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 Multigrid Multilevel Monte Carlo Method for Stokes–Darcy Model with Random Hydraulic Conductivity and Beavers–Joseph Condition
Dissertation
Sparse grid approximation with Gaussians
TL;DR: This work considers the convergence rates for multilevel quasiinterpolation of periodic functions using Gaussians on a grid, and proposes a fast, low complexity, high-dimensional quadrature formula based on Q-MuSIK interpolation of the integrand.
A validation study of a stochastic representation of composite material properties from limited experimental data
TL;DR: In this paper, a probabilistic model based on polynomial chaos expansions is implemented to represent the uncertainty due to the finite-size ensemble measurements, the coefficients of the chaos expansion are themselves considered as random variables.
Dissertation
Méthodes polynomiales parcimonieuses en grande dimension : application aux EDP paramétriques
TL;DR: In this paper, a large classe of EDP parametriques with dependance anisotrope is presented, and a set of convergences of different methodes d'approximation numerique are analyzed.
Journal ArticleDOI
Uncertainty Propagation Analysis of T/R Modules
Z. H. Wang,Chen Jiang,X. X. Ruan,Y. Q. Zhang,Zhiliang Huang,Zhiliang Huang,Congsi Wang,T. Fang +7 more
TL;DR: In this paper, an uncertainty propagation analysis method was developed to analyze transmit/receive (T/R) modules with uncertain parameters, such as variability and tolerances in the physical parameters and the variability of the transmission power distribution.
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.