Kernel principal component analysis for stochastic input model generation

Journal Article · Wed Aug 10 04:00:00 EDT 2011 · Journal of Computational Physics

OSTI ID:21592607

Xiang, Ma ^[1]; Zabaras, Nicholas ^[1]

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1. Materials Process Design and Control Laboratory, Sibley School of Mechanical and Aerospace Engineering, 101 Frank H.T. Rhodes Hall, Cornell University, Ithaca, NY 14853-3801 (United States)

Highlights: {yields} KPCA is used to construct a reduced order stochastic model of permeability. {yields} A new approach is proposed to solve the pre-image problem in KPCA. {yields} Polynomial chaos is used to provide a parametric stochastic input model. {yields} Flow in porous media with channelized permeability is considered. - Abstract: Stochastic analysis of random heterogeneous media provides useful information only if realistic input models of the material property variations are used. These input models are often constructed from a set of experimental samples of the underlying random field. To this end, the Karhunen-Loeve (K-L) expansion, also known as principal component analysis (PCA), is the most popular model reduction method due to its uniform mean-square convergence. However, it only projects the samples onto an optimal linear subspace, which results in an unreasonable representation of the original data if they are non-linearly related to each other. In other words, it only preserves the first-order (mean) and second-order statistics (covariance) of a random field, which is insufficient for reproducing complex structures. This paper applies kernel principal component analysis (KPCA) to construct a reduced-order stochastic input model for the material property variation in heterogeneous media. KPCA can be considered as a nonlinear version of PCA. Through use of kernel functions, KPCA further enables the preservation of higher-order statistics of the random field, instead of just two-point statistics as in the standard Karhunen-Loeve (K-L) expansion. Thus, this method can model non-Gaussian, non-stationary random fields. In this work, we also propose a new approach to solve the pre-image problem involved in KPCA. In addition, polynomial chaos (PC) expansion is used to represent the random coefficients in KPCA which provides a parametric stochastic input model. Thus, realizations, which are statistically consistent with the experimental data, can be generated in an efficient way. We showcase the methodology by constructing a low-dimensional stochastic input model to represent channelized permeability in porous media.

• OSTI ID: 21592607
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: 19 Vol. 230; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
ABSORPTION
CHAOS THEORY
CONVERGENCE
DIFFERENTIAL EQUATIONS
EQUATIONS
FUNCTIONS
MATERIALS
MATHEMATICAL MODELS
MATHEMATICS
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
PERMEABILITY
PHYSICAL PROPERTIES
POLAR-CAP ABSORPTION
POLYNOMIALS
POROUS MATERIALS
RANDOMNESS
SORPTION
STATISTICS
STOCHASTIC PROCESSES