A continuous approach to computing the pseudospectra of linear operators

TL;DR

This paper introduces a novel continuous method for computing the pseudospectra of linear operators, avoiding discretization errors and spectral pollution, and demonstrating high accuracy through numerical examples.

Contribution

It presents an operator-based Lanczos process approach for pseudospectra computation, which is adaptive, accurate, and free from spectral pollution.

Findings

Method is free of spectral pollution

Achieves nearly optimal accuracy

Demonstrated effectiveness through numerical examples

Abstract

We propose a continuous approach to computing the pseudospectra of linear operators with compact or compact-plus-scalar resolvent, following a 'solve-then-discretize' strategy. Instead of taking a finite section approach or using a finite-dimensional matrix to approximate the operator of interest, the new method employs an operator analogue of the Lanczos process to work with operators and functions directly. The method is shown to be free of spectral pollution and spectral invisibility, fully adaptive, and nearly optimal in accuracy. The advantages of the method are demonstrated by extensive numerical examples and comparison with the traditional method.