Fast simulation of Gaussian random fields

Fast Fourier transforms are used to develop algorithms for the fast generation of correlated Gaussian random fields on d-dimensional rectangular regions. The complexities of the algorithms are derived, simulation results and error analysis are presented.Comment: 15 pages, 8 figures. Typos corrected in Algorithm 3, Remark (4), Algorithm 4, Remark (5), and Algorithm 5, Remark (5

A stochastic fractal model of the universe related to the fractional Laplacian

A new stochastic fractal model based on a fractional Laplace equation is developed. Exact representation for the spectral and correlation functions under random boundary excitation are obtained. Randomized spectral expansion is constructed for simulation of the solution of the fractional Laplace equation. We present calculations for 2D and 3D spaces for a series of fractional parameters showing a strong memory effect: the decay of correlations is several order of magnitudes less compared to the conventional Laplace equation model

Rare event simulation for multiscale diffusions in random environments

We consider systems of stochastic differential equations with multiple scales and small noise and assume that the coefficients of the equations are ergodic and stationary random fields. Our goal is to construct provably-efficient importance sampling Monte Carlo methods that allow efficient computation of rare event probabilities or expectations of functionals that can be associated with rare events. Standard Monte Carlo algorithms perform poorly in the small noise limit and hence fast simulations algorithms become relevant. The presence of multiple scales complicates the design and the analysis of efficient importance sampling schemes. An additional complication is the randomness of the environment. We construct explicit changes of measures that are proven to be logarithmic asymptotically efficient with probability one with respect to the random environment (i.e., in the quenched sense). Numerical simulations support the theoretical results.Comment: Final version, paper to appear in SIAM Journal Multiscale Modelling and Simulatio

Stochastic analysis of an elastic 3D half-space respond to random boundary displacements: Exact results and Karhunen--Loéve expansion

A stochastic response of an elastic 3D half-space to random displacement excitations on the boundary plane is studied. We derive exact results for the case of white noise excitations which are then used to give convolution representations for the case of general finite correlation length fluctuations of displacements prescribed on the boundary. Solutions to this elasticity problem are random fields which appear to be horizontally homogeneous but inhomogeneous in the vertical direction. This enables us to construct explicitly the Karhunen-Lo\`eve (K-L) series expansion by solving the eigen-value problem for the correlation operator. Simulation results are presented and compared with the exact representations derived for the displacement correlation tensor. This paper is a complete 3D generalization of the 2D case study we presented in J. Stat. Physics, v.132 (2008), N6, 1071-1095

Elastic half-plane under random boundary excitations

We study in this paper a respond of an elastic half-plane to random boundary excitations. We treat both the white noise excitations and more generally, homogeneous random fluctuations of displacements prescribed on the boundary. Solutions to these problems are inhomogeneous random fields which are however homogeneous with respect to the longitudinal coordinate. This is used to represent the displacements as series expansions involving a complete set of deterministic functions with corresponding random coefficients. We construct the Karhunen-Lo\`eve (K-L) series expansion which is based on the eigen-decomposition of the correlation operator. The K-L expansion can be used to calculate the statistical characteristics of other functionals of interest, in particular, the strain and stress tensors and the elastic energy tensor

Stochastic simulation of flows and particle transport in porous tubes

A Monte Carlo method is developed for stochastic simulation of flows and particle transport in tubes filled with a porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modelled in a tube with prescribed boundary conditions. Numerical experiments are carried out by solving the random Darcy equation for each sample of the hydraulic conductivity by a SOR iteration method, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, diffusion coefficients, the mean and variance of Lagrangian trajectories, and discuss a ''stagnation" effect which was found in our simulations