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Title: Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs
Authors: Schwab, Christoph.
Hoang, Viet Ha.
Issue Date: 2013
Source: Hoang, V. H., & Schwab, C. (2013). Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs. Analysis and applications, 11(01), 1350001-.
Series/Report no.: Analysis and applications
Abstract: A class of second order, elliptic PDEs in divergence form with stochastic and anisotropic conductivity coefficients and n known, separated microscopic length scales εi, i = 1, …, n in a bounded domain D ⊂ ℝd is considered. Neither stationarity nor ergodicity of these coefficients is assumed. Sufficient conditions are given for the random solution to converge ℙ-a.s, as εi → 0, to a stochastic, elliptic one-scale limit problem in a tensorized domain of dimension (n + 1)d. It is shown that this stochastic limit problem admits best N-term "polynomial chaos" type approximations which converge at a rate σ > 0 that is determined by the summability of the random inputs' Karhúnen–Loève expansion. The convergence of the polynomial chaos expansion is shown to hold ℙ-a.s. and uniformly with respect to the scale parameters εi. Regularity results for the stochastic, one-scale limiting problem are established. An error bound for the approximation of the random solution at finite, positive values of the scale parameters εi is established in the case of two scales, and in the case of n > 2, scales convergence is shown, albeit without giving a convergence rate in this case.
DOI: 10.1142/S0219530513500012
Fulltext Permission: none
Fulltext Availability: No Fulltext
Appears in Collections:SPMS Journal Articles

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