigate the limit passage for a system of ordinary differential equations modeling slow and fast chemical reaction of mass-action type, where the rates of fast reactions tend to infinity. We give an elementary proof of convergence to a reduced dynamical system acting in the slow reaction directions on the manifold of fast reaction equilibria. Then we study the entropic gradient structure of these systems and prove an E-convergence result via Γ-convergence of the primary...

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

Arne Roggensack & Christiane Kraus
In this paper, we introduce and study analytically a vectorial Cahn-Hilliard reaction model coupled with rate-dependent damage processes. The recently proposed Cahn-Hilliard reaction model can e.g. be used to describe the behavior of electrodes of lithium-ion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases. The coupling with the damage process allows considering simultaneously the evolution of a damage field, a second important physical effect occurring during...

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

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

### More specific signal detection in functional magnetic resonance imaging by false discovery rate control for hierarchically structured systems of hypotheses

Konstantin Schildknecht, Karsten Tabelow & Thorsten Dickhaus
Signal detection in functional magnetic resonance imaging (fMRI) inherently involves the problem of testing a large number of hypotheses. A popular strategy to address this multiplicity is the control of the false discovery rate (FDR). In this work we consider the case where prior knowledge is available to partition the set of all hypotheses into disjoint subsets or families, e. g., by a-priori knowledge on the functionality of certain regions of interest. If the proportion...

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

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

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

Martin Eigel & Christian Merdon
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...

### Sharp-interface formation during lithium intercalation into silicon

Esteban Meca Álvarez, Andreas Münch & Barbara Wagner
In this study we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as a-Si. The governing equations couple a viscous Cahn-Hilliard-Reaction model with elasticity in the framework of the Cahn-Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in ion concentration between the initial state...

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

Esteban Meca Álvarez, Andreas Münch & Barbara Wagner
In this study we revisit experiments by Sethuraman et al. [J. Power Sources, 195, 5062 (2010)] on the stress evolution during the lithiation/delithiation cycle of a thin film of amorphous silicon. Based on recent work that show a two-phase process of lithiation of amorphous silicon, we formulate a phase-field model coupled to elasticity in the framework of Larché-Cahn. Using an adaptive nonlinear multigrid algorithm for the finite-volume discretization of this model, our two-dimensional numerical simulations...

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

Chantal Landry, Wolfgang Welz & Matthias Gerdts
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...

### From adhesive to brittle delamination in visco-elastodynamics

Riccarda Rossi & Marita Thomas
In this paper we analyze a system for brittle delamination between two visco-elastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rate-independent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak...

### Statistical Skorohod embedding problem and its generalizations

Denis Belomestny & John G. M. Schoenmakers
Given a Levy process L, we consider the so-called statistical Skorohod embedding problem of recovering the distribution of an independent random time T based on i.i.d. sample from L(T). Our approach is based on the genuine use of the Mellin and Laplace transforms. We propose consistent estimators for the density of T, derive their convergence rates and prove their optimality. It turns out that the convergence rates heavily depend on the decay of the Mellin...

### Error estimates for elliptic equations with not exactly periodic coefficients

Sina Reichelt
This note is devoted to the derivation of quantitative estimates for linear elliptic equations with coefficients that are not exactly ε-periodic and the ellipticity constant may degenerate for vanishing ε. Here ε>0 denotes the ratio between the microscopic and the macroscopic length scale. It is shown that for degenerating and non-degenerating coefficients the error between the original solution and the effective solution is of order √ε. Therefore suitable test functions are constructed via the periodic...

### Large deviations in relay-augmented wireless networks

Christian Hirsch, Benedikt Jahnel, Paul Keeler & Robert I. A. Patterson
We analyze a model of relay-augmented cellular wireless networks. The network users, who move according to a general mobility model based on a Poisson point process of continuous trajectories in a bounded domain, try to communicate with a base station located at the origin. Messages can be sent either directly or indirectly by relaying over a second user. We show that in a scenario of an increasing number of users, the probability that an atypically...

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

Dietmar Hömberg, Francesco Saverio Patacchini, Kenichi Sakamoto & Johannes Zimmer
The classical Johnson-Mehl-Avrami-Kolmogorov approach for nucleation and growth models of diffusive phase transitions is revisited and applied to model the growth of ferrite in multiphase steels. For the prediction of mechanical properties of such steels, a deeper knowledge of the grain structure is essential. To this end, a Fokker-Planck evolution law for the volume distribution of ferrite grains is developed and shown to exhibit a log-normally distributed solution. Numerical parameter studies are given and confirm...

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

The present work deals with the derivation of corrector estimates for the two-scale homogenization of a thermo-diffusion model with weak thermal coupling posed in a heterogeneous medium endowed with periodically arranged high-contrast microstructures. The terminology weak thermal coupling'' refers here to the variable scaling in terms of the small homogenization parameter ε of the heat conduction-diffusion interaction terms, while the high-contrast'' is thought particularly in terms of the heat conduction properties of the composite material....

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

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

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

Marion Dziwnik & Sebastian Jachalski
We prove existence of solutions to an anisotropic Cahn-Hilliard-type equation with degenerate diffusional mobility. In particular, the mobility vanishes at the pure phases, which is typically used to model motion by surface diffusion. The main difficulty of the present existence result is the strong non-linearity given by the fourth-order anisotropic operator. Imposing particular assumptions on the domain and assuming that the strength of the anisotropy is sufficiently small enables to establish appropriate auxiliary results which...

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

Alexander Mielke & Markus Mittnenzweig
We discuss how the recently developed energy-dissipation methods for reactiondi usion systems can be generalized to the non-isothermal case. For this we use concave entropies in terms of the densities of the species and the internal energy, where the importance is that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions we give two...

### Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave

Johannes Elschner & Masahiro Yamamoto
We consider the two dimensional inverse scattering problem of determining a sound-hard obstacle by the far field pattern. We establish the uniqueness within the class of polygonal domains by a single incoming plane wave.

### Lubrication models with small to large slip lengths

Andreas Münch, Barbara Wagner & Thomas P. Witelski
A set of lubrication models for the thin film flow of incompressible fluids on solid substrates is derived and studied.The models are obtained as asymptotic limits of the Navier-Stokes equations with the Navier-slip boundary condition for different orders of magnitude for the slip-length parameter. Specifically, the influence of slip on the dewetting behavior of fluids on hydrophobic substrates is investigated here. Matched asymptotics are used to describe the dynamic profiles for dewetting films and comparison...

### Numerical experiments on the modulation theory for the nonlinear atomic chain

Wolfgang Dreyer & Michael Herrmann
Modulation theory with periodic traveling waves is a powerful, but not rigorous tool to derive a thermodynamic description for the atomic chain. We investigate the validity of this theory by means of several numerical experiments.

### Progressively refining penalized gradient projection method for semilinear parabolic optimal control problems

Ion Chryssoverghi, Juergen Geiser & Jamil Al-Hawasy
We consider an optimal control problem defined by semilinear parabolic partial differential equations, with control and state constraints, where the state constraints and cost functional involve also the state gradient. The problem is discretized by using a finite element method in space and an implicit -scheme in time for state approximation, while the controls are approximated by blockwise constant ones. We propose a discrete penalized gradient projection method, which is applied to the continuous problem...

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#### Affiliations

• Weierstrass Institute for Applied Analysis and Stochastics
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