Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs
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Date
2011-02Type
- Report
ETH Bibliography
yes
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Abstract
A class of second order, elliptic PDEs in divergence form with stochastic and anisotropic conductivity coefficients and $n$ known, separated microscopic length scales $\epsilon_i$, $i=1,...,n$ in a bounded domain $D\subset R^d$ 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 $\epsilon_i\rightarrow 0$, 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 $\sigma>0$ 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 $\epsilon_i$. 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 $\epsilon_i$ 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. Show more
Publication status
publishedExternal links
Journal / series
SAM Research ReportVolume
Publisher
Seminar for Applied Mathematics, ETH ZurichOrganisational unit
03435 - Schwab, Christoph / Schwab, Christoph
Funding
247277 - Automated Urban Parking and Driving (EC)
Related publications and datasets
Is previous version of: https://doi.org/10.3929/ethz-a-010399583
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ETH Bibliography
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