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N-term Wiener Chaos Approximation Rates for elliptic PDEs with lognormal Gaussian random inputs

by V. H. Hoang and Ch. Schwab

(Report number 2011-59)

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 $\mathbb{R}^\mathbb{N}$. It is shown that the weak solution can be represented as Wiener-Itô Polynomial Chaos series of Hermite Polynomials of a countable number of i.i.d standard Gaussian random variables taking values in $\mathbb{R}^1$.

We establish sufficient conditions on the random inputs for weighted sequence of norms of the Wiener-Itô 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 decomposition are shown to be $p$-summable for the same value of 0<p<1.

We prove 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.

Keywords: Lognormal Gaussian Random Field, Stochastic Diffusion Equation, Wiener-Itô decomposition, polynomial chaos, random media, best N-term approximation, Hermite Polynomials

@Techreport{HS11_111,
  author = {V. H. Hoang and Ch. Schwab},
  title = {N-term Wiener Chaos Approximation Rates for elliptic PDEs with lognormal Gaussian random inputs},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-59},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-59.pdf },
  year = {2011}
}

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