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Monte Carlo and quasi-Monte Carlo methods. (English) Zbl 0949.65003

Acta Numerica 7, 1-49 (1998).
This well-written and concise survey paper brings the reader to the present state in the field of Monte Carlo quadrature. Topics discussed include the estimation of integrals via their interpretation as an expectation, the derivation of the \(O(N^{-1/2})\) convergence rate law, the efficacy of Monte Carlo for high-dimensional integrals, the presentation of methods for sampling non-uniform random variables, and the derivation of several techniques for variance reduction. The survey continues with a description of the quasi-random number technique for accelerating convergence, by stating and proving the Koksma-Hlawka Theorem, by presenting specific quasi-random number generators and by examining smoothness and dimension limitation of quasi-Monte Carlo. Dimension reduction methods such as the Brownian bridge are presented. Lastly the paper treats an application of the Monte Carlo method to the problem of rarefied gas dynamics.
For the entire collection see [Zbl 0894.00025].

MSC:

65C05 Monte Carlo methods
76N15 Gas dynamics (general theory)
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65D32 Numerical quadrature and cubature formulas
65C10 Random number generation in numerical analysis
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