Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes

Abstract

SUMMARYA random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coe cients. Karhunen-Loeve (K-L) series expansion is based on the eigen-decomposition of the covariance function. Its applicability as a simulation tool for both stationary and non-stationary Gaussian random processes is examined numerically in this paper. The study is based on ÿve common covariance models. The convergence and accuracy of the K-L expansion are investigated by comparing the second-order statistics of the simulated random process with that of the target process. It is shown that the factors a ecting convergence are: (a) ratio of the length of the process over correlation parameter, (b) form of the covariance function, and (c) method of solving for the eigen-solutions of the covariance function (namely, analytical or numerical). Comparison with the established and commonly used spectral representation method is made. K-L expansion has an edge over the spectral method for highly correlated processes. For long stationary processes, the spectral method is generally more e cient as the K-L expansion method requires substantial computational e ort to solve the integral equation. The main advantage of the K-L expansion method is that it can be easily generalized to simulate non-stationary processes with little additional e ort.

Probabilistic Analysis of a Rock Salt Cavern with Application to Energy Storage Systems

“…6.6 for an exponential auto-covariance function. In this approach, (6.9) Huang et al (2001) studied the convergence of this approach for different covariance functions using polynomials (Betz et al, 2014). The choice of the number M of terms depends on the required accuracy of the considered problem.…”

Sensitivity analysis of complex models: Coping with dynamic and static inputs

“…In this case, the representation of integral operators is made in wavelet basis and can be performed without numerical integration. The comparison of wavelet-Galerkin method with other available methods in solving the Fredholm integral equation can be found in [17].…”

“…AG Pc a nb er e p r e s e n t e da sas e r i e se x p a n s i o ni n v o l v i n gac o m p l e t es e t of deterministic functions with corresponding random coefficients [15,16], as the truncated Karhunen-Loève (KL) expansion [17,18]. KL series expansion is based on the eigen-decomposition of the covariance function, involving orthogonal deterministic basis functions and the orthogonal uncorrelated random coefficients.…”

An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

“…Arranging the eigenvalues from a KLE in decreasing order λ ω 1 ≥ λ ω 2 ≥ λ ω 3 ≥ ⋯, one may ask how quickly λ ω n decays as a function of n. Frauenfelder et al [21] obtained estimates for the decay of eigenvalues, showing the decay rate decreases with increasing spatial dimension, and that the decay is faster for analytic covariance functions than for the nonanalytic ones. Some intuition is provided by the one-dimensional case: Huang et al [35] have shown that the eigenfunctions approach sinusoids as the domain size increases and the expansion approaches a spectral representation; thus, dropping eigenpairs with small eigenvalues from the KLE corresponds to a primitive upscaling of the conductivity field, by ignoring the higher frequency modes of variation of κ.…”

“…This table shows the L 2 error of the approximate covariance function on a unit square obtained from a truncated KLE as a function of domain size in correlation lengths and number of retained variates. The data were obtained by computing the eigenfunctions Ψ ω of an isotropic Gaussian covariance (34) with correlation length ϱ on a unit square Ω and computing (35) where x 0 is a corner point of Ω, the norm is defined by spatial integration (36) and (37) is an estimate for c ψ , based on Eq. (A.4) but using only the eigenpairs corresponding to the largest eigenvalues.…”