Hierarchical Monte Carlo methods for fractal random fields |
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Authors: | Frank W Elliott Jr Andrew J Majda David J Horntrop Richard M McLaughlin |
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Institution: | (1) Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, 10012 New York, New York;(2) Department of Mathematics, University of Utah, 84112 Salt Lake City, Utah |
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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 exponentsH, 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. |
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Keywords: | Fractal random fields Monte Carlo methods successive random addition |
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