Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Multi-resolution analysis of wiener-type uncertainty propagation schemes
Journal of Computational Physics
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
An adaptive multi-element generalized polynomial chaos method for stochastic differential equations
Journal of Computational Physics
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
SIAM Journal on Scientific Computing
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
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Journal of Computational Physics
A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties
Journal of Computational Physics
Multi-element probabilistic collocation method in high dimensions
Journal of Computational Physics
Numerical approach for quantification of epistemic uncertainty
Journal of Computational Physics
Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems
Journal of Computational Physics
Time-dependent generalized polynomial chaos
Journal of Computational Physics
Probabilistic models for stochastic elliptic partial differential equations
Journal of Computational Physics
Structural and Multidisciplinary Optimization
Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids
Journal of Computational Physics
Multi-output local Gaussian process regression: Applications to uncertainty quantification
Journal of Computational Physics
A method for solving stochastic equations by reduced order models and local approximations
Journal of Computational Physics
Error Estimates for the ANOVA Method with Polynomial Chaos Interpolation: Tensor Product Functions
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Journal of Computational Physics
Simplex stochastic collocation with ENO-type stencil selection for robust uncertainty quantification
Journal of Computational Physics
An adaptive dimension decomposition and reselection method for reliability analysis
Structural and Multidisciplinary Optimization
Journal of Computational Physics
Subcell resolution in simplex stochastic collocation for spatial discontinuities
Journal of Computational Physics
Grid and basis adaptive polynomial chaos techniques for sensitivity and uncertainty analysis
Journal of Computational Physics
Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems
Journal of Scientific Computing
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Stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters. In this work, we present and examine the multi-element probabilistic collocation method (ME-PCM), which is a generalized form of the probabilistic collocation method. In the ME-PCM, the parametric space is discretized and a collocation/cubature grid is prescribed on each element. Both full and sparse tensor product grids based on Gauss and Clenshaw-Curtis quadrature rules are considered. We prove analytically and observe in numerical tests that as the parameter space mesh is refined, the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness. In addition, the L^2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided. Numerical examples demonstrating adaptive ME-PCM are shown, including low-regularity problems and long-time integration. We test the ME-PCM on two-dimensional Navier-Stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions. While the convergence rate of ME-PCM deteriorates in 50 dimensions, the error in the mean and variance is two orders of magnitude lower than the error obtained with the Monte Carlo method using only a small number of samples (e.g., 100). The computational cost of ME-PCM is found to be favorable when compared to the cost of other methods including stochastic Galerkin, Monte Carlo and quasi-random sequence methods.