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

Bachmayr, Markus; Cohen, Albert; Migliorati, Giovanni

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 321-339.

We consider the linear elliptic equation $-\text{div}\left(a\nabla u\right)=f$ on some bounded domain $D$, where $a$ has the affine form $a=a\left(y\right)=\overline{a}+{\Sigma }_{j\ge 1}{y}_j{\psi }_j$ for some parameter vector $y={\left(y_j\right)}_{j\ge 1}\in U={[-1,1]}^{\mathbb{N}}$. We study the summability properties of polynomial expansions of the solution map $y⟼u\left(y\right)\in V:=H_0^1\left(D\right)$. We consider both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11­47] show that, under a uniform ellipticity assuption, for any . 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 $\ell {}^p$ summability of the ${(\parallel \psi {}_j\parallel {}_{L^{\infty }})}_{j\ge 1}$ implies the $\ell {}^p$ 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 $s=\frac{1}{p}-1\text{in}{L}^{\infty }(U,V)\text{or}s=\frac{1}{p}-\frac{1}{2}\text{in}{L}^2(U,V)$. In this paper we considerably improve these results by providing sufficient conditions of. In this paper we considerably improve these results by providing sufficient conditions of $\ell {}^p$ summability of the coefficient $V$-norm sequences expressed in terms of the pointwise summability properties of the $\left(\right|\psi {}_j{\left|\right)}_{j\ge 1}$. 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 support of the the supports of the $\psi {}_j$. For instance, in the case of disjoint supports, these results imply that for all $0<p<2$, the $\ell {}^p$ summability of the coefficient $V$-norm sequences follows from the weaker assumption that ${(\parallel \psi {}_j\parallel {}_{L^{\infty }})}_{j\ge 1}$ is $\ell {}^q$ summable for $q=q\left(p\right):=\frac{2p}{2-p}>p$. 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.

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