### Abstract

The convergence of the Monte Carlo method for numerical integration can often be improved by replacing random numbers with more uniformly distributed numbers known as quasi-random. In this paper the convergence of Monte Carlo particle simulation is studied when these quasi-random sequences are used. For the one-dimensional heat equation discretized in both space and time, convergence is proved for a quasi-random simulation using reordering of the particles according to their position. Experimental results are presented for the spatially continuous heat equation in one and two dimensions. The results indicate that a significant improvement in both magnitude of error and convergence rate can be achieved over standard Monte Carlo simulations for certain low-dimensional problems.

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

Pages (from-to) | 1558-1573 |

Number of pages | 16 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 30 |

Issue number | 6 |

State | Published - Dec 1993 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Numerical Analysis*,

*30*(6), 1558-1573.

**Quasi-Monte Carlo approach to particle simulation of the heat equation.** / Morokoff, William J.; Caflisch, Russel.

Research output: Contribution to journal › Article

*SIAM Journal on Numerical Analysis*, vol. 30, no. 6, pp. 1558-1573.

}

TY - JOUR

T1 - Quasi-Monte Carlo approach to particle simulation of the heat equation

AU - Morokoff, William J.

AU - Caflisch, Russel

PY - 1993/12

Y1 - 1993/12

N2 - The convergence of the Monte Carlo method for numerical integration can often be improved by replacing random numbers with more uniformly distributed numbers known as quasi-random. In this paper the convergence of Monte Carlo particle simulation is studied when these quasi-random sequences are used. For the one-dimensional heat equation discretized in both space and time, convergence is proved for a quasi-random simulation using reordering of the particles according to their position. Experimental results are presented for the spatially continuous heat equation in one and two dimensions. The results indicate that a significant improvement in both magnitude of error and convergence rate can be achieved over standard Monte Carlo simulations for certain low-dimensional problems.

AB - The convergence of the Monte Carlo method for numerical integration can often be improved by replacing random numbers with more uniformly distributed numbers known as quasi-random. In this paper the convergence of Monte Carlo particle simulation is studied when these quasi-random sequences are used. For the one-dimensional heat equation discretized in both space and time, convergence is proved for a quasi-random simulation using reordering of the particles according to their position. Experimental results are presented for the spatially continuous heat equation in one and two dimensions. The results indicate that a significant improvement in both magnitude of error and convergence rate can be achieved over standard Monte Carlo simulations for certain low-dimensional problems.

UR - http://www.scopus.com/inward/record.url?scp=0027805257&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027805257&partnerID=8YFLogxK

M3 - Article

VL - 30

SP - 1558

EP - 1573

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 6

ER -