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Adaptive Monte Carlo Algorithms for General Transport Problems

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Monte Carlo and Quasi-Monte Carlo Methods 2008

Abstract

Recently there has been a concerted effort to develop adaptively modified Monte Carlo algorithms that converge geometrically to solutions of the radiative transport equation. We have concentrated on algorithms that extend to integral equations methods first proposed for matrix equations by Halton in 1962 [Halton, J., Proc. Camb. Phil. Soc., 58, 57–78 (1962)]. Geometric convergence has been rigorously demonstrated [Kong, R., and Spanier, J., J. Comp. Phys., 227(23), 9762–9777 (2008)] for these “first generation” (G1) algorithms but their practical utility is limited by computational complexities resulting from the expansion. Recently, we have developed new adaptive algorithms that overcome most of the computational restrictions of the earlier algorithms and we have also established the geometric convergence of these “second generation” (G2) algorithms [Kong, R. and Spanier, J.: Geometric convergence of second generation adaptive Monte Carlo algorithms for general transport problems based on sequential correlated sampling. In review]. In this paper we outline the main ideas involved and indicate how the resulting G2 algorithm might be optimized using information drawn from simulations of both the RTE and the dual RTE. Simple examples will illustrate these ideas and the gains in computational efficiency that the new methods can achieve.

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References

  1. Aboughantous, C.: A Contributon Monte Carlo method. Nucl. Sci. and Eng. 108, 160–177 (1994)

    Google Scholar 

  2. Bhan, K., Kong, R., Spanier, J.: Adaptive Monte Carlo Algorithms Applied to Heterogenous Transport Problems. In: Monte Carlo and Quasi-Monte Carlo Methods 2008. Springer-Verlag (to appear)

    Google Scholar 

  3. Booth, T.: Exponential convergence for Monte Carlo particle transport. Trans. Amer. Nucl. Soc. 50, 267–268 (1985)

    Google Scholar 

  4. Booth, T.: LA-10363-MS, A sample problem for variance reduction in MCNP. Tech. rep., Los Alamos National Laboratory (1985)

    Google Scholar 

  5. Booth, T.: A Monte Carlo learning/biasing experiment with intelligent random numbers. Nucl. Sci. and Eng. 92, 465–481 (1986)

    Google Scholar 

  6. Booth, T.: The intelligent random number technique in mcnp. Nucl. Sci. and Eng. 100, 248–254 (1988)

    Google Scholar 

  7. Booth, T.: Zero-variance solutions for linear Monte Carlo. Nucl. Sci. and Eng. 102(4), 332–340 (1989)

    Google Scholar 

  8. Booth, T.: Exponential convergence on a continuous Monte Carlo transport problem. Nucl. Sci. Eng. 127, 338–345 (1997)

    Google Scholar 

  9. Booth, T.: Adaptive importance sampling with a rapidly varying importance function. Nucl. Sci. and Eng. 136(3), 399–408 (2000)

    MathSciNet  Google Scholar 

  10. Booth, T., Hendricks, J.: Importance estimation in forward Monte Carlo calculations. Nucl. Technol./Fusion 5(1), 90–100 (1984)

    Google Scholar 

  11. Burn, K., Gualdrini, G., Nava, E.: Variance reduction with multiple responces. In: Advanced Monte Carlo for Radiation Physics, Particle Transport Simulation and Applications (eds. A. Kling, F. Barao, M. Nakagawa, L. Tavora and P. Vaz), Proc. MC2000 Conference, Lisbon, Portugal, 23-26 October 2000, pp. 687–695 (2002)

    Google Scholar 

  12. Burn, K., Nava, E.: Optimization of variance reduction parameters in Monte Carlo transport calculations to a number of responces of interest. In: Proc. Int. Conf. Nuclear Data for Science and Technology, Trieste, Italy, Italian Physical Society (1997)

    Google Scholar 

  13. Cooper, M., Larsen, E.: Automated weight windows for global Monte Carlo particle transport calculations. Nucl. Sci. and Eng. 137(1), 1–13 (2001)

    Google Scholar 

  14. Goertzel, G., Kalos, M.: Monte Carlo methods in transport problems. Progess in Nuclear Energy, Series I 2, 315–369 (1958)

    MathSciNet  Google Scholar 

  15. eds. Greenspan, H., Kelber, C., Okrent, D.: Computing methods in reactor physics. Gordon and Breach, New York (1968)

    MATH  Google Scholar 

  16. Halton, J.: Sequential Monte Carlo. Proc. Camb. Phil. Soc. 58, 57–78 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hammersley, J., Handscomb, D.: Monte Carlo methods. Methuen and Co., Ltd., London (1964)

    Google Scholar 

  18. Hayakawa, C.K., Spanier, J.: Comparison of Monte Carlo algorithms for obtaining geometric convergence for model transport problems. In: Monte Carlo and Quasi-Monte Carlo Methods 1998, pp. 214–226. Springer (1999)

    Google Scholar 

  19. Hayakawa, C.K., Spanier, J., Venugopalan, V.: Computational Engine for a Virtual Tissue Simulator. In: Monte Carlo and Quasi-Monte Carlo Methods 2006, A. Keller, S. Heinrich and H. Niederreiter (Eds.), pp. 431–444. Springer-Verlag (2007)

    Google Scholar 

  20. Hendricks, J.: A code-generated Monte Carlo importance function. Trans. Amer. Nucl. Soc. 41, 307 (1982)

    Google Scholar 

  21. J. Briesmeister, E.: MCNP4C, a Monte Carlo N-Particle Transport Code System, Report CCC-660. Tech. rep., Los Alamos National Laboratory (1999)

    Google Scholar 

  22. Kalos, M.: Importance sampling in Monte Carlo shielding calculations. Nucl. Sci & Eng. 16, 227–234 (1963)

    Google Scholar 

  23. Kollman, C.: Rare event simulation in radiation transport. Ph.D. thesis, University of California, Berkley (1993)

    Google Scholar 

  24. Kollman, G., Baggerly, K., Cox, D., Picard, R.: Adaptive importance sampling on discrete Markov chains. The Ann. Appl. prob. 9, 391–412 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kong, R.: Transport problems and monte carlo methods. Ph.D. thesis, Claremont Graduate University (1999)

    Google Scholar 

  26. Kong, R., Ambrose, M., Spanier, J.: Efficient, automated Monte Carlo methods for radiation transport. J. Comp. Phys. 227(22), 9463–9476 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kong, R., Spanier, J.: Geometric convergence of second generation adaptive Monte Carlo algorithms for general transport problems based on sequential correlated sampling. in review

    Google Scholar 

  28. Kong, R., Spanier, J.: Error analysis of sequential Monte Carlo methods for transport problems. In: Monte Carlo and Quasi-Monte Carlo Methods 1998, H. Niederreiter, J. Spanier (Eds.). Springer-Verlag (1999)

    Google Scholar 

  29. Kong, R., Spanier, J.: Sequential correlated sampling algorithms for some transport problems. In: Monte Carlo and Quasi-Monte Carlo Methods 1998, pp. 238–251. Springer (1999)

    Google Scholar 

  30. Kong, R., Spanier, J.: Residual Versus Error in Transport Problems. In: Monte Carlo and Quasi-Monte Carlo Methods 2000, K.-T. Fang, F. J. Hickernell and H. Niederreiter (Eds.), pp. 306–317. Springer-Verlag (2002)

    Google Scholar 

  31. Kong, R., Spanier, J.: A new proof of geometric convergence for general transport problems based on sequential correlated sampling methods. J. Comp. Phys. 227(23), 9762–9777 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. L. Liu, R.G.: A geometry-independent fine-mesh-based Monte Carlo importance generator. Nucl. Sci. and Eng. 125(2), 188–195 (1997)

    Google Scholar 

  33. Lai, Y., Spanier, J.: Adaptive importance sampling algorithms for transport problems. In: Monte Carlo and Quasi-Monte Carlo Methods 1998, pp. 273–283. Springer (1999)

    Google Scholar 

  34. Leimdorfer, M.: On the transformation of the transport equation for solving deep penetration problems by the Monte Carlo methods. Trans. Chalmers Univ. of Tech., Goteborg, Sweden 286 (1964)

    Google Scholar 

  35. Leimdorfer, M.: On the use of Monte Carlo methods for calculating the deep penetration of neutrons in shields. Trans. Chalmers Univ. of Tech., Goteborg, Sweden 287 (1964)

    Google Scholar 

  36. Li, L., Spanier, J.: Approximation of transport equations by matrix equations and sequential sampling. Monte Carlo Methods and Applications 3, 171–198 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  37. Mickael, M.: A fast, automated, semideterministic weight windows generator for mcnp. Nucl. Sci. and Eng. 119(1), 34–43 (1995)

    Google Scholar 

  38. Rubinstein, R.: Simulation and the Monte Carlo Method. Wiley (1981)

    Google Scholar 

  39. Serov, I.V., John, T.M., Hoogenboom, J.E.: A new effective Monte Carlo midway coupling method in MCNP applied to a well logging problem. Appl. Radiat. Isot. 49(12), 1737–1744 (1998)

    Article  Google Scholar 

  40. S.H. Henderson, B. Simon: Adaptive simulation using perfect control variates. J. Appl. Prob. 41, 859–876 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  41. Spanier, J.: Geometrically convergent learning algorithms for global solutions of transport problems. In: Monte Carlo and Quasi-Monte Carlo Methods 1998, pp. 98–113. Springer (1999)

    Google Scholar 

  42. Spanier, J., Gelbard, E.: Monte Carlo Principles and Neutron Transport Problems. Addison-Wesley (1969)

    Google Scholar 

  43. Spanier, J., Kong, R.: A new adaptive method for geometric convergence. In: Monte Carlo and Quasi-Monte Carlo Methods 2002, pp. 439–449. Springer (2004)

    Google Scholar 

  44. Spanier, J., Li, L.: General sequential sampling techniques for Monte Carlo simulations: Part 1 matrix problems. In: Monte Carlo and Quasi-Monte Carlo Methods 1996, pp. 382–397. Springer (1998)

    Google Scholar 

  45. Wagner, J., Haghighat, A.: Automated variance reduction of Monte Carlo shielding calculations using the discrete ordinates adjoint function. Nucl. Sci. and Eng. 128(2), 186–208 (1998)

    Google Scholar 

  46. Williams, M.L.: Generalized Contributon Response Theory. Nucl. Sci. and Eng. 108, 355–383 (1991)

    Google Scholar 

  47. Williams, M.L., Engle, W.W.: The concept of spatial channel theory applied to reactor shielding analysis. Nucl. Sci. Eng. 62, 160–177 (1977)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Beckman Laser Institute, University of California, Irvine, Irvine, California, USA

    Jerome Spanier

Authors
  1. Jerome Spanier
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  2. Rong Kong
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  3. Martin Ambrose
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Corresponding author

Correspondence to Jerome Spanier .

Editor information

Editors and Affiliations

  1. Centre de Recherches Mathématiques, Université de Montréal, Montreal, H3C 3J7, Canada

    Pierre L' Ecuyer

  2. Dept. Statistics, Stanford University, Serra Mall 390, Stanford, 94305-4065, U.S.A.

    Art B. Owen

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Spanier, J., Kong, R., Ambrose, M. (2009). Adaptive Monte Carlo Algorithms for General Transport Problems. In: L' Ecuyer, P., Owen, A. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04107-5_30

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