Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Stochastic differential equations (3rd ed.): an introduction with applications
Stochastic differential equations (3rd ed.): an introduction with applications
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Journal of Computational Physics
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
A stochastic variational multiscale method for diffusion in heterogeneous random media
Journal of Computational Physics
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 Numerical Analysis
A stochastic multiscale framework for modeling flow through random heterogeneous porous media
Journal of Computational Physics
Journal of Computational Physics
On the approximation error in high dimensional model representation
Proceedings of the 40th Conference on Winter Simulation
Multi-element probabilistic collocation method in high dimensions
Journal of Computational Physics
A stochastic mixed finite element heterogeneous multiscale method for flow in porous media
Journal of Computational Physics
Kernel principal component analysis for stochastic input model generation
Journal of Computational Physics
Adaptive ANOVA decomposition of stochastic incompressible and compressible flows
Journal of Computational Physics
An adaptive dimension decomposition and reselection method for reliability analysis
Structural and Multidisciplinary Optimization
Journal of Computational Physics
A probabilistic graphical model approach to stochastic multiscale partial differential equations
Journal of Computational Physics
A simplex-based numerical framework for simple and efficient robust design optimization
Computational Optimization and Applications
An adaptive ANOVA-based PCKF for high-dimensional nonlinear inverse modeling
Journal of Computational Physics
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A computational methodology is developed to address the solution of high-dimensional stochastic problems. It utilizes high-dimensional model representation (HDMR) technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower-order to higher-order component functions. HDMR is efficient at capturing the high-dimensional input-output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower-order terms. An adaptive version of HDMR is also developed to automatically detect the important dimensions and construct higher-order terms using only the important dimensions. The newly developed adaptive sparse grid collocation (ASGC) method is incorporated into HDMR to solve the resulting sub-problems. By integrating HDMR and ASGC, it is computationally possible to construct a low-dimensional stochastic reduced-order model of the high-dimensional stochastic problem and easily perform various statistic analysis on the output. Several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method. The cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large-input variability. The efficiency of the proposed method is examined by comparing with Monte Carlo (MC) simulation.