Skip to main content Accessibility help
×
Home

Monte Carlo and quasi-Monte Carlo methods

  • Russel E. Caflisch (a1)

Abstract

Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N−1/2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O((logN)kN−1). For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.

Copyright

References

Hide All
Acworth, P., Broadie, M. and Glasserman, P. (1997), A comparison of some Monte Carlo and quasi Monte Carlo techniques for option pricing, in Monte Carlo and Quasi-Monte Carlo Methods 1996 (Larcher, G., Niederreiter, H., Hellekalek, P. and Zinterhof, P., eds), Springer.
Babovsky, H., Gropengiesser, F., Neunzert, H., Struckmeier, J. and Wiesen, J. (1990), ‘Application of well-distributed sequences to the numerical simulation of the Boltzmann equation’, J. Comput. Appl. Math. 31, 1522.
Bardos, C., Golse, F. and Levermore, C. D. (1991), ‘Fluid dynamic limits of kinetic equations: I. Formal derivations’, J. Statist. Phys. 63, 323344.
Bardos, C., Golse, F. and Levermore, C. D. (1993), ‘Fluid dynamic limits of kinetic equations: II. Convergence proofs for the Boltzmann equation’, Comm. Pure Appl. Math. 46, 667753.
Bird, G. A. (1976), Molecular Gas Dynamics, Oxford University Press.
Bird, G. A. (1978), ‘Monte Carlo simulation of gas flows’, Ann. Rev. Fluid Mech. 10, 1131.
Borgers, C., Larsen, E. W. and Adams, M. L. (1992), ‘The asymptotic diffusion limit of a linear discontinuous discretization of a two-dimensional linear transport equationJ. Comput. Phys. 98, 285300.
Bratley, P., Fox, B. L. and Niederreiter, H. (1994), ‘Algorithm 738 – Programs to generate Niederreiter's discrepancy sequences’, ACM Trans. Math. Software 20, 494495.
Caflisch, R. E. (1983), Fluid dynamics and the Boltzmann equation, in Nonequilibrium Phenomena I: The Boltzmann equation (Montroll, E. W. and Lebowitz, J. L., eds), Vol. 10 of Studies in Statistical Mechanics, North Holland, pp. 193223.
Caflisch, R. E., Jin, S. and Russo, G. (1997a), ‘Uniformly accurate schemes for hyperbolic systems with relaxation’, SIAM J. Numer. Anal. 34, 246281.
Caflisch, R. E., Morokoff, W. and Owen, A. B. (1997b), ‘Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension’, J. Comput. Finance 1, 2746.
Cercignani, C. (1988), The Boltzmann Equation and its Applications, Springer.
Chapman, S. and Cowling, T. G. (1970), The Mathematical Theory of Non-Uniform Gases, Cambridge University Press.
De Masi, A., Esposito, R. and Lebowitz, J. L. (1989), ‘Incompressible Navier–Stokes and Euler limits of the Boltzmann equation’, Comm. Pure Appl. Math. 42, 11891214.
Faure, H. (1982), ‘Discrépance de suites associées à un système de numération (en dimension s)’, Acta Arithmetica 41, 337351.
Feller, W. (1971), An Introduction to Probability Theory and its Applications: Vol. I, Wiley.
Gabetta, E., Pareschi, L. and Toscani, G. (1997), ‘Relaxation schemes for nonlinear kinetic equations’, SIAM J. Numer. Anal. 34, 21682194.
Goldstein, D., Sturtevant, B. and Broadwell, J. E. (1988), Investigations of the motion of discrete-velocity gases, in Rarefied Gas Dynamics: Theoretical and Computational Techniques (Muntz, E. P., Weaver, D. P. and Campbell, D. H., eds), Vol. 118 of Progress in Aeronautics and Astronautics, Proceedings of the 16th International Symposium on Rarefied Gas Dynamics, pp. 100117.
Halton, J. H. (1960), ‘On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals’, Numer. Math. 2, 8490.
Hammersley, J. M. and Handscomb, D. C. (1965), Monte Carlo Methods, Wiley, New York.
Haselgrove, C. (1961), ‘A method for numerical integration’, Math. Comput. 15, 323337.
Hickernell, F. J. (1996), ‘The mean square discrepancy of randomized nets’, ACM Trans. Model. Comput. Simul. 6, 274296.
Hickernell, F. J. (1998), ‘A generalized discrepancy and quadrature error bound’, Math. Comput. 67, 299322.
Hogg, R. V. and Craig, A. T. (1995), Introduction to Mathematical Statistics, Prentice Hall.
Hua, L. K. and Wang, Y. (1981), Applications of Number Theory to Numerical Analysis, Springer, Berlin/New York.
Jin, S. and Levermore, C. D. (1993), ‘Fully-discrete numerical transfer in diffusive regimes’, Transp. Theory Statist. Phys., 22, 739791.
Jin, S., Pareschi, L. and Toscani, G. (1998), ‘Diffusive relaxation schemes for discrete-velocity kinetic equations’. Preprint.
Kalos, M. H. and Whitlock, P. A. (1986), Monte Carlo Methods, Vol. I: Basics, Wiley, New York.
Karatzas, I. and Shreve, S. E. (1991), Brownian Motion and Stochastic Calculus, Springer, New York.
Kennedy, W. J. and Gentle, J. E. (1980), Statistical Computing, Dekker, New York.
Koura, K. (1986), ‘Null-collision technique in the direct-simulation Monte Carlo method’, Phys. Fluids 29, 35093511.
Kuipers, L. and Niederreiter, H. (1974), Uniform Distribution of Sequences, Wiley, New York.
Larsen, E. W. (1984), ‘Diffusion-synthetic acceleration methods for discrete-ordinates problems’, Transp. Theory Statist. Phys. 13, 107126.
Larsen, E. W., Morel, J. E. and Miller, W. F. (1987), ‘Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes’, J. Comput. Phys. 69, 283324.
Lepage, G. (1978), ‘A new algorithm for adaptive multidimensional integration’, J. Comput. Phys. 27, 192203.
Marsaglia, G. (1991), ‘Normal (Gaussian) random variables for supercomputers’, J. Supercomputing 5, 4955.
Morokoff, W. and Caflisch, R. E. (1993), ‘A quasi-Monte Carlo approach to particle simulation of the heat equation’, SIAM J. Numer. Anal. 30, 15581573.
Morokoff, W. and Caflisch, R. E. (1994), ‘Quasi-random sequences and their discrepancies’, SIAM J. Sci. Statist. Comput. 15, 12511279.
Morokoff, W. and Caflisch, R. E. (1995), ‘Quasi-Monte Carlo integration’, J. Comput. Phys. 122, 218230.
Moskowitz, B. and Caflisch, R. E. (1996), ‘Smoothness and dimension reduction in quasi-Monte Carlo methods’, J. Math. Comput. Modeling 23, 3754.
Nanbu, K. (1986), Theoretical basis of the direct simulation Monte Carlo method, in Proceedings of the 15th International Symposium on Rarefied Gas Dynamics (Boffi, V. and Cercignani, C., eds), Vol. I, pp. 369383.
Niederreiter, H. (1992), Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, PA.
Owen, A. B. (1995), Randomly permuted (t, m, s)-nets and (t, s)-sequences, in Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (Niederreiter, H. and Shiue, P. J.-S., eds), Springer, New York, pp. 299317.
Owen, A. B. (1997), ‘Monte Carlo variance of scrambled net quadrature’, SIAM J. Numer. Anal. 34, 18841910.
Owen, A. B. (1998), ‘Scrambled net variance for integrals of smooth functions’, Ann. Statist. In press.
Press, W. H. and Teukolsky, S. A. (1992), ‘Portable random number generators’, Comput. Phys. 6, 522524.
Press, W. H., Teukolsky, S. A., Vettering, W. T. and Flannery, B. P. (1992), Numerical Recipes in C: The Art of Scientific Computing, 2nd edn, Cambridge University Press.
Pullin, D. I. (1979), ‘Generation of normal variates with given sample and mean variance’, J. Statist. Comput. Simul. 9, 303309.
Sobol, I. M.' (1967), ‘The distribution of points in a cube and the accurate evaluation of integrals’, Zh. Vychisl. Mat. Mat. Fiz. 7, 784802. In Russian.
Sobol, I. M.' (1976), ‘Uniformly distributed sequences with additional uniformity property’, USSR Comput. Math. Math. Phys. 16, 13321337.
Spanier, J. and Maize, E. H. (1994), ‘Quasi-random methods for estimating integrals using relatively small samples’, SIAM Rev. 36, 1844.
Wagner, W. (1992), ‘A convergence proof for Bird's Direct Simulation Monte Carlo method for the Boltzmann equation’, J. Statist. Phys. 66, 10111044.
Woźniakowski, H. (1991), ‘Average case complexity of multivariate integration’, Bull. Amer. Math. Soc. 24, 185194.
Xing, C. P. and Niederreiter, H. (1995), ‘A construction of low-discrepancy sequences using global function fields’, Acta Arithmetica 73, 87102.
Zaremba, S. K. (1968), ‘The mathematical basis of Monte Carlo and quasi-Monte Carlo methods’, SIAM Rev. 10, 303314.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed