Doucet, Arnaud; Johansen, Adam M.; Tadić, Vladislav B. On solving integral equations using Markov chain Monte Carlo methods. (English) Zbl 1193.65217 Appl. Math. Comput. 216, No. 10, 2869-2880 (2010). 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. Cited in 9 Documents 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 Keywords:Fredholm equation; trans-dimensional Markov chain Monte Carlo; sequential importance sampling; sequential Monte Carlo; numerical examples; second kind; von Neumann expansion PDFBibTeX XMLCite \textit{A. Doucet} et al., Appl. Math. Comput. 216, No. 10, 2869--2880 (2010; Zbl 1193.65217) Full Text: DOI References: [1] Bertsekas, D. P., Dynamic Programming and Optimal Control (1990), Athena Scientific [2] Griffiths, R. C.; Tavaré, S., Simulating probability distributions in the coalescent, Theoretical Population Biology, 46, 131-159 (1994) · Zbl 0807.92015 [3] Spanier, J.; Gelbard, E. M., Monte Carlo Principles and Neutron Transportation Problems (1969), Addison-Wesley: Addison-Wesley Reading, Massachusetts · Zbl 0244.62004 [4] Rubinstein, R., Simulation and the Monte Carlo Method (1981), Wiley: Wiley New York · Zbl 0529.68076 [5] Saberi-Nadjafi, J.; Heidari, M., Solving linear integral equations of the second kind with repeated modified trapezoid quadrature method, Applied Mathematics and Computation, 189, 1, 980-985 (2007) · Zbl 1263.65136 [6] Babolian, E.; Marsban, H. R.; Salmani, M., Using triangular orthogonal functions for solving Fredholm integral equations of the second kind, Applied Mathematics and Computation, 201, 1-2, 452-464 (2008) · Zbl 1147.65105 [7] Bertsekas, D. P.; Tsitsiklis, J., Neuro-dynamic Programming (1996), Athena Scientific · Zbl 0924.68163 [8] Del Moral, P.; Miclo, L., Branching and interacting particle systems and approximations of Feynman-Kac formulæ with applications to non-linear filtering, (Azéma, J.; Emery, M.; Ledoux, M.; Yor, M., Séminaire de Probabilités XXXIV. Séminaire de Probabilités XXXIV, Lecture Notes in Mathematics, vol. 1729 (2000), Springer-Verlag: Springer-Verlag Berlin), 1-145 · Zbl 0963.60040 [9] (Doucet, A.; de Freitas, N.; Gordon, N., Sequential Monte Carlo Methods in Practice, Statistics for Engineering and Information Science (2001), Springer-Verlag: Springer-Verlag New York) · Zbl 0967.00022 [10] Green, P. J., Reversible jump Markov Chain Monte Carlo computation and Bayesian model determination, Biometrika, 82, 711-732 (1995) · Zbl 0861.62023 [11] Green, P. J., Trans-dimensional Markov chain Monte Carlo, (Green, P. J.; Hjort, N. L.; Richardson, S., Highly Structured Stochastic Systems. Highly Structured Stochastic Systems, Oxford Statistical Science Series (2003), Oxford University Press), 179-206, Ch. 6 [12] Rust, J.; Traub, J. F.; Wózniakowski, H., Is there a curse of dimensionality for contraction fixed points in the worst case?, Econometrica, 70, 1, 285-329 (2002) · Zbl 1103.65316 [13] Veach, E.; Guibas, L. J., Metropolis light transport, (Proceedings of SIGGRAPH (1997), Addison-Wesley), 65-76 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.