Nonlinear approximation of high-dimensional anisotropic analytic functions

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 by 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 $Y≔{[-1,1]}^N$. When the function of interest is the solution of a certain type of parametric PDE, recent results (Bonito et al., 2021 [4]) 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, namely the class introduced in Bonito et al. (2021) [5]. In this way we show both where the theory developed in Bonito et al. (2021) [4] depends on being in the setting of parametric PDEs with affine diffusion coefficients, and prove a more general result outside of this setting.