### Abstract

The standard Monte Carlo approach to evaluating multidimensional integrals using (pseudo)-random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement. As an alternative to the random methods, it has been suggested that lower error and improved convergence may be obtained by replacing the pseudo-random sequences with more uniformly distributed sequences known as quasi-random. In this paper quasi-random (Halton, Sobol', and Faure) and pseudo-random sequences are compared in computational experiments designed to determine the effects on convergence of certain properties of the integrand, including variance, variation, smoothness, and dimension. The results show that variation, which plays an important role in the theoretical upper bound given by the Koksma-Hlawka inequality, does not affect convergence, while variance, the determining factor in random Monte Carlo, is shown to provide a rough upper bound, but does not accurately predict performance. In general, quasi-Monte Carlo methods are superior to random Monte Carlo, but the advantage may be slight, particularly in high dimensions or for integrands that are not smooth. For discontinuous integrands, we derive a bound which shows that the exponent for algebraic decay of the integration error from quasi-Monte Carlo is only slightly larger than 1/2 in high dimensions.

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

Pages (from-to) | 218-230 |

Number of pages | 13 |

Journal | Journal of Computational Physics |

Volume | 122 |

Issue number | 2 |

DOIs | |

State | Published - 1995 |

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### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)
- Computer Science Applications

### Cite this

*Journal of Computational Physics*,

*122*(2), 218-230. https://doi.org/10.1006/jcph.1995.1209

**Quasi-Monte carlo integration.** / Morokoff, William J.; Caflisch, Russel.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 122, no. 2, pp. 218-230. https://doi.org/10.1006/jcph.1995.1209

}

TY - JOUR

T1 - Quasi-Monte carlo integration

AU - Morokoff, William J.

AU - Caflisch, Russel

PY - 1995

Y1 - 1995

N2 - The standard Monte Carlo approach to evaluating multidimensional integrals using (pseudo)-random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement. As an alternative to the random methods, it has been suggested that lower error and improved convergence may be obtained by replacing the pseudo-random sequences with more uniformly distributed sequences known as quasi-random. In this paper quasi-random (Halton, Sobol', and Faure) and pseudo-random sequences are compared in computational experiments designed to determine the effects on convergence of certain properties of the integrand, including variance, variation, smoothness, and dimension. The results show that variation, which plays an important role in the theoretical upper bound given by the Koksma-Hlawka inequality, does not affect convergence, while variance, the determining factor in random Monte Carlo, is shown to provide a rough upper bound, but does not accurately predict performance. In general, quasi-Monte Carlo methods are superior to random Monte Carlo, but the advantage may be slight, particularly in high dimensions or for integrands that are not smooth. For discontinuous integrands, we derive a bound which shows that the exponent for algebraic decay of the integration error from quasi-Monte Carlo is only slightly larger than 1/2 in high dimensions.

AB - The standard Monte Carlo approach to evaluating multidimensional integrals using (pseudo)-random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement. As an alternative to the random methods, it has been suggested that lower error and improved convergence may be obtained by replacing the pseudo-random sequences with more uniformly distributed sequences known as quasi-random. In this paper quasi-random (Halton, Sobol', and Faure) and pseudo-random sequences are compared in computational experiments designed to determine the effects on convergence of certain properties of the integrand, including variance, variation, smoothness, and dimension. The results show that variation, which plays an important role in the theoretical upper bound given by the Koksma-Hlawka inequality, does not affect convergence, while variance, the determining factor in random Monte Carlo, is shown to provide a rough upper bound, but does not accurately predict performance. In general, quasi-Monte Carlo methods are superior to random Monte Carlo, but the advantage may be slight, particularly in high dimensions or for integrands that are not smooth. For discontinuous integrands, we derive a bound which shows that the exponent for algebraic decay of the integration error from quasi-Monte Carlo is only slightly larger than 1/2 in high dimensions.

UR - http://www.scopus.com/inward/record.url?scp=0001243365&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001243365&partnerID=8YFLogxK

U2 - 10.1006/jcph.1995.1209

DO - 10.1006/jcph.1995.1209

M3 - Article

VL - 122

SP - 218

EP - 230

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -