Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients^∗

¹ Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005 Paris, France.
bachmayr@ljll.math.upmc.fr; cohen@ljll.math.upmc.fr; migliorati@ljll.math.upmc.fr
² Department of Mathematics, Texas A&M University, College Station, TX 77840, USA.
rdevore@math.tamu.edu

Received: 27 September 2015
Revised: 9 June 2016
Accepted: 16 July 2016

Abstract

We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the form a = exp(b) with b a random function defined as b(y) = ∑ _{j ≥ 1}y_jψ_j where y = (y_j) ∈ ℝ^Nare i.i.d. standard scalar Gaussian variables and (ψ_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 y → u(y) ∈ V := H₀¹(D) , that is, expansions of the form u(y) = ∑ _{ν ∈ ℱ}u_νH_ν(y), where H_ν(y) = ∏_j≥1H_νj(y_j) 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 ≤ 1, the ℓ^p summability of the sequence (j ∥ψ_j ∥_L^∞)_{j ≥ 1} implies ℓ^p summability of the sequence (∥ u_ν∥_V)_{ν ∈ ℱ}. Such results ensure convergence rates n^{− s} 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(ℝ^N,V,γ) , where γ is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the ℓ^p summability of (∥u_ν∥_V)_{ν ∈ ℱ} expressed in terms of the pointwise summability properties of the sequence (|ψ_j|)_{j ≥ 1}. This leads to a refined analysis which takes into account the amount of overlap between the supports of the ψ_j. For instance, in the case of disjoint supports, our results imply that, for all 0 <p< 2 the ℓ^p summability of (∥u_ν∥_V)_{ν ∈ ℱ}follows from the weaker assumption that (∥ψ_j∥_L^∞)_{j ≥ 1}is ℓ^q summable for q := 2p/(2−p) . In the case of arbitrary supports, our results imply that the ℓ^p summability of (∥u_ν∥_V)_{ν ∈ ℱ} follows from the ℓ^p summability of (j^β∥ψ_j∥_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−Loève basis for the representation of b might be suboptimal compared to other representations, in terms of the resulting summability properties of (∥u_ν∥_V)_{ν ∈ ℱ}. While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.