caa a22 4500 388057351 CHVBK 20180307125110.0 cr unu---uuuuu 161130e199801 xx s 000 0 eng 10.1017/S0962492900002804 doi S0962492900002804 pii (NATIONALLICENCE)cambridge-10.1017/S0962492900002804 Caflisch Russel E. Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA E-mail: caflisch@math.ucla.edu Monte Carlo and quasi-Monte Carlo methods [Elektronische Daten] [Russel E. Caflisch] 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 © Cambridge University Press 1998 Acta Numerica Cambridge University Press 7(1998-01), 1-49 0962-4929 7<1 1998 7 ANU https://doi.org/10.1017/S0962492900002804 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0962492900002804 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Caflisch Russel E. Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA E-mail: caflisch@math.ucla.edu NATIONALLICENCE 773 0- Acta Numerica Cambridge University Press 7(1998-01), 1-49 0962-4929 7<1 1998 7 ANU CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge