## N-term Galerkin Wiener chaos approximations of elliptic PDEs with lognormal Gaussian random inputs

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Author

Hoang, V.H.

Schwab, Christoph

Date

2011-09Type

- Report

ETH Bibliography

yes
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Abstract

We consider diffusion in a random medium modeled as diffusion equation with lognormal Gaussian diffusion coefficient. Sufficient conditions on the log permeability are provided in order for a weak solution to exist in certain Bochner-Lebesgue spaces with respect to a Gaussian measure. The stochastic problem is reformulated as an equivalent deterministic parametric problem on R^N. It is shown that the weak solution can be represented as Wiener-Ito Polynomial Chaos series of Hermite Polynomials of a countable number of i.i.d standard Gaussian random variables taking values in R^1.<br/><br/>We establish sufficient conditions on the random inputs for weighted sequence of norms of the Wiener-Ito decomposition coefficients of the random solution to be p-summable for some 0 < p < 1. For random inputs with additional spatial regularity, stronger norms of the weighted coefficient sequence in the random solutions' Wiener-Ito decomposition are shown to be p-summable for the same value of 0 < p < 1.<br/><br/>We infer rates of nonlinear, best N-term Wiener Polynomial Chaos approximations of the random field, as well as for Finite Element discretizations of these approximations from a dense, nested family V_0 \subset V_1 \subset V_2 \subset ... V of finite element spaces of continuous, piecewise linear Finite Elements. Show more

Publication status

unpublishedJournal / series

Research reportsVolume

(2011/59)Publisher

Seminar für Angewandte Mathematik, ETHSubject

Lognormal Gaussian Random Field; Stochastic Diffusion Equation, Wiener-Itˆo decomposition; Polynomial chaos; Random media; Best N-term approximation; Hermite PolynomialsOrganisational unit

03435 - Schwab, Christoph / Schwab, Christoph
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