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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Stochastic Spline-Collocation Method for Constrained Optimal Control Problem Governed by Random Elliptic PDE
TL;DR: In this article, a stochastic spline-collocation approximation scheme for an optimal control problem governed by an elliptic PDE with random field coefficients is investigated. But the problem is not solved.
Book ChapterDOI
Partitioned Solution of Coupled Stochastic Problems
TL;DR: A stochastic model reduction approach based on low-rank separated representations is proposed for the parti- tioned treatment of the uncertainty space, drastically reducing the overall computational cost.
Book ChapterDOI
Stochastic Algebraic Equations
TL;DR: In this article, methods for solving approximately linear algebraic equations with random parameters, referred to as stochastic algebraic equation (SAEs), are discussed, and the authors outline potential difficulties related to the solution of these equations.
Journal ArticleDOI
Stochastic Deep-Ritz for Parametric Uncertainty Quantification
Ting Wang,Jaroslaw Knap +1 more
TL;DR: This work proposes a deep learning based numerical method for solving elliptic partial differential equations (PDE) with random coeficcients by elucidate the stochastic variational formulation for the problem by recourse to the direct method of calculus of variations.
Journal ArticleDOI
A fast discrete spectral method for stochastic partial differential equations
TL;DR: The goal of this paper is to construct an efficient numerical algorithm for computing the coefficient matrix and the right hand side of the linear system resulting from the spectral Galerkin approximation of a stochastic elliptic partial differential equation.
References
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Book
The Finite Element Method for Elliptic Problems
Philippe G. Ciarlet,J. T. Oden +1 more
TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Book
Finite Element Method for Elliptic Problems
TL;DR: In this article, Ciarlet presents a self-contained book on finite element methods for analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces.
Book
The Mathematical Theory of Finite Element Methods
TL;DR: In this article, the construction of a finite element of space in Sobolev spaces has been studied in the context of operator-interpolation theory in n-dimensional variational problems.
Book
Probability theory
TL;DR: These notes cover the basic definitions of discrete probability theory, and then present some results including Bayes' rule, inclusion-exclusion formula, Chebyshev's inequality, and the weak law of large numbers.