Adaptive approximation of nonlinear eigenproblems by minimal rational interpolation

Abstract

AbstractWe 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 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.