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

Nick Whiteley, Nikolas Kantas

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
235 Downloads (Pure)

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 Lr 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.
Original languageEnglish
Number of pages28
JournalMathematics of Operations Research
Early online date24 Mar 2017
DOIs
Publication statusE-pub ahead of print - 24 Mar 2017

Keywords

  • interacting particle methods
  • eigen-functions
  • rare events estimation
  • optimal control
  • diffusion Monte Carlo

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