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search.filters.applied.f.access: openAccess × Author: Heida, Martin × Start date: 2019 × Has files: Yes × Version: publishedVersion × WGL Institute: WIAS ×

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Now showing 1 - 10 of 27
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    Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator
    (Les Ulis : EDP Sciences, 2021) Heida, Martin; Kantner, Markus; Stephan, Artur
    We introduce a family of various finite volume discretization schemes for the Fokker–Planck operator, which are characterized by different Stolarsky weight functions on the edges. This family particularly includes the well-established Scharfetter–Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the choice of the Stolarsky weights. We show that the Scharfetter–Gummel scheme has the analytically best convergence properties but also that there exists a whole branch of Stolarsky means with the same convergence quality. We show by numerical experiments that for small convection the choice of the optimal representative of the discretization family is highly non-trivial, while for large gradients the Scharfetter–Gummel scheme stands out compared to the others.
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    Stochastic homogenization on perforated domains II – Application to nonlinear elasticity models
    (Berlin : Wiley-VCH, 2022) Heida, Martin
    Based on a recent work that exposed the lack of uniformly bounded (Formula presented.) extension operators on randomly perforated domains, we study stochastic homogenization of nonlinear p-elasticity, (Formula presented.), on such structures using instead the extension operators constructed in former works. We thereby introduce two-scale convergence methods on such random domains under the intrinsic loss of regularity and prove some generally useful calculus theorems on the probability space, for example, abstract Gauss theorems.
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    The space of bounded variation with infinite-dimensional codomain
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Heida, Martin; Patterson, Robert I.A.; Renger, D.R. Michiel
    We 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.
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    Averaging of time-periodic dissipation potentials in rate-independent processes : dedicated to Tomáš Roubícek on the occasion of his sixtieth birthday
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Heida, Martin; Mielke, Alexander
    We 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.
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    Stochastic two-scale convergence and Young measures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heida, Martin; Neukamm, Stefan; Varga, Mario
    In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.
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    Stochastic homogenization on perforated domains I: Extension operators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Heida, Martin
    This preprint is part of a major rewriting and substantial improvement of WIAS Preprint 2742. In this first part of a series of 3 papers, we set up a framework to study the existence of uniformly bounded extension and trace operators for W1,p-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. We drop the classical assumption of minimaly smoothness and study stationary geometries which have no global John regularity. For such geometries, uniform extension operators can be defined only from W1,p to W1,r with the strict inequality r
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    Stochastic homogenization on randomly perforated domains
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Heida, Martin
    We study the existence of uniformly bounded extension and trace operators for W1,p-functions on randomly perforated domains, where the geometry is assumed to be stationary ergodic. Such extension and trace operators are important for compactness in stochastic homogenization. In contrast to former approaches and results, we use very weak assumptions on the geometry which we call local (δ, M)-regularity, isotropic cone mixing and bounded average connectivity. The first concept measures local Lipschitz regularity of the domain while the second measures the mesoscopic distribution of void space. The third is the most tricky part and measures the ''mesoscopic'' connectivity of the geometry. In contrast to former approaches we do not require a minimal distance between the inclusions and we allow for globally unbounded Lipschitz constants and percolating holes. We will illustrate our method by applying it to the Boolean model based on a Poisson point process and to a Delaunay pipe process. We finally introduce suitable Sobolev spaces on Rd and Ω in order to construct a stochastic two-scale convergence method and apply the resulting theory to the homogenization of a p-Laplace problem on a randomly perforated domain.
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    Stochastic unfolding and homogenization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Heida, Martin; Neukamm, Stefan; Varga, Mario
    The 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.
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    Stochastic homogenization of Lambda-convex gradient flows
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Heida, Martin; Neukamm, Stefan; Varga, Mario
    In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen--Cahn type equations and evolutionary equations driven by the p-Laplace operator with p ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals.
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    Estimation of the infinitesimal generator by square-root approximation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Donati, Luca; Heida, Martin; Weber, Marcus; Keller, Bettina
    For 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.