Sparse polynomial approximation of parametric elliptic pdes. part II: Lognormal coefficients

Abstract

We consider the linear elliptic equation ?div(aVu) = f on some bounded domain D, where a has the form a = exp(b) with b a random function defined as b(y) = σj≥1 yjψj where y = (yj) € RN are i.i.d. standard scalar Gaussian variables and (ψj )j≥1 is a given sequence of functions in L ∞ (D). We study the summability properties of Hermite-Type expansions of the solution map y →u(y) || V := H1 0 (D), that is, expansions of the form u(y) = σF u-H-(y), where Hp (y) = || j≥1 H?j (yj) are the tensorized Hermite polynomials indexed by the set F of finitely supported sequences of nonnegative integers. Previous results [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797-826] have demonstrated that, for any 0 < p < 2 the Lp summability of (||uv||V )F€ follows from the weaker assumption that (||ψj||L∞)j≥1 is Lq summable for q := 2p 2?p < p. In the case of arbitrary supports, our results imply that the Lqsummability of (||uv||V )F€ follows from the Lpsummability of (d1β||ψj||L∞)j≥1 for some β > 1 2 , which still represents an improvement over the condition in [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797?826]. We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen-Loeve basis for the representation of b might be suboptimal compared to other representations, in terms of the resulting summability properties of (||uv||V )F€. While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.