A global quadratic speed-up for computing the principal eigenvalue of Perron-like operators

TL;DR

This paper introduces a new algorithm based on the min-max Collatz-Wielandt formalism for efficiently computing the principal eigenvalue and eigenfunction of Perron-like operators, with proven quadratic convergence and practical numerical demonstrations.

Contribution

The paper presents a novel algorithm with quadratic convergence for Perron-like operators, extending eigenvalue computation methods to PDE-related operators with rigorous convergence proofs.

Findings

Quadratic convergence of the proposed algorithm

Effective computation of principal eigenvalues for PDE operators

Numerical examples demonstrating scheme effectiveness

Abstract

We consider a new algorithm in light of the min-max Collatz-Wielandt formalism to compute the principal eigenvalue and the eigenvector (eigen-function) for a class of positive Perron-Frobenius-like operators. Such operators are natural generalizations of the usual nonnegative primitive matrices. These have nontrivial applications in PDE problems such as computing the principal eigenvalue of Dirichlet Laplacian operators on general domains. We rigorously prove that for general initial data the corresponding numerical iterates converge globally to the unique principal eigenvalue with quadratic convergence. We show that the quadratic convergence is sharp with compatible upper and lower bounds. We demonstrate the effectiveness of the scheme via several illustrative numerical examples.