On efficient algorithms for computing near-best polynomial approximations to high-dimensional, Hilbert-valued functions from limited samples

TL;DR

This paper develops robust, efficient algorithms with theoretical guarantees for computing near-best polynomial approximations of high-dimensional, Hilbert-valued functions from limited samples, advancing the understanding of sparse approximation methods in computational science.

Contribution

It introduces new algorithms and theoretical analysis that achieve near-best approximation rates for high-dimensional functions, including Hilbert-valued functions, with robustness to errors.

Findings

Algorithms achieve exponential or algebraic convergence rates.

Theoretical guarantees ensure robustness to sampling and discretization errors.

Numerical experiments validate the effectiveness of the proposed methods.

Abstract

Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in uncertainty quantification where the function is the solution map of a parametric or stochastic differential equation (DE). Yet, sparse polynomial approximation lacks a complete theory. On the one hand, there is a well-developed theory of best $s$-term polynomial approximation, which asserts exponential or algebraic rates of convergence for holomorphic functions. On the other, there are increasingly mature methods such as (weighted) $\ell^1$-minimization for computing such approximations. While the sample complexity of these methods has been analyzed with compressed sensing, whether they achieve best $s$-term approximation rates is not fully understood. Furthermore, these methods are not algorithms per se, as they involve exact minimizers of nonlinear optimization problems. This paper closes these gaps. Specifically, we consider the following question: are there robust, efficient algorithms for computing approximations to finite- or infinite-dimensional, holomorphic and Hilbert-valued functions from limited samples that achieve best $s$-term rates? We answer this affirmatively by introducing algorithms and theoretical guarantees that assert exponential or algebraic rates of convergence, along with robustness to sampling, algorithmic, and physical discretization errors. We tackle both scalar- and Hilbert-valued functions, this being key to parametric or stochastic DEs. Our results involve significant developments of existing techniques, including a novel restarted primal-dual iteration for solving weighted $\ell^1$-minimization problems in Hilbert spaces. Our theory is supplemented by numerical experiments demonstrating the efficacy of these algorithms.