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

TL;DR

This work extends adaptive rational approximation to multivariate settings via the p-AAA framework and tackles the curse of dimensionality by enforcing a low-rank, separable representation of the barycentric coefficients. By expressing the barycentric form with separable components and coupling it to CP tensor decompositions, the authors develop a low-rank p-AAA algorithm that replaces large LS problems with contracted, ALS-based solves, dramatically reducing memory and compute requirements. The approach is validated on synthetic, thermal, trigonometric, and mass–spring–damper models, demonstrating accurate surrogates with substantially improved scalability and regularization effects. The results indicate strong potential for parametric reduced-order modeling in high-dimensional settings, with future work addressing non-grid data and data-tensor scalability.

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.

Figures (7)

• Figure 1: Sample data errors for example \ref{['sec:synthetic']}. The relative (nonlinear) LS and relative maximum errors are depicted for the standard p-AAA algorithm and low-rank p-AAA with various choices for $r$.
• Figure 2: Validation data errors for example \ref{['sec:synthetic']}. The relative (nonlinear) LS and relative maximum errors are depicted for the standard p-AAA algorithm and low-rank p-AAA with various choices for $r$.
• Figure 3: Surface plot for $\textbf{f}$ and the error $\textbf{e} := \lvert \textbf{f} - \textbf{r} \rvert$ for example \ref{['sec:thermalblock']}. Here, we consider $p^{(3)} = 10^{-7}$ and $p^{(4)} = 10^{3}$ which are an order of magnitude below/above the sampling points used for computing $\textbf{r}$.
• Figure 4: Surface plot for $\textbf{f}$ and the error $\textbf{e} := \lvert \textbf{f} - \textbf{r} \rvert$ for example \ref{['sec:trigexample3']}. The figure depicts $\textbf{f}$ and $\textbf{e}$ with the fixed value $z^{(3)} = 0$.
• Figure 5: Linearized and nonlinear LS approximation errors for example \ref{['sec:trigexample3']}. The errors are depicted for each ALS iteration step performed throughout the low-rank p-AAA algorithm. Hence, each cumulative ALS iteration depicted on the $x$-axis represents one complete loop of Algorithm \ref{['alg:als']}. Additionally, the outer iterations (i.e., the iterations of Algorithm \ref{['alg:lrpaaa']}) where new interpolation points were added through a greedy selection are indicated via markers in the plots. The final order of the rational approximant is $(12,12,13)$ for this example.

Theorems & Definitions (4)

• Lemma 1
• Lemma 2
• Lemma 3
• proof