Multivariate Rational Approximation of Scattered Data Using the p-AAA Algorithm

TL;DR

This paper extends the p-AAA algorithm for multivariate rational approximation from grid data to scattered data, enabling effective approximation without uniform sampling by formulating and solving structured least-squares problems.

Contribution

The work introduces a novel extension of the p-AAA algorithm to handle scattered data, incorporating interpolation constraints and structured matrix solutions.

Findings

Effective approximation of scattered data demonstrated

Structured matrices enable closed-form solutions

Algorithm performs well on various examples

Abstract

Many algorithms for approximating data with rational functions are built on interpolation or least-squares approximation. Inspired by the adaptive Antoulas-Anderson (AAA) algorithm for the univariate case, the parametric adaptive Antoulas-Anderson (p-AAA) algorithm extends this idea to the multivariate setting, combining least-squares and interpolation formulations into a single effective approximation procedure. In its original formulation p-AAA operates on grid data, requiring access to function samples at every combination of discrete sampling points in each variable. In this work we extend the p-AAA algorithm to scattered data sets, without requiring uniform/grid sampling. In other words, our proposed p-AAA formulation operates on a set of arbitrary sampling points and is not restricted to a grid structure for the sampled data. Towards this goal, we introduce several formulations for rational least-squares optimization problems that incorporate interpolation conditions via constraints. We analyze the structure of the resulting optimization problems and introduce structured matrices whose singular value decompositions yield closed-form solutions to the underlying least-squares problems. Several examples illustrate computational aspects and the effectiveness of our proposed procedure.