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

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Title: Calculating principal eigen-functions of non-negative integral kernels: particle approximations and applications
Authors: Whiteley, N
Kantas, N
Item Type: Journal Article
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.
Issue Date: 24-Mar-2017
Date of Acceptance: 12-Sep-2016
URI: http://hdl.handle.net/10044/1/40800
DOI: https://dx.doi.org/10.1287/moor.2016.0834
ISSN: 1526-5471
Publisher: INFORMS (Institute for Operations Research and Management Sciences)
Start Page: 1007
End Page: 1034
Journal / Book Title: Mathematics of Operations Research
Volume: 42
Issue: 4
Copyright Statement: Copyright © 2017, INFORMS
Keywords: Science & Technology
Technology
Physical Sciences
Operations Research & Management Science
Mathematics, Applied
Mathematics
interacting particle methods
eigenfunctions
rare events estimation
optimal control
diffusion Monte Carlo
MARKOV ADDITIVE PROCESSES
FEYNMAN-KAC FORMULAS
LIMIT-THEOREMS
SNELL ENVELOPE
MODELS
FILTERS
stat.CO
0102 Applied Mathematics
0103 Numerical And Computational Mathematics
0802 Computation Theory And Mathematics
Operations Research
Publication Status: Published
Open Access location: http://arxiv.org/pdf/1202.6678v3.pdf
Appears in Collections:Mathematics
Statistics
Faculty of Natural Sciences



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