Nonlinear approximation of high-dimensional anisotropic analytic functions

TL;DR

This paper extends the theoretical understanding of library approximation methods for high-dimensional anisotropic analytic functions, particularly those arising from parametric PDEs, by providing new bounds and generalizations beyond previous PDE-specific results.

Contribution

It generalizes existing approximation bounds from parametric PDEs to a broader class of anisotropic analytic functions, highlighting the dependence on PDE structure and expanding applicability.

Findings

Proves bounds on the number of local Taylor approximations needed for accuracy.

Shows the dependence of approximation theory on PDE structure.

Extends previous results to more general parametric PDEs.

Abstract

Motivated by nonlinear approximation results for classes of parametric partial differential equations (PDEs), we seek to better understand so-called library approximations to analytic functions of countably infinite number of variables. Rather than approximating a function of interest in a single space, a library approximation uses a collection of spaces and the best space may be chosen for any point in the domain. In the setting of this paper, we use a specific library which consists of local Taylor approximations on sufficiently small rectangular subdomains of the (rescaled) parameter domain, $Y:=[-1,1]^\mathbb{N}$. When the function of interest is the solution of a certain type of parametric PDE, recent results (Bonito et al, 2020, arXiv:2005.02565) prove an upper bound on the number of spaces required to achieve a desired target accuracy. In this work, we prove a similar result for a more general class of functions with anisotropic analyticity. In this way we show both where the theory developed in (Bonito et al 2020) depends on being in the setting of parametric PDEs with affine diffusion coefficients, and also expand the previous result to include more general types of parametric PDEs.