Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
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We consider the linear elliptic equation div(au) = f on some bounded domain D, where a has the form a = exp(b) with b a random function defined as b(y) = j 1yjj where y = (yj)Nare i.i.d. standard scalar Gaussian variables and (j)j 1 is a given sequence of functions in L(D). We study the summability properties of Hermite-type expansions of the solution map yu(y)V:=H01(D) , that is, expansions of the form u(y) = uH(y), where H(y) = j1Hj(yj) are the tensorized Hermite polynomials indexed by the set of finitely supported sequences of nonnegative integers. Previous results [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797826] have demonstrated that, for any 0 <p 1, the p summability of the sequence (jjL)j 1 implies p summability of the sequence ( uV). Such results ensure convergence rates ns with s = (1/p)(1/2) of polynomial approximations obtained by best n-term truncation of Hermite series, where the error is measured in the mean-square sense, that is, in L2(N,V,) , where is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the p summability of (uV) expressed in terms of the pointwise summability properties of the sequence (|j|)j 1. This leads to a refined analysis which takes into account the amount of overlap between the supports of the j. For instance, in the case of disjoint supports, our results imply that, for all 0 <p< 2 the p summability of (uV)follows from the weaker assumption that (jL)j 1is q summable for q:=2p/(2p) . In the case of arbitrary supports, our results imply that the p summability of (uV) follows from the
