Journal ArticleDOI
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
TLDR
This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms (input data of the model) and provides a rigorous convergence analysis of the fully discrete problem.Abstract:
This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms (input data of the model). The method consists of a Galerkin approximation in the space variables and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem. These procedures seem to be very effective for the problems under study. The proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh approximately equally in the solution, while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables, as in the case where input random variables are Karhunen-Loeve truncations of “smooth” random fields. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates (sub)exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime, with respect to the total number of collocation points. It also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo. In particular, for moderately large-dimensional problems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods.read more
Citations
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Journal ArticleDOI
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
TL;DR: A rigorous convergence analysis is provided and exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space is demonstrated, under some regularity assumptions on the random input data.
ReportDOI
DAKOTA : a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis. Version 5.0, user's manual.
Michael S Eldred,Keith R. Dalbey,William J. Bohnhoff,Brian M. Adams,Laura Painton Swiler,Patricia Diane Hough,John Eddy,Karen H. Haskell +7 more
TL;DR: This report serves as a reference manual for the commands specification for the DAKOTA software, providing input overviews, option descriptions, and example specifications.
Journal ArticleDOI
Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics
TL;DR: This review describes the use of PC expansions for the representation of random variables/fields and discusses their utility for the propagation of uncertainty in computational models, focusing on CFD models.
Journal Article
Fast numerical methods for stochastic computations: A review
TL;DR: This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology.
Journal ArticleDOI
An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations
Xiang Ma,Nicholas Zabaras +1 more
TL;DR: An adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions and 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.
References
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Journal ArticleDOI
Numerical integration using sparse grids
Thomas Gerstner,Michael Griebel +1 more
TL;DR: The usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction is suggested and their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules is shown.
Journal ArticleDOI
A method for numerical integration on an automatic computer
C. W. Clenshaw,A. R. Curtis +1 more
TL;DR: In this paper, a method for numerical integration of a well-behaved function over a finite range of argument is described, which consists essentially of expanding the integrand in a series of Chebyshev polynomials, and integrating this series term by term.
Journal ArticleDOI
Galerkin finite element approximations of stochastic elliptic partial differential equations
TL;DR: A priori error estimates for the computation of the expected value of the solution are given and a comparison of the computational work required by each numerical approximation is included to suggest intuitive conditions for an optimal selection of the numerical approximation.
Journal ArticleDOI
High dimensional polynomial interpolation on sparse grids
TL;DR: The error bounds show that the polynomial interpolation on a d-dimensional cube, where d is large, is universal, i.e., almost optimal for many different function spaces.
ReportDOI
Latin Hypercube Sampling and the Propagation of Uncertainty in Analyses of Complex Systems
Jon C. Helton,Freddie J. Davis +1 more
TL;DR: The following techniques for uncertainty and sensitivity analysis are briefly summarized: Monte Carlo analysis, differential analysis, response surface methodology, Fourier amplitude sensitivity test, Sobol’ variance decomposition, and fast probability integration.