Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Nonlinear component analysis as a kernel eigenvalue problem
Neural Computation
Kernel PCA and de-noising in feature spaces
Proceedings of the 1998 conference on Advances in neural information processing systems II
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Modeling uncertainty in flow simulations via generalized polynomial chaos
Journal of Computational Physics
Kernel Methods for Pattern Analysis
Kernel Methods for Pattern Analysis
Journal of Computational Physics
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
Sparse grid collocation schemes for stochastic natural convection problems
Journal of Computational Physics
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
A non-linear dimension reduction methodology for generating data-driven stochastic input models
Journal of Computational Physics
Journal of Computational Physics
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Journal of Computational Physics
Polynomial chaos representation of spatio-temporal random fields from experimental measurements
Journal of Computational Physics
Identification of Bayesian posteriors for coefficients of chaos expansions
Journal of Computational Physics
Journal of Computational Physics
SIAM Journal on Scientific Computing
A stochastic mixed finite element heterogeneous multiscale method for flow in porous media
Journal of Computational Physics
The pre-image problem in kernel methods
IEEE Transactions on Neural Networks
A stochastic mixed finite element heterogeneous multiscale method for flow in porous media
Journal of Computational Physics
Hi-index | 31.45 |
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.