Simulation Of Gaussian Random Fields Research Articles

A fast direct solver for nonlocal operators in wavelet coordinates

In this article, we consider fast direct solvers for nonlocal operators. The pivotal idea is to combine a wavelet representation of the system matrix, yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The latter drastically reduces the fill-in during the factorization of the system matrix by means of a Cholesky decomposition or an LU decomposition, respectively. This way, we end up with the exact inverse of the compressed system matrix with only a moderate increase of the number of nonzero entries in the matrix.To illustrate the efficacy of the approach, we conduct numerical experiments for different highly relevant applications of nonlocal operators: We consider (i) the direct solution of boundary integral equations in three spatial dimensions, issuing from the polarizable continuum model, (ii) a parabolic problem for the fractional Laplacian in integral form and (iii) the fast simulation of Gaussian random fields.

Fast and exact simulation of Gaussian random fields defined on the sphere cross time

We provide a method for fast and exact simulation of Gaussian random fields on the sphere having isotropic covariance functions. The method proposed is then extended to Gaussian random fields defined over the sphere cross time and having covariance functions that depend on geodesic distance in space and on temporal separation. The crux of the method is in the use of block circulant matrices obtained working on regular grids defined over longitude and latitude.

Conditional Latin Hypercube Simulation of (Log)Gaussian Random Fields

In earth and environmental sciences applications, uncertainty analysis regarding the outputs of models whose parameters are spatially varying (or spatially distributed) is often performed in a Monte Carlo framework. In this context, alternative realizations of the spatial distribution of model inputs, typically conditioned to reproduce attribute values at locations where measurements are obtained, are generated via geostatistical simulation using simple random (SR) sampling. The environmental model under consideration is then evaluated using each of these realizations as a plausible input, in order to construct a distribution of plausible model outputs for uncertainty analysis purposes. In hydrogeological investigations, for example, conditional simulations of saturated hydraulic conductivity are used as input to physically-based simulators of flow and transport to evaluate the associated uncertainty in the spatial distribution of solute concentration. Realistic uncertainty analysis via SR sampling, however, requires a large number of simulated attribute realizations for the model inputs in order to yield a representative distribution of model outputs; this often hinders the application of uncertainty analysis due to the computational expense of evaluating complex environmental models. Stratified sampling methods, including variants of Latin hypercube sampling, constitute more efficient sampling aternatives, often resulting in a more representative distribution of model outputs (e.g., solute concentration) with fewer model input realizations (e.g., hydraulic conductivity), thus reducing the computational cost of uncertainty analysis. The application of stratified and Latin hypercube sampling in a geostatistical simulation context, however, is not widespread, and, apart from a few exceptions, has been limited to the unconditional simulation case. This paper proposes methodological modifications for adopting existing methods for stratified sampling (including Latin hypercube sampling), employed to date in an unconditional geostatistical simulation context, for the purpose of efficient conditional simulation of Gaussian random fields. The proposed conditional simulation methods are compared to traditional geostatistical simulation, based on SR sampling, in the context of a hydrogeological flow and transport model via a synthetic case study. The results indicate that stratified sampling methods (including Latin hypercube sampling) are more efficient than SR, overall reproducing to a similar extent statistics of the conductivity (and subsequently concentration) fields, yet with smaller sampling variability. These findings suggest that the proposed efficient conditional sampling methods could contribute to the wider application of uncertainty analysis in spatially distributed environmental models using geostatistical simulation.

Статистичне моделювання сейсмічного шуму в багатовимірній області змінних для визначення частотних характеристик геологічного середовища

The paper deals with the theory and methods of statistical simulation of random processes and fields based on their spectral decomposition and Kotelnikov-Shennon modified interpolation sums, as well as applying these methods for environmental geophysical monitoring. Statistical simulation of multivariate random fields (those homogeneous in time and homogeneous isotropic in n other variables) are considered to be essential for seismological research into frequency characteristics of geological media. A statistical model and a numerical algorithm of simulating random fields are built on the basis of Kotelnikov-Shennon modified interpolation decomposition to generate adequate realizations of seismic noise. The paper examines real-valued random fields ξ(t,x),tϵÎR,xÎRn, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. It also considers approximation of random fields by the random fields with a bounded spectrum. There is made an analogue of the Kotelnikov–Shannon theorem for random fields with a bounded spectrum. Besides, there are obtained estimates of the mean-square approximation of random fields in the space R´Rn by a model constructed with the help of spectral decomposition and Kotelnikov–Shannon interpolation formula. The paper provides a mechanism for statistical simulation of Gaussian random fields with a bounded spectrum; namely, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. Proved have been the theorems of the mean-square approximation of random fields (those homogeneous in time and homogeneous isotropic ones relative to n- other variables) by special partial sums. A simulation method was used to formulate an algorithm of numerical simulation by means of these theorems. There are also considered ways to carry out spectral analysis of generated seismic noise realizations. Finally, there have been developed universal methods of statistical simulation (Monte Carlo methods) of multi-parameter seismology data for generating seismic noise on 2D and 3D grids of the required detail and regularity.

Erratum: Fast simulation of Gaussian random fields [Monte Carlo Methods Appl. 17 (2011), 195–214

In the paper “Fast simulation of Gaussian random fields”, a typo occurred. Instead of it should read in Algorithm 3.1, Remark d). For convenience of the reader we reproduce below the complete corrected algorithms.

Simulation of Gaussian Random Fields With One Derivative

Circulant embedding provides a fast and exact algorithm for simulating stationary Gaussian random fields on grids. However, the method tends to work poorly for differentiable processes with correlation ranges comparable to the size of the simulation domain. This work describes an algorithm for simulating differentiable stationary Gaussian processes, with the focus on processes possessing exactly one derivative. The algorithm works by first simulating a filtered version of the process using circulant embedding and then recovering the original process from the filtered version. This recovery step is the slowest part of the computation, requiring many solutions of large systems of linear equations. Preconditioned conjugate gradient provides a viable method for obtaining essentially exact solutions to these linear equations. The memory requirements are linear in the number of simulation points and if more than one simulation under a model is needed, subsequent simulations are much faster than the first. The algorithm can be modified to simulate certain differentiable nonstationary intrinsic random functions, to which existing approaches to circulant embedding do not apply. A supplemental file contains code for simulating these intrinsic random functions.

Multi-Gaussian kriging and simulation in the presence of an uncertain mean value

The multi-Gaussian model is used in geostatistical applications to predict functions of a regionalized variable and to assess uncertainty by determining local (conditional to neighboring data) distributions. The model relies on the assumption that the regionalized variable can be represented by a transform of a Gaussian random field with a known mean value, which is often a strong requirement. This article presents two variations of the model to account for an uncertain mean value. In the first one, the mean of the Gaussian random field is regarded as an unknown non-random parameter. In the second model, the mean of the Gaussian field is regarded as a random variable with a very large prior variance. The properties of the proposed models are compared in the context of non-linear spatial prediction and uncertainty assessment problems. Algorithms for the conditional simulation of Gaussian random fields with an uncertain mean are also examined, and problems associated with the selection of data in a moving neighborhood are discussed.

Generalized Sequential Gaussian Simulation on Group Size and Screen-Effect Approximations for Large Field Simulations

The modelling of spatial uncertainty in attributes of geological phenomena is frequently based on the stochastic simulation of Gaussian random fields. This paper presents a generalization of the sequential Gaussian simulation method founded upon the group decomposition of the posterior probability density function of a stationary and ergodic Gaussian random field into posterior probability densities of a set of groups of nodes of size ν. The method, which is termed “generalized sequential Gaussian simulation on group size ν,” relies computationally on sharing the neighborhood of adjacent nodes and simulates groups of ν nodes at a time, instead of the traditional node-by-node simulation. The new method is computationally efficient and suitable for simulation on large grids of nodes. Results suggest that, in terms of computing cost (time), an optimal size ν of a group is about 80% of the optimal neighborhood used for sequential Gaussian simulation and that computation can be up to 50 times faster than the regular sequential Gaussian method, with little loss in accuracy. The effectiveness of the method is assessed by using a general measure of accuracy, “screen-effect approximation loss” (SEA loss), defined herein as the mean-square difference between the simulated value posterior to the information in the neighborhood and the simulated value posterior to all information, and shown to be determined by the corresponding posterior variances. The results presented show that both the exponential and the spherical models perform well and can meet the less-than 5% relative SEA loss requirement for any grid setup using a relatively small neighborhood. The Gaussian covariance model was found to have a relatively high relative SEA loss in most cases.

A Hybrid Markov Chain Monte Carlo Method for Generating Permeability Fields Conditioned to Multiwell Pressure Data and Prior Information

Summary It is necessary to construct and sample the a posteriori probability density functions (pdf's) for the rock property fields to properly evaluate the uncertainty in reservoir performance predictions. In this work, the a posteriori pdf is constructed based on prior means and variograms (covariance function) for log-permeability and multiwell pressure data. Within the context of sampling the probability density function, we argue that the notion of equally probable realizations is the wrong paradigm for reservoir characterization. If the simulation of Gaussian random fields with a known variogram is the objective, it is shown that the variogram should not be incorporated directly into the objective function if simulated annealing (SA) is applied either to sample the a posteriori pdf or to estimate a global minimum of the associated objective function. It is shown that the hybrid Markov chain Monte Carlo (MCMC) method provides a way to explore more fully the set of plausible log-permeability fields and does not suffer from the high rejection rates of more standard MCMC methods.

Conditional simulation of non-Gaussian random fields for earthquake monitoring systems

The problem of conditional simulation of random fields gained a significant interest recently due to its applications to urban earthquake monitoring. In this paper, for the first time in the literature, the method of conditional simulation of non-Gaussian random fields is developed. It combines previous techniques of iterative procedure of unconditional simulation of non-Gaussian fields, and the procedure of conditional simulation of Gaussian random fields. To contrast the agreement between the simulated correlation function and targeted correlation function, the numerical error is decomposed into two parts, namely, into simulation error and mapping error. Simulation error can be reduced by increasing number of samples while mapping error is eliminated by the suitable iteration procedure. In this paper univariate and time-independent random fields are considered. Numerical example shows that the correlation structure and probability distribution of the simulated random field have excellent agreements with given correlation structure and probability distribution, respectively.

Conditional simulation of non-Gaussian random fields

The method of conditional simulation of homogeneous or inhomogeneous non-Gaussian random fields is developed. It represents a combination of two previously developed methods, namely of the iterative method for unconditional simulation of non-Gaussian random fields put forward by Shinozuka and Yamazaki, and of the Hoshiya method for conditional simulation of Gaussian random fields. To gain physical insight, several numerical examples are evaluated. High accuracy of the proposed method is demonstrated.

Improving Stochastic Relaxation for Gussian Random Fields

In this paper, we are concerned with the simulation of Gaussian random fields by means of iterative stochastic algorithms, which are compared in terms of rate of convergence. A parametrized class of algorithms, which includes stochastic relaxation (Gibbs sampler), is proposed and its convergence properties are established. A suitable choice for the parameter improves the rate of convergence with respect to stochastic relaxation for special classes of covariance matrices. Some examples and numerical experiments are given.

Simulation of space-random fields for solution of stochastic boundary-value problems

The technique of digital simulation of Gaussian random fields (which are nonhomogeneous in space or are part of homogeneous random fields in space) is presented to a solution of stochastic boundary-value problems. The method consists of expanding the simulated field, with known mean and autocorrelation function, in series in terms of the structural ’’natural’’ mode shapes, and the Fourier coefficients of the truncated series are then simulated as random normal vectors. The method is applicable to static or dynamic stochastic two-point boundary-value problems in mechanics of solids.