Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations
TLDR
In this paper, the mathematical setting of stationary systems modelled by elliptic partial differential equations with stochastic coefficients (random fields) is investigated and stability with respect to stability.About:
This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2005-04-08 and is currently open access. It has received 590 citations till now. The article focuses on the topics: Stochastic partial differential equation & Polynomial chaos.read more
Figures
Figure 8: Mean and Variance of Solution on an L-shaped Region. 
Figure 2: Realisation ofκ. 
Figure 7: Realisations of Material and Solution on an L-shaped Region. 
Figure 12: Solution Errors by Direct Computation 
Figure 3: KL-Eigenfunctions Nos. 1 and 15. 
Figure 10: Error in Mean for PC-Galerkin and Example Statistic.
Citations
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Journal ArticleDOI
Global sensitivity analysis using polynomial chaos expansions
TL;DR: In this article, generalized polynomial chaos expansions (PCE) are used to build surrogate models that allow one to compute the Sobol' indices analytically as a post-processing of the PCE coefficients.
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.
Journal ArticleDOI
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
TL;DR: This work 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, indicating for which problems the sparse grid stochastic collocation method is more efficient than Monte Carlo.
Journal ArticleDOI
Adaptive sparse polynomial chaos expansion based on least angle regression
Géraud Blatman,Bruno Sudret +1 more
TL;DR: A non intrusive method that builds a sparse PC expansion, which may be obtained at a reduced computational cost compared to the classical ''full'' PC approximation.
Journal ArticleDOI
The stochastic finite element method: Past, present and future
TL;DR: A state-of-the-art review of past and recent developments in the SFEM area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future are provided.
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
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
TL;DR: In this paper, Schnabel proposed a modular system of algorithms for unconstrained minimization and nonlinear equations, based on Newton's method for solving one equation in one unknown convergence of sequences of real numbers.
Book
Numerical Solution of Stochastic Differential Equations
Peter E. Kloeden,Eckhard Platen +1 more
TL;DR: In this article, a time-discrete approximation of deterministic Differential Equations is proposed for the stochastic calculus, based on Strong Taylor Expansions and Strong Taylor Approximations.