Affine Decompositions of Parametric Stochastic Processes for Application within Reduced Basis Methods ?
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چکیده
We 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. e.g. Patera and Rozza (2006); Haasdonk et al. (2012)). 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), Loève (1978)) where the terms involving 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)). Furthermore, we introduce a combination of these methods with least-squares approximations. We emphasize that the presented methods are not limited to stochastic functions but work analogously on noisy input data or on other hardly decomposable functions.
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تاریخ انتشار 2012