### 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)^{k}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.

Original language | English (US) |
---|---|

Pages (from-to) | 1-49 |

Number of pages | 49 |

Journal | Acta Numerica |

Volume | 7 |

DOIs | |

State | Published - 1998 |

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### ASJC Scopus subject areas

- Numerical Analysis
- Mathematics(all)

### Cite this

*Acta Numerica*,

*7*, 1-49. https://doi.org/10.1017/S0962492900002804

**Monte Carlo and quasi-Monte Carlo methods.** / Caflisch, Russel.

Research output: Contribution to journal › Article

*Acta Numerica*, vol. 7, pp. 1-49. https://doi.org/10.1017/S0962492900002804

}

TY - JOUR

T1 - Monte Carlo and quasi-Monte Carlo methods

AU - Caflisch, Russel

PY - 1998

Y1 - 1998

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85011514513&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85011514513&partnerID=8YFLogxK

U2 - 10.1017/S0962492900002804

DO - 10.1017/S0962492900002804

M3 - Article

VL - 7

SP - 1

EP - 49

JO - Acta Numerica

JF - Acta Numerica

SN - 0962-4929

ER -