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Books Year : 2020

## Monte Carlo and Quasi-Monte Carlo Methods

Bruno Tuffin
• Function : Author
• PersonId : 830166
Pierre L'Écuyer
• Function : Author
• PersonId : 833462

#### 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((log N^N'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.

#### Domains

Operations Research [cs.RO]

### Dates and versions

hal-02560816 , version 1 (02-05-2020)

### Identifiers

• HAL Id : hal-02560816 , version 1
• DOI :

### Cite

Bruno Tuffin, Pierre L'Écuyer. Monte Carlo and Quasi-Monte Carlo Methods: MCQMC 2018, Rennes, France, July 1–6. Springer International Publishing, 324, pp.539, 2020, Springer Proceedings in Mathematics & Statistics, ⟨10.1007/978-3-030-43465-6⟩. ⟨hal-02560816⟩

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