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

Bachmayr, Markus; Cohen, Albert; DeVore, Ronald; Migliorati, Giovanni

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 341-363.

We consider the linear elliptic equation $-\text{div}\left(a\nabla u\right)=f$ on some bounded domain $D$, where $a$ has the form $a=\text{exp}\left(b\right)$ with $b$ a random function defined as $b\left(y\right)={\sum }_{j\ge 1}{y}_j\psi {}_j$ where $y=\left(y_j\right)\in {ℝ}^{\mathbb{N}}$ are i.i.d. standard scalar Gaussian variables and ${\left(\psi {}_j\right)}_{j\ge 1}$ is a given sequence of functions in $L^{\infty }\left(D\right)$. We study the summability properties of Hermite-type expansions of the solution map $y⟼u\left(y\right)\in V:=H_0^1\left(D\right)$, that is, expansions of the form(D) , that is, expansions of the form $u\left(y\right)={\sum }_{\nu \in F}{u}_{\nu }{H}_{\nu }\left(y\right)$, where $H_{\nu }\left(y\right){=}_{j\ge 1}{H}_{{\nu }_j}\left(y_j\right)$ are the tensorized Hermite polynomials indexed by the set H$ℱ$ 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 $0<p\le 1$, the $\ell {}^p$ summability of the sequence ${(j\parallel \psi {}_j\parallel {}_{L^{\infty }})}_{j\ge 1}$ implies $\ell {}^p$ summability of the sequence ${(\parallel u_{\nu }\parallel {}_V)}_{\nu \in ℱ}$. Such results ensure convergence rates $n^{-s}$ with $s=\frac{1}{p}-\frac{1}{2}$ 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 $L^2({ℝ}^N,V,\gamma )$, where $\gamma$ is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the $\ell {}^p$ summability of ${(\parallel u_{\nu }\parallel {}_V)}_{\nu \in ℱ}$ expressed in terms of the pointwise summability properties of the sequence $\left(\right|\psi {}_j{\left|\right)}_{j\ge 1}$. This leads to a refined analysis which takes into account the amount of overlap between the supports of the $\psi {}_j$. For instance, in the case of disjoint supports, our results imply that, for all $0<p<2$ the $\ell {}^p$ summability of ${(\parallel u_{\nu }\parallel {}_V)}_{\nu \in ℱ}$follows from the weaker assumption that ${(\parallel \psi {}_j\parallel {}_{L^{\infty }})}_{j\ge 1}$is $\ell {}^q$ summable for $q:=\frac{2p}{2-p}>p$. In the case of arbitrary supports, our results imply that the $\ell {}^p$ summability of ${(\parallel u_{\nu }\parallel {}_V)}_{\nu \in ℱ}$ follows from the $\ell {}^p$ summability of ${(j^{\beta }\parallel \psi {}_j\parallel {}_{L^{\infty }})}_{j\ge 1}$ for some $>\frac{1}{2}$ which ch 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 might be suboptimal compared to other representations, in terms of the resulting summability properties of ${(\parallel u_{\nu }\parallel {}_V)}_{\nu \in ℱ}$. While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.

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