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Applications of QR-based Vector-Valued Rational Approximation

Simon Dirckx

TL;DR

QR-AAA delivers a fast, robust vector-valued rational approximation framework by first extracting a small basis $\mathbf{Q}$ for component functions and then applying SV-AAA to obtain a shared-pole vector-valued rational approximant $\mathbf{r}_m$. The method is demonstrated across Hedgehog-accelerated Stokes boundary integrals, vector-valued rational quadrature, simple multivariate rational approximation, multivariate analytic extension, and parallel near-field compression (PQR-AAA), achieving substantial speedups and accuracy gains. Across these applications, QR-AAA enables efficient high-dimensional, vector-valued rational modeling with shared poles, facilitating scalable boundary integral methods, quadrature construction, and multivariate function handling. The results highlight practical impact in fast Stokes computations, reliable quadrature design, flexible multivariate approximation, analytic extensions, and scalable near-field compression for Helmholtz BEM.

Abstract

Several applications of the QR-AAA algorithm, a greedy scheme for vector-valued rational approximation, are presented. The focus is on demonstrating the flexibility and practical effectiveness of QR-AAA in a variety of computational settings, including Stokes flow computation, multivariate rational approximation, function extension, the development of novel quadrature methods and near-field approximation in the boundary element method.

Applications of QR-based Vector-Valued Rational Approximation

TL;DR

QR-AAA delivers a fast, robust vector-valued rational approximation framework by first extracting a small basis for component functions and then applying SV-AAA to obtain a shared-pole vector-valued rational approximant . The method is demonstrated across Hedgehog-accelerated Stokes boundary integrals, vector-valued rational quadrature, simple multivariate rational approximation, multivariate analytic extension, and parallel near-field compression (PQR-AAA), achieving substantial speedups and accuracy gains. Across these applications, QR-AAA enables efficient high-dimensional, vector-valued rational modeling with shared poles, facilitating scalable boundary integral methods, quadrature construction, and multivariate function handling. The results highlight practical impact in fast Stokes computations, reliable quadrature design, flexible multivariate approximation, analytic extensions, and scalable near-field compression for Helmholtz BEM.

Abstract

Several applications of the QR-AAA algorithm, a greedy scheme for vector-valued rational approximation, are presented. The focus is on demonstrating the flexibility and practical effectiveness of QR-AAA in a variety of computational settings, including Stokes flow computation, multivariate rational approximation, function extension, the development of novel quadrature methods and near-field approximation in the boundary element method.
Paper Structure (15 sections, 2 theorems, 33 equations, 15 figures, 2 tables, 2 algorithms)

This paper contains 15 sections, 2 theorems, 33 equations, 15 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The QR-AAA method
  3. Vector-valued rational approximation and interpolative decompositions
  4. Applications of QR-AAA
  5. A vector-valued Hedgehog method (with J. Bagge)
  6. Rational quadrature (with D. Huybrechs)
  7. Computing the quadrature weights
  8. Numerical experiments
  9. Simple multivariate rational approximation
  10. Numerical experiments
  11. A two-step procedure
  12. Multivariate function extension
  13. Oscillatory function extension
  14. Extending the norm function
  15. Near-field compression using parallel QR-AAA (from PQRAAA)

Key Result

Theorem 2.1

Suppose $\mathbf{f}:\mathbb{F}\to\mathbb{F}^n$ and $\mathbf{r}_m$ is an SV-AAA approximation of $\mathbf{f}$ with support points $Z_m=\{\zeta_1,\ldots,\zeta_m\} \subset Z=\{z_1,\ldots,z_{N}\}$, such that for $\epsilon>0$ Let $\mathbf{F}:=\mathbf{F}(Z)$ and $\mathbf{R}_m:=\mathbf{R}_m(Z)$ be as in equation eq:defFmat. Then the matrix $\mathbf{R}_m$ is of rank at most $m$, and can be written as whe

Figures (15)

  • Figure 1: Diagram illustrating the Hedgehog set-up under consideration.
  • Figure 2: Numerical results for QR-AAA as compared to AAA for $512$ Hedgehog spikes, with $[\alpha,\beta]=[.1,.2]$. Left: relative errors for the Hedgehog spikes that have the largest relative interpolation error in $[0,\alpha]$, for $|Z|=100$. Right: timings for the regular Hedgehog method (each spike approximated separately) and for the QR-AAA accelerated version.
  • Figure 3: Quadrature error for the QR-AAA based quadrature (class $\Psi_1$) over the requested tolerance $\textbf{tol}$ for various function classes. Quadrature errors with weights computed using exact integrals (the second method from Section \ref{['sec:weightComp']}) are shown in red.
  • Figure 4: Quadrature error for the QR-AAA based quadrature (class $\Psi_2$) over the requested tolerance $\textbf{tol}$ for various function classes. Quadrature errors with weights computed using exact integrals (the second method from Section \ref{['sec:weightComp']}) are shown in red.
  • Figure 5: Scatter plot of the QR-AAA poles selected for the approximation of $\Psi_1$ and $\Psi_2$, with the QR-AAA tolerance set to $10^{-10}$.
  • ...and 10 more figures

Theorems & Definitions (4)

  • Theorem 2.1
  • Proposition 3.1
  • Remark 3.2
  • Remark 3.3