Adaptive approximation of nonlinear eigenproblems by minimal rational interpolation

TL;DR

This paper introduces an adaptive rational interpolation method for efficiently solving nonlinear eigenproblems by selectively sampling data points, leading to accurate eigenvalue and eigenvector approximations.

Contribution

It presents a novel adaptive minimal rational interpolation approach that optimally selects sampling points for solving nonlinear eigenproblems.

Findings

Effective eigenvalue approximation demonstrated on numerical examples

Adaptive sampling reduces computational cost

Method accurately recovers eigenvalues and eigenvectors

Abstract

We describe a strategy for solving nonlinear eigenproblems numerically. Our approach is based on the approximation of a vector-valued function, defined as solution of a non-homogeneous version of the eigenproblem. This approximation step is carried out via the minimal rational interpolation method. Notably, an adaptive sampling approach is employed: the expensive data needed for the approximation is gathered at locations that are optimally chosen by following a greedy error indicator. This allows the algorithm to employ computational resources only where "most of the information" on not-yet-approximated eigenvalues can be found. Then, through a post-processing of the surrogate, the sought-after eigenvalues and eigenvectors are recovered. Numerical examples are used to showcase the effectiveness of the method.