Multivariate Rational Approximation via Low-Rank Tensors and the p-AAA Algorithm

TL;DR

This paper extends the AAA rational approximation algorithm to multivariate functions using low-rank tensor decompositions, enabling efficient high-dimensional approximation and applications in parametric reduced-order modeling.

Contribution

The paper introduces the low-rank p-AAA algorithm that combines barycentric forms with tensor decompositions for multivariate rational approximation.

Findings

Effective in high-dimensional parametric modeling

Reduces computational and memory demands

Demonstrated success on four numerical examples

Abstract

Approximations based on rational functions are widely used in various applications across computational science and engineering. For univariate functions, the adaptive Antoulas-Anderson algorithm (AAA), which uses the barycentric form of a rational approximant, has established itself as a powerful tool for efficiently computing such approximations. The p-AAA algorithm, an extension of the AAA algorithm specifically designed to address multivariate approximation problems, has been recently introduced. A common challenge in multivariate approximation methods is that multivariate problems with a large number of variables often pose significant memory and computational demands. To tackle this hurdle in the setting of p-AAA, we first introduce barycentric forms that are represented in the terms of separable functions. This then leads to the low-rank p-AAA algorithm which leverages low-rank tensor decompositions in the setting of barycentric rational approximations. We discuss various theoretical and practical aspects of the proposed computational framework and showcase its effectiveness on four numerical examples. We focus specifically on applications in parametric reduced-order modeling for which higher-dimensional data sets can be tackled effectively with our novel procedure.