Efficient Function Approximation in Enriched Approximation Spaces

TL;DR

This paper introduces an efficient algorithm for function approximation using enriched bases, which are overcomplete and can be ill-conditioned, demonstrating stability and accuracy in various computational examples.

Contribution

It proposes a simplified AZ algorithm tailored for enriched bases, providing a stable and efficient method for high-accuracy approximations.

Findings

The algorithm is stable and accurate in practice.

It effectively handles overcomplete and ill-conditioned systems.

Applications include non-standard and spectral approximation methods.

Abstract

An enriched approximation space is the span of a conventional basis with a few extra functions included, for example to capture known features of the solution to a computational problem. Adding functions to a basis makes it overcomplete and, consequently, the corresponding discretized approximation problem may require solving an ill-conditioned system. Recent research indicates that these systems can still provide highly accurate numerical approximations under reasonable conditions. In this paper we propose an efficient algorithm to compute such approximations. It is based on the AZ algorithm for overcomplete sets and frames, which simplifies in the case of an enriched basis. In addition, analysis of the original AZ algorithm and of the proposed variant gives constructive insights on how to achieve optimal and stable discretizations using enriched bases. We apply the algorithm to examples of enriched approximation spaces in literature, including a few non-standard approximation problems and an enriched spectral method for a 2D boundary value problem, and show that the simplified AZ algorithm is indeed stable, accurate and efficient.