Hierarchical Monte Carlo methods for fractal random fields

Frank W. Elliott, Andrew J. Majda, David J. Horntrop, Richard M. McLaughlin

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)717-736
Number of pages20
JournalJournal of Statistical Physics
Volume81
Issue number3-4
DOIs
StatePublished - Nov 1995

Fingerprint

Monte Carlo method
Random Field
Fractal
fractals
Multiwavelet
Hurst Exponent
Power Law
Mathematical reasoning
exponents
Scaling
scaling
Gaussian Random Field
expansion
Appeal
Cancellation
Inconsistent
cancellation
Velocity Field
Averaging
Intuitive

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

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

In: Journal of Statistical Physics, Vol. 81, No. 3-4, 11.1995, p. 717-736.

Research output: Contribution to journalArticle

Elliott, Frank W. ; Majda, Andrew J. ; Horntrop, David J. ; McLaughlin, Richard M. / Hierarchical Monte Carlo methods for fractal random fields. In: Journal of Statistical Physics. 1995 ; Vol. 81, No. 3-4. pp. 717-736.
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