Finite sections of random Jacobi operators

TL;DR

This paper investigates a finite section approach for approximating solutions to infinite-dimensional equations involving random Jacobi operators, focusing on truncation techniques for non-selfadjoint and self-adjoint cases.

Contribution

It introduces a novel finite section method tailored for random Jacobi operators, enabling effective approximation of infinite difference equations with stochastic coefficients.

Findings

Developed a truncation technique for random Jacobi operators

Extended analysis to both non-selfadjoint and self-adjoint cases

Provided insights into the numerical approximation of stochastic difference equations

Abstract

This article is about a problem in the numerical analysis of random operators. We study a version of the finite section method for the approximate solution of equations $Ax=b$ in infinitely many variables, where $A$ is a random Jacobi operator. In other words, we approximately solve infinite second order difference equations with stochastic coefficients by reducing the infinite volume case to the (large) finite volume case via a particular truncation technique. For most of the paper we consider non-selfadjoint operators $A$ but we also comment on the self-adjoint case when simplifications occur.