Preprints der Fakultät für Mathematik
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Item Interface conditions for Maxwell’s equations by homogenization of thin inclusions: transmission, reflection or polarization(2025-01-29) Schweizer, Ben; Wiedemann, DavidWe consider the time-harmonic Maxwell equations in a complex geometry. We are interested in geometries that model polarization filters or Faraday cages. We study the situation that the underlying domain contains perfectly conducting inclusions, the inclusions are distributed in a periodic fashion along a surface. The periodicity is η > 0 and the typical scale of the inclusion is η, but we allow also the presence of even smaller scales, e.g. when thin wires are analyzed. We are interested in the limit η → 0 and in effective equations. Depending on geometric properties of the inclusions, the effective system can imply perfect transmission, perfect reflection or polarization.Item Concentration inequalities in random Schrödinger operators(2019-01-21) Schumacher, ChristophItem Time harmonic Maxwell's equations in periodic waveguides(2024-01-31) Kirsch, Andreas; Schweizer, BenWe study Maxwell’s equations with periodic coefficients in a closed waveguide. A functional analytic approach is used to formulate and to solve the radiation problem. We furthermore characterize the set of all bounded solutions to the homogeneous problem. The case of a compact perturbation of the medium is included, the scattering problem and the limiting absorption principle are discussed.Item Inf-sup theory for the quasi-static Biot's equations in poroelasticity(2024-07-01) Zanotti, PietroWe analyze the two-field formulation of the quasi-static Biot's equations by means of the inf-sup theory. For this purpose, we exploit an equivalent four-field formulation of the equations, introducing the so-called total pressure and total fluid content as independent variables. We establish existence, uniqueness and stability of the solution. Our stability estimate is two-sided and robust, meaning that the regularity established for the solution matches the regularity requirements for the data and the involved constants are independent of all material parameters. We prove also that additional regularity in space of the data implies, in some cases, corresponding additional regularity in space of the solution. These results are instrumental to the design and the analysis of discretizations enjoying accurate stability and error estimates.Item Effective sound absorbing boundary conditions for complex geometries(2024-05-22) Schweizer, BenWe analyze a system of equations that describes the propagation of sound waves. We are interested in complex constructions along a part of the boundary of the domain, for example constructions with small chambers that are connected to the domain. We also allow that different flow equations are used in the chambers, e.g., modelling a damping material. In addition to the complex geometry, we assume that the viscosity vanishes in the limit. The limiting system is given by wave equations, we derive these equations and determine the effective boundary conditions. The effective boundary conditions replace the large number of small chambers. We provide examples for sound absorbing constructions and their Dirichlet-to-Neumann boundary conditions.Item The time horizon for stochastic homogenization of the one-dimensional wave equation(2023-07-28) Schäffner, Mathias; Schweizer, BenThe wave equation with stochastic coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε, classical homogenization fails for times of the order ε−2. We consider the one-dimensional wave equation and are interested in the critical time scale ε−β from where on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε−1Item Periodic wave-guides revisited: Radiation conditions, limiting absorption principles, and the space of bounded solutions(2023-07-28) Kirsch, Andreas; Schweizer, BenWe study the Helmholtz equation with periodic coefficients in a closed wave-guide. A functional analytic approach is used to formulate and to solve the radiation problem in a self-contained exposition. In this context, we simplify the non-degeneracy assumption on the frequency. Limiting absorption principles (LAPs) are studied and the radiation condition corresponding to the chosen LAP is derived; we include an example to show different LAPs lead, in general, to different solutions of the radiation problem. Finally, we characterize the set of all bounded solutions to the homogeneous problem.Item A radiation box domain truncation scheme for the wave equation(2022-03-24) Schäffner, Mathias; Schweizer, Ben; Tjandrawidjaja, YohanesWe consider the wave equation in an unbounded domain and are interested in domain truncation methods. Our aim is to develop a numerical scheme that allows calculations for truncated waveguide geometries with periodic coefficient functions. The scheme is constructed with radiation boxes that are attached to the artificially introduced boundaries. A Dirichlet-to-Neumann operator N is calculated in these radiation boxes. Efficiency of the scheme is obtained by calculating N not with an iteration, but with a single run through the time interval. We observe speed-up factors of up to 20 in comparison to calculations without domain truncation.Item A data driven framework for evolutionary problems in solid mechanics(2021-11-18) Poelstra, Klaas; Bartel, Thorsten; Schweizer, BenData driven schemes introduced a new perspective in elasticity: While certain physical principles are regarded as invariable, material models for the relation between strain and stress are replaced by data clouds of admissible pairs of these variables. A data driven approach is of particular interest for plasticity problems, since the material modelling is even more unclear in this field. Unfortunately, so far, data driven approaches to evolutionary problems are much less understood. We try to contribute in this area and propose an evolutionary data driven scheme. We presenta first analysis of the scheme regarding existence and data convergence. Encouraging numerical tests are also included.Item Domain truncation methods for the wave equation in a homogenization limit(2021-09-29) Schäffner, Mathias; Schweizer, Ben; Tjandrawidjaja, YohanesItem Travelling wave solutions for gravity fingering in porous media flows(2020-11-10) Mitra, Koondanibha; Schweizer, Ben; Rätz, AndreasWe study an imbibition problem for porous media. When a wetted layer is above a dry medium, gravity leads to the propagation of the water downwards into the medium. In experiments, the occurence of fingers was observed, a phenomenon that can be described with models that include hysteresis. In the present paper we describe a single finger in a moving frame and set up a free boundary problem to describe the shape and the motion of one finger that propagates with a constant speed. We show the existence of solutions to the travelling wave problem and investigate the system numerically.Item Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials(2020-09-29) Andraus, Sergio; Hermann, Kilian; Voit, MichaelItem Limit theorems for Bessel and Dunkl processes of large dimensions and free convolutions(2020-09-28) Voit, Michael; Woerner, Jeannette H. C.Item Sound absorption by perforated walls along boundaries(2020-06-03) Donato, Patrizia; Lamacz, Agnes; Schweizer, BenWe analyze the Helmholtz equation in a complex domain. A sound absorbing structure at a part of the boundary is modelled by a periodic geometry with periodicity ε > 0. A resonator volume of thickness ε is connected with thin channels (opening ε^3) with the main part of the macroscopic domain. For this problem with three different scales we analyze solutions in the limit ε → 0 and find that the effective system can describe sound absorption.Item Representation of solutions to wave equations with profile functions(2019-05-17) Lamacz, Agnes; Schweizer, BenSolutions to the wave equation with constant coefficients in $\mathbb{R}^d$ ca be represented explicitly in Fourier space. We investigate a reconstruction formula, which provides an approximation of solutions $u(., t)$ to initial data $u_0(.)$ for large times. The reconstruction consists of three steps: 1) Given $u_0$, initial data for a profile equation are extracted. 2) A profile evolution equation determines the shape of the profile at time $\tau = \varepsilon^2 t$. 3) A shell reconstruction operator transforms the profile to a function on $\mathbb{R}^d$. The sketched construction simplifies the wave equation, since only a one-dimensional problem in an $O(1)$ time span has to be solved. We prove that the construction provides a good approximation to the wave evolution operator for times $t$ of order $\varepsilon^{-2}$.Item Some central limit theorems for random walks associated with hypergeometric functions of type BC(2018-02) Artykov, Merdan; Voit, MichaelThe spherical functions of the noncompact Grassmann manifolds $G_{p,q}(\mathbb F)=G/K$ over $\mathbb F=\mathbb R, \mathbb C, \mathbb H$ with rank $q\ge1$ and dimension parameter $p>q$ are Heckman-Opdam hypergeometric functions of type BC, when the double coset spaces $G//K$ are identified with the Weyl chamber $C_q^B\subset \mathbb R^q$ of type B. The associated double coset hypergroups on $ C_q^B$ can be embedded into a continuous family of commutative hypergroups $(C_q^B,*_p)$ with $p\in[2q-1,\infty[$ associated with these hypergeometric functions by Rösler. Several limit theorems for random walks on these hypergroups were recently derived by Voit (2017). We here present further limit theorems when the time as well as $p$ tend to $\infty$. For integers $p$, this admits interpretations for group-invariant random walks on the Grassmannians $G/K$.Item Continuous Association Schemes and Hypergroups(2018-02) Voit, MichaelClassical finite association schemes lead to finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes can be easily extended to the possibly infinite case. Moreover, this notion can be relaxed slightly by using suitably deformed families of stochastic matrices by skipping the integrality conditions. This leads to a larger class of examples which are again associated to discrete hypergroups. In this paper we propose a topological generalization of association schemes by using a locally compact basis space $X$ and a family of Markov-kernels on $X$ indexed by some locally compact space $D$ where the supports of the associated probability measures satisfy some partition property. These objects, called continuous association schemes, will be related to hypergroup structures on $D$. We study some basic results for this notion and present several classes of examples. It turns out that for a given commutative hypergroup the existence of an associated continuous association scheme implies that the hypergroup has many features of a double coset hypergroup. We in particular show that commutative hypergroups, which are associated with commutative continuous association schemes, carry dual positive product formulas for the characters. On the other hand, we prove some rigidity results in particular in the compact case which say that for given spaces $X,D$ there are only a few continuous association schemes.Item Existence results for the Helmholtz equation in periodic wave-guides with energy methods(2019-05-10) Schweizer, BenThe Helmholtz equation $ - \nabla \cdot (a \nabla u) - \omega^2 u = f$ is considered in an unbounded wave-guide $\Omega := \mathbb{R} \times S \subset \mathbb{R}^d$, where $S \subset \mathbb{R}^{d-1}$ is a bounded domain. The coefficient $a$ is strictly elliptic and (locally) periodic in the unbounded direction $x_1\in \mathbb{R}$. For non-singular frequencies $\omega$, we show the existence of a solution $u$. While previous proofs of such results were based on operator theory, our proof uses only energy methods.Item Functional central limit theorems for multivariate Bessel processes in the freezing regime(2019-01) Voit, Michael; Woerner, Jeannette H.C.Multivariate Bessel processes $(X_{t,k})_{t\ge0}$ describe interacting particle systems of Calogero-Moser-Sutherland type and are related with $\beta$-Hermite and $\beta$-Laguerre ensembles. They depend on a root system and a multiplicity $k$ which corresponds to the parameter $\beta$ in random matrix theory. In the recent years, several limit theorems were derived for $k\to\infty$ with fixed $t>0$ and fixed starting point. Only recently, Andraus and Voit used the stochastic differential equations of $(X_{t,k})_{t\ge0}$ to derive limit theorems for $k\to\infty$ with starting points of the form $\sqrt k\cdot x$ with $x$ in the interior of the corresponding Weyl chambers.Here we provide associated functional central limit theorems which are locally uniform in $t$.The Gaussian limiting processes admit explicit representations in terms of matrix exponentials and the solutions of the associated deterministic dynamical systems.Item Relaxation analysis in a data driven problem with a single outlier(2019-07-11) Röger, Matthias; Schweizer, BenWe study a scalar elliptic problem in the data driven context. Our interest is to study the relaxation of a data set that consists of the union of a linear relation and single outlier. The data driven relaxation is given by the union of the linear relation and a truncated cone that connects the outlier with the linear subspace.