Summary
In this paper I present some new approaches to the random field simulation, and show in four different examples how this simulation technique works. The first example deals with a transport in turbulent flows, where the Lagrangian trajectories are described by a stochastic differential equation whose drift term involves the Eulerian velocity as a random field with a given spectral tensor. Studies of the second example concern with the flows in porous medium governed by the Darcy equation with random hydraulic conductivity. Elasticity system of elliptic Lamé equations with random loads is considered in the third example. Finally, in the fourth example we solve a nonlinear Smoluchowski equation which is used to model the process of crystal growth.
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References
C.K. Chui. An Introduction to Wavelets. Academic Press, Inc., 1992.
G. Dagan. Flow and Transport in Porous Formations. Springer‐Verlag, Berlin‐Heidelberg, Germany, 1989.
G. Dagan. Spatial moments, Ergodicity, and Effective Dispersion. Water Resour. Res., 26(6):1281-1290, 1990.
G. Dagan, A. Fiori, and I. Janković. Flow and transport in highly heterogeneous formations: 1. Conceptual frameework and validity of first‐order approximations. Water Resour. Res., 39(9):1268, 2003.
F.W. ElliottJr, and A.J. Majda. A wavelet Monte Carlo method for turbulent diffusion with many spatial scales. J. Comp. Phys, 113(1):82-111,1994.
F.W. ElliottJr, and A.J. Majda. A new algorithm with plane waves and wavelets for random velocity fields with many spatial scales. J. Comp. Phys, 117:146-162, 1995.
L.W. Gelhar. Stochastic Subsurface Hydrology. Prentice‐Hall, Englewood Cliffs, N.J., 1993.
P. Kramer, O. Kurbanmuradov and K. Sabelfeld. Extensions of multiscale Gaussian random field simulation algorithms. Preprint No. 1040, WIAS, Berlin. To appear in J. Comp. Physics, 2007.
V.M. Kaganer, K.H. Ploog, and K.K. Sabelfeld. Dynamic coalescence kinetics of facetted 2D islands. Physical Review B, 73, N 11 (2006).
O. Kurbanmuradov and K. Sabelfeld. Coagulation of aerosol particles in intermittent turbulent flows. Monte Carlo Methods and Applications. 6 (2000), N3, 211-253.
D. Kolyukhin and K. Sabelfeld. Stochastic Eulerian model for the flow simulation in porous media. Monte Carlo Methods and Applications, 9, No. 3, 2003, 271-290.
A. Kolodko and K. Sabelfeld. Stochastic particle methods for Smolu‐chowski coagulation equation: variance reduction and error estimations. Monte Carlo Methods and Applications. vol. 9, N4, 2003, 315-340.
D. Kolyukhin and K. Sabelfeld. Stochastic flow simulation in 3D Porous media. Monte Carlo Methods and Applications, 11, No. 1, 2005, 15-38.
O. Kurbanmuradov and K. Sabelfeld. Stochastic spectral and Fourierwavelet methods for vector Gaussian random field. Monte Carlo Methods and Applications, 12 (2006), N 5-6, 395-445.
O. Kurbanmuradov, K. Sabelfeld, O. Smidts and H.A. Vereecken. Lagrangian stochastic model for transport in statistically homogeneous porous media. Monte Carlo Methods and Applications. 9, No. 4, 2003, 341-366.
O. Kurbanmuradov. Weak Convergence of Approximate Models of Ran‐dom Fields. Russian Journal of Numerical Analysis and Mathematical Modelling, 10 (1995), N6, 500-517.
Y. Meyer. Ondelettes et opérateurs. I: Ondelettes, Hermann, Paris, 1990. (English translation: Wavelets and operators, Cambridge University Press, 1992.)
G.A. Mikhailov. Approximate models of random processes and fields. Russian J. Comp. Mathem. and mathem. Physics, vol. 23 (1983), N3, 558-566. (in Russian).
A.S. Monin and A.M. Yaglom. Statistical Fluid Mechanics: Mechanics of Turbulence, Volume 2. The M.I.T. Press, 1981.
F. Poirion and C. Soize. Numerical methods and mathematical aspects for simulation of homogenous and non homogenous Gaussian vector fields. In Paul Kree and Walter Wedig, editors, Probabilistic methods in applied physics, volume 451 of Lecture Notes in Physics, pages 17-53. Springer‐Verlag, Berlin, 1995.
K. Sabelfeld. Monte Carlo methods in boundary value problems, chapter 1,5, pages 31-47, 228-238. Springer Series in Computational Physics. Springer‐Verlag, Berlin, 1991.
M. Shinozuka. Simulation of multivariate and multidimensional random processes. J. of Acoust. Soc. Am. 49 (1971), 357-368.
K. Sabelfeld, A. Levykin, and T. Privalova. A fast stratified sampling simulation of coagulation processes. Monte Carlo Methods and Applications, 13 (2007), N1, 63-84.
P.D. Spanos and V. Ravi S. Rao. Random Field Representation in a Biorthogonal Wavelet Basis. Journal of Engineering Mechanics, 127, No. 2 (2001), 194-205.
K. Sabelfeld, I. Shalimova and A.I. Levykin. Stochastic simulation method for a 2D elasticity problem with random loads. Submitted to “Probabilistic Engineering Mechanics”.
D.J. Thomson and B.J. Devenish. Particle pair in kinematic simulations. Journal of Fluid Mechanics, 526 (2005), 277-302.
W. Wagner. Post‐gelation behavior of a spatial coagulation model. Electron. J. Probab., 893-933. / (WIAS preprint N 1128, 2006).
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Sabelfeld, K. (2008). Random Field Simulation and Applications. In: Keller, A., Heinrich, S., Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74496-2_7
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DOI: https://doi.org/10.1007/978-3-540-74496-2_7
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