### Abstract

Two hierarchical Monte Carlo methods for the generation of self-similar fractal random fields are compared and contrasted. The first technique, successive random addition (SRA), is currently popular in the physics community. Despite the intuitive appeal of SRA, rigorous mathematical reasoning reveals that SRA cannot be consistent with any stationary power-law Gaussian random field for any Hurst exponent; furthermore, there is an inherent ratio of largest to smallest putative scaling constant necessarily exceeding a factor of 2 for a wide range of Hurst exponents H, with 0.30<H<0.85. Thus, SRA is inconsistent with a stationary power-law fractal random field and would not be useful for problems that do not utilize additional spatial averaging of the velocity field. The second hierarchical method for fractal random fields has recently been introduced by two of the authors and relies on a suitable explicit multiwavelet expansion (MWE) with high-moment cancellation. This method is described briefly, including a demonstration that, unlike SRA, MWE is consistent with a stationary power-law random field over many decades of scaling and has low variance.

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

Pages (from-to) | 717-736 |

Number of pages | 20 |

Journal | Journal of Statistical Physics |

Volume | 81 |

Issue number | 3-4 |

DOIs | |

State | Published - Nov 1995 |

### Fingerprint

### Keywords

- Fractal random fields
- Monte Carlo methods
- successive random addition

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*81*(3-4), 717-736. https://doi.org/10.1007/BF02179254

**Hierarchical Monte Carlo methods for fractal random fields.** / Elliott, Frank W.; Majda, Andrew J.; Horntrop, David J.; McLaughlin, Richard M.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 81, no. 3-4, pp. 717-736. https://doi.org/10.1007/BF02179254

}

TY - JOUR

T1 - Hierarchical Monte Carlo methods for fractal random fields

AU - Elliott, Frank W.

AU - Majda, Andrew J.

AU - Horntrop, David J.

AU - McLaughlin, Richard M.

PY - 1995/11

Y1 - 1995/11

N2 - Two hierarchical Monte Carlo methods for the generation of self-similar fractal random fields are compared and contrasted. The first technique, successive random addition (SRA), is currently popular in the physics community. Despite the intuitive appeal of SRA, rigorous mathematical reasoning reveals that SRA cannot be consistent with any stationary power-law Gaussian random field for any Hurst exponent; furthermore, there is an inherent ratio of largest to smallest putative scaling constant necessarily exceeding a factor of 2 for a wide range of Hurst exponents H, with 0.30<H<0.85. Thus, SRA is inconsistent with a stationary power-law fractal random field and would not be useful for problems that do not utilize additional spatial averaging of the velocity field. The second hierarchical method for fractal random fields has recently been introduced by two of the authors and relies on a suitable explicit multiwavelet expansion (MWE) with high-moment cancellation. This method is described briefly, including a demonstration that, unlike SRA, MWE is consistent with a stationary power-law random field over many decades of scaling and has low variance.

AB - Two hierarchical Monte Carlo methods for the generation of self-similar fractal random fields are compared and contrasted. The first technique, successive random addition (SRA), is currently popular in the physics community. Despite the intuitive appeal of SRA, rigorous mathematical reasoning reveals that SRA cannot be consistent with any stationary power-law Gaussian random field for any Hurst exponent; furthermore, there is an inherent ratio of largest to smallest putative scaling constant necessarily exceeding a factor of 2 for a wide range of Hurst exponents H, with 0.30<H<0.85. Thus, SRA is inconsistent with a stationary power-law fractal random field and would not be useful for problems that do not utilize additional spatial averaging of the velocity field. The second hierarchical method for fractal random fields has recently been introduced by two of the authors and relies on a suitable explicit multiwavelet expansion (MWE) with high-moment cancellation. This method is described briefly, including a demonstration that, unlike SRA, MWE is consistent with a stationary power-law random field over many decades of scaling and has low variance.

KW - Fractal random fields

KW - Monte Carlo methods

KW - successive random addition

UR - http://www.scopus.com/inward/record.url?scp=21844508439&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21844508439&partnerID=8YFLogxK

U2 - 10.1007/BF02179254

DO - 10.1007/BF02179254

M3 - Article

AN - SCOPUS:21844508439

VL - 81

SP - 717

EP - 736

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -