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

Linus Balicki; Serkan Gugercin

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arXiv · 2510.22861

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

Linus Balicki·Serkan Gugercin

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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.

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Linus Balicki, Serkan Gugercin. 2025-10-26. Multivariate Rational Approximation of Scattered Data Using the p-AAA Algorithm. https://arxiv.org/abs/2510.22861

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