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
Journal of Computational Physics
A fast adaptive wavelet collocation algorithm for multidimensional PDEs
Journal of Computational Physics
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
Uncertainty propagation using Wiener-Haar expansions
Journal of Computational Physics
Multi-resolution analysis of wiener-type uncertainty propagation schemes
Journal of Computational Physics
Using stochastic analysis to capture unstable equilibrium in natural convection
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
Algorithm 847: Spinterp: piecewise multilinear hierarchical sparse grid interpolation in MATLAB
ACM Transactions on Mathematical Software (TOMS)
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
Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network
Journal of Computational Physics
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Journal of Computational Physics
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications
Journal of Computational Physics
Journal of Computational Physics
A stochastic multiscale framework for modeling flow through random heterogeneous porous media
Journal of Computational Physics
A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties
Journal of Computational Physics
Journal of Computational Physics
Numerical approach for quantification of epistemic uncertainty
Journal of Computational Physics
Journal of Computational Physics
A non-adapted sparse approximation of PDEs with stochastic inputs
Journal of Computational Physics
Structural and Multidisciplinary Optimization
Characterization of discontinuities in high-dimensional stochastic problems on adaptive sparse grids
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
Supporting dynamic parameter sweep in adaptive and user-steered workflow
Proceedings of the 6th workshop on Workflows in support of large-scale science
Multiscale Stochastic Preconditioners in Non-intrusive Spectral Projection
Journal of Scientific Computing
Journal of Computational Physics
Multi-output local Gaussian process regression: Applications to uncertainty quantification
Journal of Computational Physics
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
Journal of Computational Physics
Journal of Computational Physics
A probabilistic graphical model approach to stochastic multiscale partial differential equations
Journal of Computational Physics
Journal of Computational Physics
Subcell resolution in simplex stochastic collocation for spatial discontinuities
Journal of Computational Physics
A one-time truncate and encode multiresolution stochastic framework
Journal of Computational Physics
Grid and basis adaptive polynomial chaos techniques for sensitivity and uncertainty analysis
Journal of Computational Physics
Preconditioned Bayesian Regression for Stochastic Chemical Kinetics
Journal of Scientific Computing
Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems
Journal of Scientific Computing
Hi-index | 31.57 |
In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method.