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- ItemOn convergences of the squareroot approximation scheme to the Fokker-Planck operator(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2017) Heida, MartinWe study the qualitative convergence properties of a finite volume scheme that recently was proposed by Lie, Fackeldey and Weber [SIAM Journal on Matrix Analysis and Applications 2013 (34/2)] in the context of conformation dynamics. The scheme was derived from physical principles and is called the squareroot approximation (SQRA) scheme. We show that solutions to the SQRA equation converge to solutions of the Fokker-Planck equation using a discrete notion of G-convergence. Hence the squareroot approximation turns out to be a usefull approximation scheme to the Fokker-Planck equation in high dimensional spaces. As an example, in the special case of stationary Voronoi tessellations we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property. In particular, the class of tessellations for which the G-convergence result holds is not trivial.
- ItemThe space of bounded variation with infinite-dimensional codomain(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2016) Heida, Martin; Patterson, Robert I.A.; Renger, D.R. MichielWe study functions of bounded variation with values in a Banach or in a metric space. We provide several equivalent notions of variations and provide the notion of a time derivative in this abstract setting. We study four distinct topologies on the space of bounded variations and provide some insight into the structure of these topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical Aubin-Lions theorem. We finally provide some useful applications to stochastic processes.
- ItemFractal homogenization of a multiscale interface problem(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2017) Heida, Martin; Kornhuber, Ralf; Podlesny, JoschaInspired from geological problems, we introduce a new geometrical setting for homogenization of a well known and well studied problem of an elliptic second order differential operator with jump condition on a multiscale network of interfaces. The geometrical setting is fractal and hence neither periodic nor stochastic methods can be applied to the study of such kind of multiscale interface problem. Instead, we use the fractal nature of the geometric structure to introduce smoothed problems and apply methods from a posteriori theory to derive an estimate for the order of convergence. Computational experiments utilizing an iterative homogenization approach illustrate that the theoretically derived order of convergenceof the approximate problems is close to optimal.
- ItemAveraging of time-periodic dissipation potentials in rate-independent processes : dedicated to Tomáš RoubĂcek on the occasion of his sixtieth birthday(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2016) Heida, Martin; Mielke, AlexanderWe study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit → 0 and show that the effective dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable qui-continuity of the solutions in the limit → 0.
- ItemStochastic unfolding and homogenization(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2017) Heida, Martin; Neukamm, Stefan; Varga, MarioThe notion of periodic two-scale convergence and the method of periodic un- folding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coecients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coecients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in dierent ways to the stochastic case. In this work we introduce a stochastic unfolding method that fea- tures many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a stochastic homogeniza- tion result for an non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method descibed in the present paper extends to the continuum setting the notion of discrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition.
- ItemEstimation of the infinitesimal generator by square-root approximation(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2017) Donati, Luca; Heida, Martin; Weber, Marcus; Keller, BettinaFor the analysis of molecular processes, the estimation of time-scales, i.e., tran- sition rates, is very important. Estimating the transition rates between molecular conformations is - from a mathematical point of view - an invariant subspace projec- tion problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is ap- proximated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the ge- ometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correlation between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide.
- ItemStochastic homogenization of rate-dependent models of monotone type in plasticity(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2017) Heida, Martin; Nesenenko, SergiyIn this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatricks function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Gamma-convergence theory.
- ItemThe fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2018) Flegel, Franziska; Heida, MartinWe study a general class of discrete p-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional p-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Lapalace operator to the continuous fractional Laplace operator.
- ItemHomogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Flegel, Franziska; Heida, Martin; Slowik, MartinWe study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for the normalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian’s Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence.
- ItemStochastic homogenization of rate-independent systems(Berlin : WeierstraĂź-Institut fĂĽr Angewandte Analysis und Stochastik, 2016) Heida, MartinWe study the stochastic and periodic homogenization 1-homogeneous convex functionals. We proof some convergence results with respect to stochastic twoscale convergence, which are related to classical Gamma-convergence results. The main result is a general lim inf-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rate-independent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandlt-Reuss plasticity, Coulomb friction on a macroscopic surface and Coulomb friction on microscopic fissures.