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Randomized sketching of nonlinear eigenvalue problems

Stefan Güttel, Daniel Kressner, Bart Vandereycken

TL;DR

This work proposes and analyzes a new sketching approach for large-scale vector- and matrix-valued functions called sketchAAA that leads to much better approximants than previously suggested approaches while retaining efficiency.

Abstract

Rational approximation is a powerful tool to obtain accurate surrogates for nonlinear functions that are easy to evaluate and linearize. The interpolatory adaptive Antoulas--Anderson (AAA) method is one approach to construct such approximants numerically. For large-scale vector- and matrix-valued functions, however, the direct application of the set-valued variant of AAA becomes inefficient. We propose and analyze a new sketching approach for such functions called sketchAAA that, with high probability, leads to much better approximants than previously suggested approaches while retaining efficiency. The sketching approach works in a black-box fashion where only evaluations of the nonlinear function at sampling points are needed. Numerical tests with nonlinear eigenvalue problems illustrate the efficacy of our approach, with speedups above 200 for sampling large-scale black-box functions without sacrificing on accuracy.

Randomized sketching of nonlinear eigenvalue problems

TL;DR

This work proposes and analyzes a new sketching approach for large-scale vector- and matrix-valued functions called sketchAAA that leads to much better approximants than previously suggested approaches while retaining efficiency.

Abstract

Rational approximation is a powerful tool to obtain accurate surrogates for nonlinear functions that are easy to evaluate and linearize. The interpolatory adaptive Antoulas--Anderson (AAA) method is one approach to construct such approximants numerically. For large-scale vector- and matrix-valued functions, however, the direct application of the set-valued variant of AAA becomes inefficient. We propose and analyze a new sketching approach for such functions called sketchAAA that, with high probability, leads to much better approximants than previously suggested approaches while retaining efficiency. The sketching approach works in a black-box fashion where only evaluations of the nonlinear function at sampling points are needed. Numerical tests with nonlinear eigenvalue problems illustrate the efficacy of our approach, with speedups above 200 for sampling large-scale black-box functions without sacrificing on accuracy.
Paper Structure (18 sections, 4 theorems, 34 equations, 3 figures, 10 tables, 2 algorithms)

This paper contains 18 sections, 4 theorems, 34 equations, 3 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Surrogate functions and the AAA algorithm
  3. Analysis of random sketching for function approximation
  4. Preliminaries
  5. Full (non-tensorized) probing vectors
  6. Tensorized probing vectors
  7. Applications and numerical experiments
  8. NLEVP benchmark collection
  9. Small NLEVPs
  10. Large NLEVPs
  11. An artificial example on the split form
  12. Impact on numerical solution of NLEVP
  13. Scattering problem
  14. Boundary element matrices
  15. Storage possible
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $\mathbf h \in L^2(\Omega, {\mathbb K}^N)$. Then there exist orthonormal vectors $\mathbf u_1, \ldots, \mathbf u_N \in {\mathbb K}^N$, orthonormal functions $v_1,\ldots, v_N\in L^2(\Omega, \mathbb K)$, and scalars $\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_N \ge 0$ such that

Figures (3)

  • Figure 1: Approximation errors of the rational interpolant for the buckling_plate example as functions of the degree $d$. The error of the surrogate functions ($\ell=1$ or $\ell=2$ probing vectors, tensorized or not) is shown in the left panel, and on the right the error of the full interpolants is shown. While the errors of the surrogates all converge to machine precision (left), the error of the full interpolants stagnates when scalar surrogates ($\ell=1$) are used (right). With $\ell=2$ probing vectors the surrogate and full approximants converge similarly. On the right, the error of the set-valued AAA approximant applied the nine components of the original function $F$ is indicated with dark grey dots.
  • Figure 2: Bands for the 90 and 50 percentiles of 100 random initializations of the surrogates (non-tensorized) for the buckling_plate and nep2 examples. The dashed line is the minimal error achieved by the set-valued AAA method applied to the original function $F$.
  • Figure 3: Scattering problem. Left: the Euclidean norm of $\mathbf f(z)$ as a function of $z$. Right: the relative error $\| \mathbf f(z) - \mathbf r^{(d)}(z) \|_{\infty} / \max_z \| \mathbf f(z) \|_\infty$ for the rational approximants $\mathbf r^{(d)}(z)$ with $\ell = 24$ probing vectors. The dark grey lines indicate the tolerances requested, namely, $10^{-8}$ and $10^{-12}$.

Theorems & Definitions (7)

  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 4.1