Numerical Fredholm determinants for matrix-valued kernels on the real line

TL;DR

This paper develops and analyzes a numerical method for approximating Fredholm determinants of matrix-valued kernels on the real line, extending previous scalar-focused work and providing error estimates and numerical validation.

Contribution

It introduces a truncation and quadrature-based numerical approach for matrix-valued kernels on the real line, with rigorous error estimates and practical numerical results.

Findings

Error estimates for truncation and quadrature approximations

Extension of scalar kernel analysis to matrix-valued kernels

Numerical validation on stability analysis of nonlinear wave equations

Abstract

We analyze a numerical method for computing Fredholm determinants of trace class and Hilbert Schmidt integral operators defined in terms of matrix-valued kernels on the entire real line. With this method, the Fredholm determinant is approximated by the determinant of a matrix constructed by truncating the kernel of the operator to a finite interval and then applying a quadrature rule. Under the assumption that the kernel decays exponentially, we derive an estimate relating the Fredholm determinant of the operator on the real line to that of its truncation to a finite interval. Then we derive a quadrature error estimate relating the Fredholm determinant of a matrix-valued kernel on a finite interval to its numerical approximation obtained via an adaptive composite Simpson's quadrature rule. These results extend the analysis of Bornemann which focused on Fredholm determinants of trace class operators defined by scalar-valued kernels on a finite interval. Numerical results are provided for a Birman-Schwinger operator that characterizes the stability of stationary solutions of nonlinear wave equations.