Randomized sketching of nonlinear eigenvalue problems

TL;DR

This paper introduces sketchAAA, a novel sketching method for nonlinear eigenvalue problems that significantly improves efficiency and accuracy over existing methods, enabling faster approximations of large-scale functions.

Contribution

The paper presents a new sketching approach called sketchAAA for nonlinear eigenvalue problems, offering better approximants and efficiency in large-scale black-box function evaluations.

Findings

Achieves over 200x speedup in sampling large-scale functions

Maintains high accuracy in nonlinear eigenvalue problem approximations

Outperforms previous approaches in efficiency and quality

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.