A refined nonlinear least-squares method for the rational approximation problem

TL;DR

This paper introduces a refined nonlinear least-squares approach to improve the AAA algorithm for rational approximation, focusing on achieving minimal degree approximations with guaranteed error convergence.

Contribution

It proposes new refinement strategies for the AAA algorithm's linear step, ensuring smaller degrees and monotonic error reduction in rational approximation.

Findings

Enhanced approximation accuracy with smaller degrees

Theoretical analysis of gradient-based minimization

Numerical validation in function approximation and model reduction

Abstract

The adaptive Antoulas-Anderson (AAA) algorithm for rational approximation is a widely used method for the efficient construction of highly accurate rational approximations to given data. While AAA can often produce rational approximations accurate to any prescribed tolerance, these approximations may have degrees larger than what is actually required to meet the given tolerance. In this work, we consider the adaptive construction of interpolating rational approximations while aiming for the smallest feasible degree to satisfy a given error tolerance. To this end, we introduce refinement approaches to the linear least-squares step of the classical AAA algorithm that aim to minimize the true nonlinear least-squares error with respect to the given data. Furthermore, we theoretically analyze the derived approaches in terms of the corresponding gradients from the resulting minimization problems and use these insights to propose a new greedy framework that ensures monotonic error convergence. Numerical examples from function approximation and model order reduction verify the effectiveness of the proposed algorithm to construct accurate rational approximations of small degrees.