Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients^∗

Sorbonne Universités, UPMC Univ. Paris 6, 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

Received: 27 September 2015
Revised: 26 May 2016
Accepted: 9 June 2016

Abstract

We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the affine form a = a(y) = ā + ∑_j≥1y_jψ_j for some parameter vector y = (y_j)_{j ≥ 1} ∈ U = [−1,1]^N. We study the summability properties of polynomial expansions of the solution map y → u(y) ∈ V := H₀¹(D) . We consider both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47] show that, under a uniform ellipticity assuption, for any 0 <p< 1, the ℓ^p summability of the (∥ψ_j∥_L^∞)_{j ≥ 1} implies the ℓ^p summability of the V-norms of the Taylor or Legendre coefficients. Such results ensure convergence rates n^{− s} of polynomial approximations obtained by best n-term truncation of such series, with s = (1/p)−1 in L^∞(U,V) or s = (1/p)−(1/2) in L²(U,V). In this paper we considerably improve these results by providing sufficient conditions of ℓ^p summability of the coefficient V-norm sequences expressed in terms of the pointwise summability properties of the (|ψ_j|)_{j ≥ 1}. The approach in the present paper strongly differs from that of [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47], which is based on individual estimates of the coefficient norms obtained by the Cauchy formula applied to a holomorphic extension of the solution map. Here, we use weighted summability estimates, obtained by real-variable arguments. While the obtained results imply those of [7] as a particular case, they lead 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, these results imply that for all 0 <p< 2, the ℓ^p summability of the coefficient V-norm sequences follows from the weaker assumption that (∥ψ_j∥_L^∞)_{j ≥ 1} is ℓ^q summable for q = q(p) := 2p/(2−p) . We provide a simple analytic example showing that this result is in general optimal and illustrate our findings by numerical experiments. The analysis in the present paper applies to other types of linear PDEs with similar affine parametrization of the coefficients, and to more general Jacobi polynomial expansions.