The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications
Journal of Computational Physics
Journal of Computational Physics
A least-squares approximation of partial differential equations with high-dimensional random inputs
Journal of Computational Physics
Padé-Legendre approximants for uncertainty analysis with discontinuous response surfaces
Journal of Computational Physics
Multi-element probabilistic collocation method in high dimensions
Journal of Computational Physics
Journal of Computational Physics
Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems
Journal of Computational Physics
Journal of Computational Physics
A non-adapted sparse approximation of PDEs with stochastic inputs
Journal of Computational Physics
Sparse Tensor Discretization of Elliptic sPDEs
SIAM Journal on Scientific Computing
Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications
Journal of Computational Physics
Efficient stochastic structural analysis using Guyan reduction
Advances in Engineering Software
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
A Stochastic Mortar Mixed Finite Element Method for Flow in Porous Media with Multiple Rock Types
SIAM Journal on Scientific Computing
Multi-output local Gaussian process regression: Applications to uncertainty quantification
Journal of Computational Physics
SIAM Journal on Scientific Computing
A Posteriori Error Analysis of Parameterized Linear Systems Using Spectral Methods
SIAM Journal on Matrix Analysis and Applications
Strong and Weak Error Estimates for Elliptic Partial Differential Equations with Random Coefficients
SIAM Journal on Numerical Analysis
A Sparse Composite Collocation Finite Element Method for Elliptic SPDEs.
SIAM Journal on Numerical Analysis
Uncertainty Quantification and Weak Approximation of an Elliptic Inverse Problem
SIAM Journal on Numerical Analysis
Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems
SIAM Journal on Matrix Analysis and Applications
An upscaling method using coefficient splitting and its applications to elliptic PDEs
Computers & Mathematics with Applications
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
Combination technique based k-th moment analysis of elliptic problems with random diffusion
Journal of Computational Physics
Uncertainty quantification for integrated circuits: stochastic spectral methods
Proceedings of the International Conference on Computer-Aided Design
Journal of Computational Physics
Computers & Mathematics with Applications
Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems
Journal of Scientific Computing
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This work proposes and analyzes a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems as in the Monte Carlo method. If the number of random variables needed to describe the input data is moderately large, full tensor product spaces are computationally expensive to use due to the curse of dimensionality. In this case the sparse grid approach is still expected to be competitive with the classical Monte Carlo method. Therefore, it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than Monte Carlo. This work provides error estimates for the fully discrete solution using $L^q$ norms and analyzes the computational efficiency of the proposed method. In particular, it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem (number of input random variables) in the final estimates. The derived estimates are then used to compare the method with Monte Carlo, indicating for which problems the former is more efficient than the latter. Computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and Monte Carlo approaches.