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On solving integral equations using Markov chain Monte Carlo methods

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Doucet, Arnaud, Johansen, Adam M. and Tadić, Vladislav B.. (2010) On solving integral equations using Markov chain Monte Carlo methods. Applied Mathematics and Computation, Vol.216 (No.10). pp. 2869-2880. ISSN 0096-3003

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Official URL: http://dx.doi.org/10.1016/j.amc.2010.03.138

Abstract

In this paper, we propose an original approach to the solution of Fredholm equations of the second kind. We interpret the standard Von Neumann expansion of the solution as an expectation with respect to a probability distribution defined on a union of subspaces of variable dimension. Based on this representation, it is possible to use trans-dimensional Markov chain Monte Carlo (MCMC) methods such as Reversible Jump MCMC to approximate the solution numerically. This can be an attractive alternative to standard Sequential Importance Sampling (SIS) methods routinely used in this context. To motivate our approach, we sketch an application to value function estimation for a Markov decision process. Two computational examples are also provided.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Fredholm equations, Markov processes, Monte Carlo method
Journal or Publication Title:	Applied Mathematics and Computation
Publisher:	Elsevier Science Inc
ISSN:	0096-3003
Date:	15 July 2010
Volume:	Vol.216
Number:	No.10
Number of Pages:	12
Page Range:	pp. 2869-2880
Identification Number:	10.1016/j.amc.2010.03.138
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
References:	[1] D. P. Bertsekas, Dynamic Programmming and Optimal Control, Athena Scientific, 1990. [2] R. C. Griffiths, S. Tavare, Simulating probability distributions in the coalescent, Theoretical Population Biology 46 (1994) 131{159. [3] J. Spanier, E. M. Gelbard, Monte Carlo Principles and Neutron Transportation Problems, Addison-Wesley, Reading, Massachusetts, 1969. [4] R. Rubinstein, Simulation and the Monte Carlo Method, Wiley, New York, 1981. [5] J. Saberi-Nadja, M. Heidari, Solving linear integral equations of the second kind with repeated modified trapezoid quadrature method, Applied Mathematics and Computation 189 (1) (2007) 980{985. [6] E. Babolian, H. R. Marsban, M. Salmani, Using triangular orthogonal functions for solving Fredholm integral equations of the second kind, Applied Mathematics and Computation 201 (1{2) (2008) 452{464. [7] D. P. Bertsekas, J. Tsitsiklis, Dynamic Programmming and Optimal Control, Athena Scientific, 1996. [8] P. Del Moral, L. Miclo, Branching and interacting particle systems and approximations of Feynman-Kac formulae with applications to non-linear filtering, in: J. Azema, M. Emery, M. Ledoux, M. Yor (Eds.), Seminaire de Probabilites XXXIV, Vol. 1729 of Lecture Notes in Mathematics, Springer Verlag, Berlin, 2000, pp. 1{145. [9] A. Doucet, N. de Freitas, N. Gordon (Eds.), Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Science, Springer Verlag, New York, 2001. [10] P. J. Green, Reversible jump Markov Chain Monte Carlo computation and Bayesian model determination, Biometrika 82 (1995) 711{732. [11] P. J. Green, Trans-dimensional Markov chain Monte Carlo, in: P. J. Green, N. L. Hjort, S. Richardson (Eds.), Highly Structured Stochastic Systems, Oxford Statistical Science Series, Oxford University Press, 2003, Ch. 6, pp. 179{206. [12] J. Rust, J. F. Traub, H. Wozniakowski, Is there a curse of dimensionality for contraction fixed points in the worst case?, Econometrica 70 (1) (2002) 285{329. [13] E. Veach, L. J. Guibas, Metropolis light transport, in: Proceedings of SIGGRAPH, Addison-Wesley, 1997, pp. 65{76.
URI:	http://wrap.warwick.ac.uk/id/eprint/5735