Affine Decompositions of Parametric Stochastic Processes for Application within Reduced Basis Methods

Abstract

AbstractWe consider parameter dependent spatial stochastic processes in the context of partial differential equations (PDEs) and model order reduction. For a given parameter, a random sample of such a process specifies a sample coefficient function of a PDE, e.g. characteristics of porous media such as Li-ion batteries or random influences in biomechanical systems. To apply the Reduced Basis Method (RBM) to parametrized systems (with stochastic or deterministic parameter dependencies), it is necessary to obtain affine decompositions of the systems in parameter and space (cf. Patera and Rozza (2006); Wieland (2010)). For deterministic problems, it is common to use the Empirical Interpolation Method (EIM) (cf. Barrault et al. (2004)). For stochastic coefficients one can apply the Karhunen-Loève (KL) expansion (cf. Karhunen (1947)) where the terms with stochastic dependencies are assumed to satisfy certain distributions and are modeled using polynomial chaos (PC) expansions (cf. Ghanem and Spanos (1991)).In this paper, we extend the EIM to parametrized spatial stochastic processes. The goal is to develop efficiently computable affine decompositions of not only parameter dependent but also stochastic systems that separate spatial dependencies from parametric and probabilistic influences without any assumptions on the distribution of non-spatial terms. We will use the basic concept of the EIM together with ideas from Proper Orthogonal Decomposition (POD) as well as from the Discrete Empirical Interpolation Method (DEIM) (cf. Chaturantabut and Sorensen (2010)), and we use least-squares approxmations.