In modeling particle transport through a medium, the path of a
particle behaves as a transient Markov chain. We are interested in
characteristics of the particle's movement conditional on its starting state,
which take the form of a "score" accumulated with each transition. Importance
sampling is an essential variance reduction technique in this setting, and we
provide an adaptive (iteratively updated) importance sampling algorithm that
converges exponentially to the solution. Examples illustrating this phenomenon
are provided.
References
Apostol, T. A. (1957). Mathematical Analy sis. Addison-Wesley, Reading, MA.
Booth, T. E. (1985). Exponential convergence for Monte Carlo particle transport? Trans. Amer. Nuclear Soc. 50 267-268.
Booth, T. E. (1986). A Monte Carlo learning/biasing experiment with intelligent random numbers. Nuclear Science and Engineering 92 465-481.
Booth, T. E. (1988). The intelligent random number technique in MCNP. Nuclear Science and Engineering 100 248-254.
Booth, T. E. (1989). Zero-variance solutions for linear Monte Carlo. Nuclear Science and Engineering 102 332-340.
Brieman, L. (1968). Probability. Addison-Wesley, Reading, MA.
Briesmeister, J. F., ed. (1993). MCNP-A general Monte Carlo N-particle transport code. Version 4A, Los Alamos National Laboratory Report LA-12625-MS.
Carter, L. L. and Cashwell, E. D. (1975). Particle Transport Simulation and the Monte Carlo Method. Technical Information Center, Springfield, VA, Energy Research and Development Administration. Glasserman, P. (1993a). Stochastic monotonicity and conditional Monte Carlo for likelihood ratios. Adv. in Appl. Probab. 25 103-115.
412 KOLLMAN, BAGGERLY, COX AND PICARD
Halton, J. H. (1962). Sequential Monte Carlo. Proc. Cambridge Philos. Soc. 58 57-78.
Hammersley, J. M. and Handscomb, D. C. (1964). Monte Carlo Methods. Chapman and Hall, London.
Kalos, M. H. and Whitlock, P. A. (1986). Monte Carlo Methods. Wiley, New York.
Kollman, C. (1993). Rare event simulation in radiation transport. Ph.D. dissertation, Dept. Statistics, Stanford Univ.
Lux, I. and Koblinger, L. (1991). Monte Carlo Particle Transport Methods: Neutron and Photon Calculations. CRC Press, Boca Raton, FL.
Marshall, A. W. (1956). The use of multi-stage sampling schemes in Monte Carlo computation. In University of Florida Sy mposium on Monte Carlo Methods 123-140. Wiley, New York.
Parzen, E. (1954). On uniform convergence of families of sequences of random variables. Univ. California Publications in Statistics 2 23-54.
