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On solving integral equations using Markov chain Monte Carlo methods. (English) Zbl 1193.65217

Summary: 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 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.

MSC:

65R20 Numerical methods for integral equations
45B05 Fredholm integral equations
65C05 Monte Carlo methods
65C40 Numerical analysis or methods applied to Markov chains
60J22 Computational methods in Markov chains
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