A Numerical Technique for Simulating Linear Operations on Random Fields (original) (raw)
Related papers
Efficiency and accuracy in simulation of random fields
Probabilistic Engineering Mechanics, 1996
A direct method for the conditional simulation of a stationary, Gaussian scalar random field is compared with an alternative formulation which uses frequency domain probability density functions. In both cases the random field is described by given correlation or spectral density functions, and no restrictions are placed on these functions, except that they must be positive definite. Efficient implementation techniques are investigated for both general methods. The major computational effort in the most efficient implementations of both procedures is in the solution of linear algebraic equations in which the coefficients are spectral densities. The direct method is shown to be significantly more efficient than existing methods for applying the probability density function technique. However, a new implementation method for the latter technique is also presented, and it equals the efficiency of the direct method. Problems of numerical accuracy due to ill-conditioned matrices are shown not to be severe except when using an inherently problematic form for the spectral density. Numerical examples demonstrate that either method can simulate highly coherent time histories.
Simulation of simply cross correlated random fields by series expansion methods
Structural Safety, 2008
A practical framework for generating cross correlated fields with a specified marginal distribution function, an autocorrelation function and cross correlation coefficients is presented in the paper. The approach relies on well known series expansion methods for simulation of a Gaussian random field. The proposed method requires all cross correlated fields over the domain to share an identical autocorrelation function and the cross correlation structure between each pair of simulated fields to be simply defined by a cross correlation coefficient. Such relations result in specific properties of eigenvectors of covariance matrices of discretized field over the domain. These properties are used to decompose the eigenproblem which must normally be solved in computing the series expansion into two smaller eigenproblems. Such a decomposition represents a significant reduction of computational effort. Non-Gaussian components of a multivariate random field are proposed to be simulated via memoryless transformation of underlying Gaussian random fields for which the Nataf model is employed to modify the correlation structure. In this method, the autocorrelation structure of each field is fulfilled exactly while the cross correlation is only approximated. The associated errors can be computed before performing simulations and it is shown that the errors happen especially in the cross correlation between distant points and that they are negligibly small in practical situations.
2018
This paper presents an approach of one-and two-dimensional random field simulation methods using a correlated random vector and the Karhunen-Loève expansion. Comparison of the authors' analytical solution of the Fredholm integral equation of the second kind with the numerical solution using the finite element method and the inverse vector iteration technique is presented. Numerical approach and sample realizations of one-and two-dimensional random fields are presented using described techniques as well as generated probability distribution functions for chosen point of the analysed domain.
Simulation of Homogeneous and Partially Isotropic Random Fields
Journal of Engineering Mechanics, 1999
A rigorous methodology for the simulation of homogeneous and partially isotropic multidimensional random fields is introduced. The property of partial isotropy of the random field is explicitly incorporated in the derivation of the algorithm. This consideration reduces significantly the computational effort associated with the generation of sample functions, as compared with the case when only the homogeneity in the field is taken into account. The approach is based on the spectral representation method, utilizes the fast Fourier transform, and generates simulations with random variability in both their amplitudes and phases, or in their phases only. Spatially variable seismic ground motions experiencing loss of coherence are generated as an example application of the developed approach.
An efficient and accurate algorithm for generating spatially-correlated random fields
Communications in Numerical Methods in Engineering, 2003
This paper presents a new computer algorithm for generating spatially-correlated random ÿelds. Such ÿelds are often encountered in hydrology and hydrogeology and in the earth sciences and used as inputs for Monte Carlo simulations. The algorithm is designed by using a multilevel grid strategy and combining the matrix decomposition (MD) method and the screening sequential simulation (SSS) method. The idea originates from the facts: (i) the MD method accounts for all possible nodal correlation values and hence the accuracy of the method is high, but it can be extremely computationally intensive for ÿne meshes with large number of nodes, and (ii) the SSS method is more e cient because a small search neighbourhood for conditioning can be used due to the screening e ect of measurements, however, for large separation distances, correlation values from the SSS method is signiÿcantly inferior to the values obtained by the MD method. Numerical examples are presented to demonstrate the new method. It is shown that the presented method is much more e cient than the MD method and more accurate than the SSS method. The new algorithm is also versatile: it can directly simulate ÿelds of irregular geometry without additional e orts.
Computational Geosciences, 2008
This article presents a variant of the spectral turning bands method that allows fast and accurate simulation of intrinsic random fields with power, spline, or logarithmic generalized covariances. The method is applicable in any workspace dimension and is not restricted in the number and configuration of the locations where the random field is simulated; in particular, it does not require these locations to be regularly spaced. On the basis of the central limit and Berry-Esséen theorems, an upper bound is derived for the Kolmogorov distance between the distributions of generalized increments of the simulated random fields and the normal distribution.
Polynomial chaos representation of spatio-temporal random fields from experimental measurements
Journal of Computational Physics, 2009
require a manageable amount of information and thus often provide a rational modeling alternative. These challenges notwithstanding, it remains a recognized fact that many processes representing physical phenomena rarely satisfy the assumptions and constraints associated with a Gaussian process. The current work focuses on the construction of a probability model of a non-stationary and non-Gaussian random process by using a set of measurement data and the associated simulation technique based on the constructed model. The resulting mathematical model readily lends itself to the generation of consistent samples of the process.
This study presents an efficient practical method for the generation of sequential conditional simulation of a Gaussian two-dimensional random field which we frequently encounter in GIS spatial analysis problems such as DEM's generation from a limited number of data. The many realizations typically correspond to many reasons such as the geospatial uncertainty, the morphological perturbations over the surface having a complex structure or the inadequate representation of the triangulated network TIN or grid. These realizations with simulation-based concept enable the performance and uncertainty assessment that tunes to various geospatial (GIS) applications. For DEM generation and implementation of the conditional simulation, we need to decompose the covariance matrix of the data points and grid nodes by Cholesky Decomposition. Conditional simulation respect data values and transfers those values into the grid nodes. With the Incomplete Cholesky decomposition of the covariance matrix, we can produces as many simulations as needed in a single step with an accuracy, in a global sense, much better than the Moving Window Kriging method. In other words, we don't need to repeat covariance matrix generation and decomposition many times. On the other hand, there is the problem of producing covariance matrices in the case of large dataset, which proved to be time consuming and may take several hours on PC. The present paper presents a solution to this problem using Sparse Matrices Technique and Cholesky decomposition to achieve conditional simulation, reducing the time required for computations dramatically, as well as decreasing the demand of large amount of computer memory. For the purpose of this study and testing all algorithms, a MATLAB Programs was made by the author. They have been used in all computation stages and applied using real data. The study has shown that we can reduce computation time by 85%-95% according to the scale of the problem yet saving a considerable space in memory needed to store matrices.
On Fractional Gaussian Random Fields Simulations
2007
To simulate Gaussian fields poses serious numerical problems: storage and computing time. The midpoint displacement method is often used for simulating the fractional Brownian fields because it is fast. We propose an effective and fast method, valid not only for fractional Brownian fields, but for any Gaussian fields. First, our method is compared with midpoint for fractional Brownian fields. Second,