Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

QR-based Parallel Set-Valued Approximation with Rational Functions

Simon Dirckx, Karl Meerbergen, Daan Huybrechs

TL;DR

The paper tackles efficient, accurate set-valued rational approximation for vector-valued functions by introducing a QR-based acceleration (QR-AAA) and a parallel variant (PQR-AAA). By exploiting a transitive row-interpolative decomposition (RID) via rank-revealing QR, QR-AAA preserves the accuracy of SV-AAA while markedly reducing computational cost, enabling parallelization with limited communication. The authors demonstrate comparable accuracy to SV-AAA across challenging nonlinear eigenvalue problems (NLEVPs) and show substantial speedups, including a large-scale BEM near-field wavenumber compression using PQR-AAA. These advances facilitate scalable, high-fidelity rational approximation for large sets of matrix-valued functions, with practical impact in NLEVP analysis and boundary element computations. The work provides a rigorous framework, implementation details, and extensive numerical experiments highlighting robustness, efficiency, and parallel scalability.

Abstract

In this article a fast and parallelizable algorithm for rational approximation is presented. The method, called (P)QR-AAA, is a (parallel) set-valued variant of the AAA algorithm for scalar functions. It builds on the set-valued AAA framework introduced by Lietaert, Meerbergen, P{é}rez and Vandereycken, accelerating it by using an approximate orthogonal basis obtained from a truncated QR decomposition. We demonstrate both theoretically and numerically this method's accuracy and efficiency. We show how it can be parallelized while maintaining the desired accuracy, with minimal communication cost.

QR-based Parallel Set-Valued Approximation with Rational Functions

TL;DR

The paper tackles efficient, accurate set-valued rational approximation for vector-valued functions by introducing a QR-based acceleration (QR-AAA) and a parallel variant (PQR-AAA). By exploiting a transitive row-interpolative decomposition (RID) via rank-revealing QR, QR-AAA preserves the accuracy of SV-AAA while markedly reducing computational cost, enabling parallelization with limited communication. The authors demonstrate comparable accuracy to SV-AAA across challenging nonlinear eigenvalue problems (NLEVPs) and show substantial speedups, including a large-scale BEM near-field wavenumber compression using PQR-AAA. These advances facilitate scalable, high-fidelity rational approximation for large sets of matrix-valued functions, with practical impact in NLEVP analysis and boundary element computations. The work provides a rigorous framework, implementation details, and extensive numerical experiments highlighting robustness, efficiency, and parallel scalability.

Abstract

In this article a fast and parallelizable algorithm for rational approximation is presented. The method, called (P)QR-AAA, is a (parallel) set-valued variant of the AAA algorithm for scalar functions. It builds on the set-valued AAA framework introduced by Lietaert, Meerbergen, P{é}rez and Vandereycken, accelerating it by using an approximate orthogonal basis obtained from a truncated QR decomposition. We demonstrate both theoretically and numerically this method's accuracy and efficiency. We show how it can be parallelized while maintaining the desired accuracy, with minimal communication cost.
Paper Structure (16 sections, 4 theorems, 45 equations, 9 figures, 1 table, 1 algorithm)

This paper contains 16 sections, 4 theorems, 45 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Set-Valued AAA
  3. SV-AAA as a Row Interpolative Decomposition
  4. QR-based SV-AAA
  5. Parallel QR-AAA
  6. The accumulate support problem
  7. Schematic description and communication costs
  8. Experiments
  9. NLEVP problems
  10. Problem 1: A Clamped Sandwich Beam
  11. Problem 2: Photonic Crystals
  12. Problem 3: Schrödinger's Equation in a Canyon Well
  13. Problem 4: Time Delay Systems with random delays
  14. Numerical results
  15. Comparison to weighted AAA
  16. ...and 1 more sections

Key Result

Theorem 2.1

Suppose $\mathbf{f}:\mathbb{C}\to\mathbb{C}^N$ can be approximated on $Z\subset\mathbb{C}$ by the set-valued rational function $\mathbf{r}_m$ with support points $Z_m \subset Z$ as in Equation eq:sv-aaa-approx, such that for some $\epsilon>0$ Let $F:=F(Z)$ and $\widetilde{F}_m:=\widetilde{F}_m(Z)$ be as in equations eq:defFmat and eq:defFmmat. Then the matrix $\widetilde{F}_m$ is of rank $m$, and

Figures (9)

  • Figure 1: Diagram showing the principle of QR-AAA. Here $\Gamma=\text{diag}(R)$ and $\widetilde{Q}=H_mQ(Z_m,:)$ with $H_m$ as in Theorem \ref{['thm:obs1']}. The dashed SV-AAA arrow indicated that the final $(Z_m,W_m)$ constitute a set-valued rational approximation $F\approx H_m F(Z_m,:)$.
  • Figure 2: The parallel QR based set-valued AAA approach. The original $F$ is distributed over the chosen nodes, and on each node an RRQR decomposition is executed in parallel ('pRRQR'). Then, in parallel, each (weighted) $Q$ is approximated using SV-AAA ('pSV-AAA'). Finally, these are accumulated ('accumul') by applying steps (3)-(4) outlined in the above.
  • Figure 3: Diagram illustrating the PQR-AAA pairwise accumulation stage. At each stage the '$\uplus$' sum $\hat{Q}_{\mu_1}\uplus\hat{Q}_{\mu_2}$ of the approximations in two nodes is constructed, after which SV-AAA$(\hat{Q}_{\mu_1}\uplus\hat{Q}_{\mu_2})$ is computed.
  • Figure 4: Residue $\textbf{res}_m$ for SV-AAA (left) and QR-AAA (right) over the degree $m$, for the first two selected problems.
  • Figure 5: Residue $\textbf{res}_m$ for SV-AAA (left) and QR-AAA (right) over the degree $m$, for the second two selected problems.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Theorem 2.1
  • proof
  • Theorem 2.2: transitive property of RIDs
  • proof
  • Remark 2.3
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Definition 4.2
  • Example 4.3
  • ...and 2 more