Parallel Importance Separation and Adaptive Monte Carlo Algorithms for Multiple Integrals

  • Ivan Dimov
  • Aneta Karaivanova
  • Rayna Georgieva
  • Sofiya Ivanovska
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2542)


Monte Carlo Method (MCM) is the only viable method for many high-dimensional problems since its convergence is independent of the dimension. In this paper we develop an adaptive Monte Carlo method based on the ideas and results of the importance separation, a method that combines the idea of separation of the domain into uniformly small subdomains with the Kahn approach of importance sampling. We analyze the error and compare the results with crude Monte Carlo and importance sampling which is the most widely used variance reduction Monte Carlo method. We also propose efficient parallelizations of the importance separation method and the studied adaptive Monte Carlo method. Numerical tests implemented on PowerPC cluster using MPI are provided.


Adaptive Algorithm Importance Sampling Adaptive Method Subdivision Strategy Monte Carlo Variance 
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  1. 1.
    Bahvalov, N. S.: On the optimal estimations of convergence of the quadrature processes and integration methods. Numerical Methods for Solving Differential and Integral Equations, Nauka, Moscow (1964) 5–63 (in Russian).Google Scholar
  2. 2.
    Berntsen, J., Espelid, T. O., and Genz, A.: An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Softw., 17 (1991) 437–451.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bull, J. M. and Freeman, T. L.: Parallel globally adaptive quadrature on the KSR-1. Advances in Comp. Mathematics, 2 (1994) 357–373.zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bull, J. M. and Freeman, T. L.: Parallel algorithms for multi-dimensional integration. Parallel and Distributed Computing Practices, 1(1) (1998) 89–102.Google Scholar
  5. 5.
    Caflisch, R. E.: Monte Carlo and quasi-Monte Carlo methods. Acta Numerica, 7 (1998) 1–49.MathSciNetCrossRefGoogle Scholar
  6. 6.
    van Dooren, P. and de Ridder, L.: An adaptive algorithm for numerical integration over an N-dimensional cube. Journal of Computational and Applied Mathematics, 2 (1976) 207–217.zbMATHCrossRefGoogle Scholar
  7. 7.
    Freeman, T. L. and Bull, J. M.: A comparison of parallel adaptive algorithms for multi-dimensional integration. Proceedings of 8th SIAM Conference on Parallel Processing for Scientificc Computing (1997)Google Scholar
  8. 8.
    Hesterberg, T.: Weighted average importance sampling and defensive mixture distributions. Technometrics, 37(2) (1995) 185–194.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Karaivanova, A.: Adaptive Monte Carlo methods for numerical integration. Mathematica Balkanica, 11 (1997) 391–406.zbMATHMathSciNetGoogle Scholar
  10. 10.
    Karaivanova, A. and Dimov, I.: Error analysis of an adaptive Monte Carlo method for numerical integration. Mathematics and Computers in Simulation, 47 (1998) 201–213.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Owen, A. and Zhou, Y.: Safe and effiective importance sampling. Technical report, Stanford University, Statistics Department (1999)Google Scholar
  12. 12.
    Sobol', I. M.: Monte Carlo Numerical Methods. Nauka, Moscow (1973) (in Russian).Google Scholar
  13. 13.
    Veach, E. and Guibas, L. J.: Optimally combining sampling techniques for Monte Carlo rendering. Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH’ 95 (1995) 419–428.Google Scholar
  14. 14.
    Veach, E.: Robust Monte Carlo Methods for Light Transport Simulation. Ph.D. dissertation, Stanford Universty (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ivan Dimov
    • 1
  • Aneta Karaivanova
    • 1
  • Rayna Georgieva
    • 1
  • Sofiya Ivanovska
    • 1
  1. 1.CLPP - Bulgarian Academy of SciencesAcad. G. Bonchev St.SofiaBulgaria

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