Calculating principal eigen-functions of non-negative integral kernels: particle approximations and applications

TL;DR

This paper introduces particle approximation methods to numerically compute principal eigen-functions and eigen-values of non-negative integral kernels, with applications in rare events estimation and optimal control.

Contribution

It develops a generic interacting particle algorithm for approximating eigen-quantities and analyzes their properties, including error estimates and relevance to applications.

Findings

Particle algorithms effectively approximate eigen-functions and eigen-values.

Error bounds and mean properties of the algorithms are established.

Numerical examples demonstrate applications in importance sampling and stochastic control.

Abstract

Often in applications such as rare events estimation or optimal control it is required that one calculates the principal eigen-function and eigen-value of a non-negative integral kernel. Except in the finite-dimensional case, usually neither the principal eigen-function nor the eigen-value can be computed exactly. In this paper, we develop numerical approximations for these quantities. We show how a generic interacting particle algorithm can be used to deliver numerical approximations of the eigen-quantities and the associated so-called "twisted" Markov kernel as well as how these approximations are relevant to the aforementioned applications. In addition, we study a collection of random integral operators underlying the algorithm, address some of their mean and path-wise properties, and obtain $L_{r}$ error estimates. Finally, numerical examples are provided in the context of importance sampling for computing tail probabilities of Markov chains and computing value functions for a class of stochastic optimal control problems.