What Monte Carlo models can do and cannot do efficiently?

Applied Mathematical Modelling

Volumn 32, Issue 8, 2008, Pages 1477-1500

  Atanassov, Emanouil ^a   Dimov, Ivan T ^b  

^a BULGARIAN ACADEMY OF SCIENCES   (Bulgaria)

^b UNIVERSITY OF READING   (United Kingdom)

Author keywords

Deterministic algorithms; Integral equations; Monte Carlo algorithms; Multidimensional integration; Unimprovable rate of convergence

Indexed keywords

ALGORITHMS; COMPUTATIONAL COMPLEXITY; COMPUTATIONAL METHODS; CONVERGENCE OF NUMERICAL METHODS; INTEGRAL EQUATIONS; INTEGRATION; MATHEMATICAL MODELS;

DETERMINISTIC ALGORITHM; LINEAR FUNCTION; MULTIDIMENSIONAL INTEGRATION; THEORETICAL PERFORMANCE ANALYSIS;

MONTE CARLO METHODS;

ALGORITHMS; COMPUTATIONAL COMPLEXITY; COMPUTATIONAL METHODS; CONVERGENCE OF NUMERICAL METHODS; INTEGRAL EQUATIONS; INTEGRATION; MATHEMATICAL MODELS; MONTE CARLO METHODS;

EID: 41949139021     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2007.04.010     Document Type: Article

References (75)

1. Shreider Yu.A. The Monte Carlo method. Method of Statistical Testing (1966), Pergamon Press Ltd., Oxford, London, Edinburgh, New York, Paris, Frankfurt
2. Metropolis N., and Ulam S. The Monte Carlo method. J. Am. Statist. Assoc. 44 247 (1949) 335-341
3. Ermakov S.M. Monte Carlo Methods and Mixed Problems (1985), Nauka, Moscow
4. G. Comte de Buffon, Essai d'arithmetique morale, Suppl. l'Histoire Naturalle, 4, 1777.
5. M.H. Kalos, P.A. Whitlock, Monte Carlo Methods, Basics, vol. 1, Wiley, New York, 1986.
6. Dimov I., and Tonev O. Random walk on distant mesh points Monte Carlo methods. J. Statist. Phys. 70 5-6 (1993) 1333-1342
7. Dimov I., and Tonev O. Monte Carlo algorithms: performance analysis for some computer architectures. J. Comput. Appl. Math. 48 (1993) 253-277
8. Dimov I., Karaivanova A., Kuchen H., and Stoltze H. Monte Carlo algorithms for elliptic differential equations. data parallel functional approach. J. Parallel Algorithms Appl. 9 (1996) 39-65
9. Dimov I., and Dimov O. Criteria estimators for speed-up and efficiency of some parallel Monte Carlo algorithms for different computer architectures. Proc. WP&DP'91 (Sofia, April 16-19) (1992), North-Holland Publ. Co., Amsterdam 243-262
10. Sobol I.M. Monte Carlo Numerical Methods (1973), Nauka, Moscow
11. Stewart L. Bayesian analysis using Monte Carlo integration - a powerful methodology for handling some difficult problems. Statistician 32 (1983) 195-200
12. Stewart L. Multiparameter Bayesian inference using Monte Carlo integration - some techniques for bivariate analysis. In: Bernardo J.M., de Groot M.H., Lindley D.V., and Smith A.F. (Eds). Bayesian Statistics vol. 2 (1985), North-Holland, Amsterdam
13. Haber S. A modified Monte Carlo quadrature. Math. Comput. 20 95 (1966) 361-368
14. Haber S. A modified Monte Carlo quadrature. Math. Comput. 21 99 (1967) 388-397
15. Atanassov E., and Dimov I. A new Monte Carlo method for calculating integrals of smooth functions. Monte Carlo Methods Appl. 5 2 (1999) 149-167
16. Heinrich S. Complexity theory of Monte Carlo algorithms. Lect. Appl. Math. 32 (1996) 405-419
17. Novak E. Deterministic and stochastic error bound in numerical analysis. Lect. Notes Math. vol. 1349 (1988), Springer, Berlin, Heidelberg, New York, London, Paris, Tokyo
18. Traub J.F., Wasilkowski G.W., and Woźniakowski H. Information-based Complexity (1988), Academic Press, New York
19. Ko K.-I. Computational Complexity of Real Functions (1991), Birkhauser Boston, Boston, MA
20. Ermakov S.M., and Mikhailov G.A. Statistical Modeling (1982), Nauka, Moscow
21. Ermakov S.M., Nekrutkin V.V., and Sipin A.S. Random Processes for Solving Classical Equations of Mathematical Physics (1984), Nauka, Moscow
22. Hammersley J.M., and Handscomb D.C. Monte Carlo Methods (1964), John Wiley & Sons, inc., New York, London, Sydney, Methuen
23. Shreider Yu.A. Method of Statistical Testing. Monte Carlo Method (1964), Elsevier Publishing Co., 335 Jan Van Galenstraat, P.O. Box 211, Amsterdam, Netherlands
24. Novak E. The real number model in numerical analysis. J. Complex. 11 (1995) 57-73
25. Bakhvalov N.S. On the approximate computation of multiple integrals. Vestnik Moscow State University, Ser. Mat., Mech. 4 (1959) 3-18
26. Bakhvalov N.S. Average estimation of the remainder term of quadrature formulas. USSR Comput. Math. Math. Phys. 1 1 (1961) 64-77
27. Bakhvalov N.S. On the optimal estimations of convergence of the quadrature processes and integration methods. Numerical Methods for Solving Differential and Integral Equations (1964), Nauka, Moscow
28. Korobov N.M. Approximate computation of multiple integrals with the aid of methods of the theory of numbers. Dokl. Acad. Nauk. SSSR 115 6 (1957) 1062-1065 (in Russian)
29. N.M. Korobov, On the Approximate Computation of Multiple Integrals, vol. 4, Vestnik Moskow State University, 1959 (in Russian).
30. Bakhvalov N.S., Zhidkov N.P., and Kobelkov G.M. Numerical Methods (1987), Nauka, Moscow
31. Dimov I. Efficient and overconvergent Monte Carlo methods. Parallel algorithms. In: Dimov I., and Tonev O. (Eds). Advances in Parallel Algorithms (1994), IOS Press, Amsterdam 100-111
32. Halton J.H., and Handscome D.C. A method for increasing the efficiency of Monte Carlo integrations. J. Assoc. Comput. Machinery 4 3 (1957) 329-340
33. Novak E., and Ritter K. Optimal stochastic quadrature formulas for convex functions. BIT 34 (1994) 288-294
34. Sabelfeld K. Algorithms of the Method Monte Carlo for Solving Boundary Value Problems (1989), Nauka, Moscow
35. Sendov Bl., Andreev A., and Kjurkchiev N. Numerical solution of polynomial equations. In: Ciarlet P.G., and Lions J.L. (Eds). Handbook of Numerical Analysis. Proc. WP&DP'91 (Sofia, April 16-19) vol. 3 (1994), North-Holland, Amsterdam, NY
36. Smolyak S.A. Quadrature and interpolation formulas for tensor products of certain classes of functions. Sov. Math. Dokl. 4 (1963) 240-243
37. Sobol I.M. On quadratic formulas for functions of several variables satisfying a general Lipschitz condition. USSR Comput. Math. Math. Phys. 29 6 (1989) 936-941
38. Dupach V. Stochasticke pocetni metody. Cas. Pro. Pest. Mat. 81 1 (1956) 55-68
39. E. Novak, K. Ritter, High dimensional integration of smooth functions over cubes, Numer. Math. (1996) 1-19.
40. Tanaka H., and Nagata H. Quasi-Random number method for the numerical integration. Suppl. Progr. Theoret. Phys. 56 (1974) 121-131
41. V.N. Temlyakov, On a way of obtaining lower estimates for the errors of quadrature formulas, Math. Sbornik 181 (1990) 1403-1413 (in Russian); English translation: Math. USSR Sbornik, 71 (1992) 247-257.
42. Temlyakov V.N. Approximation of Periodic Functions (1993), Nova Science Publishers
43. Cranley R., and Patterson T.N.L. Randomization of number theoretic methods for multiple integration. SIAM J. Numer. Anal. 13 (1976) 904-914
44. L'Ecuyer P., Lecot C., and Tuffin B. Randomized quasi-Monte Carlo simulation of Markov chains with an ordered state space. In: Niederreiter H., and Talay D. (Eds). Monte Carlo and Quasi-Monte Carlo Methods, 2004 (2006), Springer-Verlag 331-342
45. Sloan I.H., and Woniakowski H. Tractability of multivariate integration for weighted Korobov classes. J. Complexity 17 4 (2001) 697-721
46. P. Davis, P. Rabinowitz, Methods of numerical integration, second edition, Comp. Sci. Appl. Math., Academic Press, 1984.
47. Beckers M., and Haegmans A. Transformations of integrands for lattice rules. In: Espelid T.O., and Genz A. (Eds). Numerical Integration - Recent Developments, Software and Applications (1992), Kluwer, Dordrecht, the Netherlands 329-340
48. Krommer A.R., and Ueberhuber C.W. Computational Integration (1998), SIAM
49. Maire S., and De Luigi C. Quasi-Monte Carlo quadratures for multivariate smooth functions. Appl. Numer. Math. 56 (2006) 146-162
50. J.H. Curtiss, Sampling methods applied to differential and difference equations, in: Proc. Seminar on Sci. Comput. IBM, New York, 1949.
51. I.T. Dimov, Minimization of the probable error for some Monte Carlo methods, in: Proc. of the Summer School on Mathematical Modelling and Scientific Computations, Bulgaria, Sofia, Publ. House of the Bulg. Acad. Sci., 1991, pp. 159-170.
52. Kalos M.H. Importance sampling in Monte Carlo calculations of thick shield penetration. Nucl. Sci. Eng. 2 1 (1959) 34-35
53. Sabelfeld K.K., and Shalimova I.A. Spherical Means for PDE's (1994), VSP, Utrecht, The Netherlands
54. Levitan Y., Markovich N., Rozin S., and Sobol I. On quasi-random sequences for numerical computations. USSR Comput. Math. Math. Phys. 28 5 (1988) 755-759
55. Niederreiter H. Quasi-Monte Carlo methods and pseudorandom numbers. Bull. Am. Math. Soc. 84 (1978) 957-1041
56. Niederreiter H. Point sets and sequences with small discrepancy. Monatsh. Math. 104 (1987) 273-337
57. Niederreiter H. Random number generation and Quasi-Monte Carlo methods, Number 63. CBMS-NSF Series in Applied Mathematics (1992), SIAM, Philadelphia
58. I.M. Sobol, Quasi-Monte Carlo Methods, in: Bl. Sendov, I. Dimov (Eds.), International Youth Workshop on Monte Carlo Methods and Parallel Algorithms - Primorsko, World Scientific, Singapore, 1990, pp. 75-81.
59. Pages G. A space vector quantization for numerical integration. J. Comput. Appl. Math. 89 (1997) 1-38
60. Maire S. Reducing variance using iterated control variates. J. Statist. Comput. Simulat. 73 1 (2003) 1-29
61. Maire S. An iterative computation of approximatons on Korobov-like spaces. J. Comput. Appl. Math. 157 (2003) 261-281
62. Morton K.W. A generalization of the antithetic variate technique for evaluating integrals. J. Math. Phys. 36 3 (1957) 289-293
63. Press W.P., and Farrar G.R. Recursive stratified sampling for multidimensional Monte Carlo integration. Comput. Phys. 4 (1990) 190-195
64. Stewart L. Hierarchical Bayesian analysis using Monte Carlo integration computing posterior distributions when there are many possible models. Statistician 36 (1987) 211-219
65. Stewart L., and Davis W. Bayesian posterior distribution over sets of possible models with inferences computed by Monte Carlo integration. Statistician 35 (1986) 175-182
66. I. Dimov, T. Gurov, Parallel Monte Carlo algorithms for calculation integrals, in: K. Boyanov (Ed.), Proc. WP&DP 1993, Sofia, pp. 426-434.
67. I.T. Dimov, O.I. Tonev, Monte Carlo numerical methods with overconvergent probable error, Numerical methods and applications, in: Proc. II Int. Conf. on Numerical Methods and Applications, Sofia, Publ. House of the Bulg. Acad. Sci., Sofia, 1989, pp. 116-120.
68. Nikolski S.M. Quadrature Formulas (1988), Nauka, Moscow
69. Schurer R. A comparison between (Quasi-)Monte Carlo and cubature rule based methods for solving high-dimensional integration problems. Math. Comput. Simulat. 62 (2003) 509-517
70. Emelyanov K.V., and Il'in A.M. On the number of arithmetic operations necessary for the approximate solution of Fredholm integral equations of second kind. J. Vychisl. Math. i Math. Phys. 7 (1967) 905-910 (in Russian)
71. I. Dimov, R. Georgieva, Complexity of a class of Monte Carlo algorithms for integral equations, in: Proc. of the Int. Conf. on MC&QMC Methods, June 7-10, 2004 Juan-les-Pins, Côte d'Azur, France, 2004.
72. I.T. Dimov, A.N. Karaivanova, Iterative Monte Carlo algorithms for linear algebra problems, in: L. Vulkov, J. Wasniewski, P. Yalamov (Ed.), Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1196, pp. 150-160.
73. Dimov I.T., Alexandrov V.N., and Karaivanova A.N. Parallel resolvent Monte Carlo algorithms for linear algebra problems. Math. Comput. Simulat. 55 (2001) 25-35
74. Krylov V.I., Bobkov V.G., and Monastyrni P. Numerical Methods (1977), Nauka, Moscow
75. Kantorovich L.V., and Krylov V.I. Approximate Methods of Higher Analysis (1962), Physical and Mathematical State Publishing House, Leningrad