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.
email@example.com; firstname.lastname@example.org; email@example.com
2 Department of Mathematics, Texas A&M University, College Station, TX 77840, USA.
Revised: 9 June 2016
Accepted: 16 July 2016
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 ≥ 1yjψj where y = (yj) ∈ ℝNare 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 := H01(D) , that is, expansions of the form u(y) = ∑ ν ∈ ℱuνHν(y), where Hν(y) = ∏j≥1Hνj(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 ≤ 1, the ℓp summability of the sequence (j ∥ψj ∥L∞)j ≥ 1 implies ℓp summability of the sequence (∥ uν∥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(ℝN,V,γ) , where γ is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the ℓp summability of (∥uν∥V)ν ∈ ℱ expressed in terms of the pointwise summability properties of the sequence (|ψj|)j ≥ 1. This leads to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, our results imply that, for all 0 <p< 2 the ℓp summability of (∥uν∥V)ν ∈ ℱfollows from the weaker assumption that (∥ψj∥L∞)j ≥ 1is ℓq summable for q := 2p/(2−p) . In the case of arbitrary supports, our results imply that the ℓp summability of (∥uν∥V)ν ∈ ℱ follows from the ℓp summability of (jβ∥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−Loève basis for the representation of b might be suboptimal compared to other representations, in terms of the resulting summability properties of (∥uν∥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.
Mathematics Subject Classification: 41A10 / 41A58 / 41A63 / 65N15 / 65T60
Key words: Stochastic PDEs / lognormal coefficients / n-term approximation / Hermite polynomials
© EDP Sciences, SMAI 2016