The AZ algorithm for least squares systems with a known incomplete generalized inverse

TL;DR

This paper presents the AZ algorithm for solving least squares problems with ill-conditioned matrices, leveraging a known approximate inverse to achieve high-order solutions in complex settings.

Contribution

The paper introduces the AZ algorithm that utilizes a known generalized inverse to efficiently solve ill-conditioned least squares systems with low-rank corrections.

Findings

Effective in function approximation on irregular domains

Handles weighted least squares with skewed weights

Achieves high-order accuracy in spectral approximation of singular functions

Abstract

We introduce an algorithm for the least squares solution of a rectangular linear system $Ax=b$, in which $A$ may be arbitrarily ill-conditioned. We assume that a complementary matrix $Z$ is known such that $A - AZ^*A$ is numerically low rank. Loosely speaking, $Z^*$ acts like a generalized inverse of $A$ up to a numerically low rank error. We give several examples of $(A,Z)$ combinations in function approximation, where we can achieve high-order approximations in a number of non-standard settings: the approximation of functions on domains with irregular shapes, weighted least squares problems with highly skewed weights, and the spectral approximation of functions with localized singularities. The algorithm is most efficient when $A$ and $Z^*$ have fast matrix-vector multiplication and when the numerical rank of $A - AZ^*A$ is small.