Hierarchical Monte Carlo methods for fractal random fields. (English) Zbl 1107.82326

Summary: 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.


82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
60K40 Other physical applications of random processes
Full Text: DOI


[1] M. Lesieur,Trubulence in Fluids (Kluwer, Boston, 1990), Chapter 8, and references therein. · Zbl 0748.76004
[2] W. McComb,The Physics of Fluid Turbulence (Clarendon Press, Oxford, 1990), Chapters 12 and 13, and references therein. · Zbl 0748.76005
[3] G. CsanadyTurbulent Diffusion in the Environment (Reidel, Dordrecht, Holland, 1973).
[4] G. Dagan, Theory of solute transport by ground water,Annu. Rev. Fluid Mech. 19:183–215 (1987). · Zbl 0687.76091
[5] F. Williams, Turbulent combustion, inThe Mathematics of Combustion, J. Buckmaster, ed. (SIAM, Philadelphia, 1985), pp. 97–131.
[6] J. Feder,Fractals (Plenum Press, New York, 1988), Chapters 9–14. · Zbl 0648.28006
[7] R. F. Voss, Random fractal forgeries, inFundamental Algorithms in Computer Graphics, R. A. Earnshaw, ed. (Springer-Verlag, Berlin), pp. 805–835.
[8] J. Viecelli and E. Canfield, Functional representation of power-law random fields and time series,J. Comp. Phys. 95:29–39 (1991). · Zbl 0729.60115
[9] F. Elliott and A. Majda, A wavelet Monte Carlo method for turbulent diffusion with many spatial scales.J. Comp. Phys. 113:82–109 (1994). · Zbl 0853.76058
[10] F. Elliott and A. Majda, A new algorithm with plane waves and wavelets for random velocity fields with many spatial scales,J. Comp. Phys. 117:146–162 (1995). · Zbl 0821.65097
[11] F. Elliott and A. Majda, Monte Carlo simulation of pair dispersion over an inertial range with many decades,Phys. Fluids (1995), submitted.
[12] B. Alpert, Sparse representation of smooth linear operators, Ph.D. thesis, Department of Computer Science, Yale University (December 1990).
[13] F. Elliott and A. Majda, The convergence of multi-wavelet Monte Carlo methods for fractal random fields, in preparation.
[14] J. Eggers and S. Grossman, Effect of dissipation fluctuations on anomalous velocity scaling in turbulence,Phys. Rev. A 45:2360–2369 (1992).
[15] A. Juneja, D. Lathrop, K. Sreenivasan, and G. Stolovitzky, Synthetic turbulence,Phys. Rev. E 49:5179–5194 (1994).
[16] R. Benzi, L. Biferale, A. Crisanti, G. Paladin, M. Vergassola, and A. Vulpiani, A random process for the construction of multiaffine fields,Physica D 65:352–358 (1993). · Zbl 0772.60093
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.