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

Linus Balicki, Serkan Gugercin

TL;DR

This work extends the multivariate p-AAA framework to scattered data by developing linear least-squares formulations with interpolation constraints, enabling rational approximation on arbitrary sampling sets. It leverages a barycentric representation for multivariate rational functions, uses structured matrices and SVD to obtain closed-form LS solutions, and introduces a scattered interpolation variant that preserves interpolation at chosen points while avoiding spurious singularities. The approach is demonstrated through grid-based and scattered-data examples, including a MEMS transfer function and a stationary thermal model, showing accurate approximations even with substantial data gaps. The results provide a practical, data-driven method for high-dimensional rational approximation with scalable computation, and outline future directions toward exact recovery and extensions of Loewner-type frameworks to scattered data.

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.

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

TL;DR

This work extends the multivariate p-AAA framework to scattered data by developing linear least-squares formulations with interpolation constraints, enabling rational approximation on arbitrary sampling sets. It leverages a barycentric representation for multivariate rational functions, uses structured matrices and SVD to obtain closed-form LS solutions, and introduces a scattered interpolation variant that preserves interpolation at chosen points while avoiding spurious singularities. The approach is demonstrated through grid-based and scattered-data examples, including a MEMS transfer function and a stationary thermal model, showing accurate approximations even with substantial data gaps. The results provide a practical, data-driven method for high-dimensional rational approximation with scalable computation, and outline future directions toward exact recovery and extensions of Loewner-type frameworks to scattered data.

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.
Paper Structure (10 sections, 4 theorems, 70 equations, 3 figures, 1 algorithm)

This paper contains 10 sections, 4 theorems, 70 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Two-variable barycentric forms
  3. Rational approximation with grid data
  4. Linear least-squares for rational approximation
  5. Interpolation constraints for grid data
  6. The p-AAA algorithm
  7. Rational approximation with scattered data
  8. Linear least-squares for scattered data
  9. Singularities of sparse barycentric forms
  10. Interpolation constraints for scattered data

Key Result

Proposition 1

Consider the grid sampling points $\textbf{S}$ in eq:samplingPoints, the corresponding sample data $\textbf{D}$ in eq:samples, fixed sets of nodes $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$ as in eq:barycentricNodes, and the set of admissible rational functions $\mathcal{R}$ in eq:unconstrainedAd where

Figures (3)

  • Figure 1: Various settings for the constrained LS problem \ref{['eq:introPAAALS']} arising in rational approximation that we consider in this manuscript. Red crosses visualize the sampling data set $\textbf{S}$ and cyan circles the interpolation set $\textbf{I}$. An arrow originating at an illustrated setting indicates that it is a special case of the setting that it is pointing at.
  • Figure 2: Algorithm \ref{['alg:paaa']} applied to the peaks function which is depicted in the left subfigure. On the right, we see the p-AAA approximation error after $23$ iterations. Additionally, locations of sampling points (red dots) and interpolation points selected by the algorithm (cyan dots) are depicted.
  • Figure 3: Zeros of the numerator and denominator, singularities, and interpolation points discussed in Example \ref{['ex:singularities']}.

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Example 1
  • Proposition 3
  • Example 2
  • Theorem 1
  • proof