Monte Carlo and quasi-Monte Carlo methods | Acta Numerica | Cambridge Core (original) (raw)

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.

References

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.Google Scholar

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, 15–22.CrossRefGoogle Scholar

Bardos, C., Golse, F. and Levermore, C. D. (1991), ‘Fluid dynamic limits of kinetic equations: I. Formal derivations’, J. Statist. Phys. 63, 323–344.CrossRefGoogle Scholar

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, 667–753.CrossRefGoogle Scholar

Bird, G. A. (1976), Molecular Gas Dynamics, Oxford University Press.Google Scholar

Bird, G. A. (1978), ‘Monte Carlo simulation of gas flows’, Ann. Rev. Fluid Mech. 10, 11–31.CrossRefGoogle Scholar

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 equation’ J. Comput. Phys. 98, 285–300.CrossRefGoogle Scholar

Bratley, P., Fox, B. L. and Niederreiter, H. (1994), ‘Algorithm 738 – Programs to generate Niederreiter's discrepancy sequences’, ACM Trans. Math. Software 20, 494–495.CrossRef1967), ‘The distribution of points in a cube and the accurate evaluation of integrals’, Zh. Vychisl. Mat. Mat. Fiz. 7, 784–802. In Russian.Google Scholar

Sobol, I. M.' (1976), ‘Uniformly distributed sequences with additional uniformity property’, USSR Comput. Math. Math. Phys. 16, 1332–1337.CrossRefGoogle Scholar

Spanier, J. and Maize, E. H. (1994), ‘Quasi-random methods for estimating integrals using relatively small samples’, SIAM Rev. 36, 18–44.CrossRefGoogle Scholar

Wagner, W. (1992), ‘A convergence proof for Bird's Direct Simulation Monte Carlo method for the Boltzmann equation’, J. Statist. Phys. 66, 1011–1044.CrossRefGoogle Scholar

Woźniakowski, H. (1991), ‘Average case complexity of multivariate integration’, Bull. Amer. Math. Soc. 24, 185–194.CrossRefGoogle Scholar

Xing, C. P. and Niederreiter, H. (1995), ‘A construction of low-discrepancy sequences using global function fields’, Acta Arithmetica 73, 87–102.CrossRefGoogle Scholar

Zaremba, S. K. (1968), ‘The mathematical basis of Monte Carlo and quasi-Monte Carlo methods’, SIAM Rev. 10, 303–314.CrossRefGoogle Scholar