Abstract
In this paper we present error and performance analysis of a Monte Carlo variance reduction method for solving multidimensional integrals and integral equations. This method, called importance separation, combines the idea of separation of the domain into uniformly small subdomains with the approach of importance sampling. The importance separation method is originally described in our previous works, here we generalize our results and discuss the performance in comparison with crude Monte Carlo and importance sampling. Based on our previous investigation we propose efficient parallelizations of the importance separation method. Numerical tests implemented on PowerPC cluster using MPI are provided. The considered algorithms are carried out using pseudorandom numbers.
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Bahvalov, N.S.: On the optimal estimations of convergence of the quadrature processes and integration methods. In: Numerical Methods for Solving Differential and Integral Equations, Nauka, Moscow, pp. 5–63 (1964) (in Russian)
Dimov, I.: Minimization of the probable error for some Monte Carlo methods. In: Andreev, Dimov, Markov, Ulrich (eds.) Mathematical Modelling and Scientific Computations, pp. 159–170. Bulgarian Academy of Sciences, Sofia (1991)
Dimov, I., Karaivanova, A., Georgieva, R., Ivanovska, S.: Parallel Importance Separation and Adaptive Monte Carlo Algorithms for Multiple Integrals. In: Dimov, I.T., Lirkov, I., Margenov, S., Zlatev, Z. (eds.) NMA 2002. LNCS, vol. 2542, pp. 99–107. Springer, Heidelberg (2003)
Dupach, V.: Stochasticke pocetni metody. Cas. pro pest. mat. 81(1), 55–68 (1956)
Georgieva, R., Ivanovska, S.: Importance Separation for Solving Integral Equations. In: Lirkov, I., Margenov, S., Waśniewski, J., Yalamov, P. (eds.) LSSC 2003. LNCS, vol. 2907, pp. 144–152. Springer, Heidelberg (2004)
Hesterberg, T.: Weighted average importance sampling and defensive mixture distributions. Technometrics 37(2), 185–194 (1995)
Kahn, H.: Random sampling (Monte Carlo) techniques in neutron attenuation problems. Nucleonics 6(5), 27–33, 6(6), 60–65 (1950)
Karaivanova, A.: Adaptive Monte Carlo methods for numerical integration. Mathematica Balkanica 11, 391–406 (1997)
Mikhailov, G.A.: Optimization of the ”weight” Monte Carlo methods, Moskow (1987)
Moskowitz, B., Caflisch, R.E.: Smoothness and dimension reduction in quasi- Monte Carlo methods. J. Math. Comput. Modeling 23, 37–54 (1996)
Owen, A., Zhou, Y.: Safe and effective importance sampling, Technical report, Stanford University, Statistics Department (1999)
Sobol, I.M.: Monte Carlo Numerical Methods, Nauka, Moscow (1973) (in Russian)
Veach, E., Guibas, L.J.: Optimally combining sampling techniques for Monte Carlo rendering. In: Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH 1995, pp. 419–428 (1995)
Veach, E.: Robust Monte Carlo Methods for Light Transport Simulation, Ph.D. dissertation, Stanford University (1997)
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Ivanovska, S., Karaivanova, A. (2004). Parallel Importance Separation for Multiple Integrals and Integral Equations. In: Bubak, M., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds) Computational Science - ICCS 2004. ICCS 2004. Lecture Notes in Computer Science, vol 3039. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-25944-2_65
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DOI: https://doi.org/10.1007/978-3-540-25944-2_65
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