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
More filters
Journal ArticleDOI
Weighted Approximate Fekete Points: Sampling for Least-Squares Polynomial Approximation
TL;DR: A weighted greedy scheme for computing deterministic sample configurations in multidimensional space for performing least-squares polynomial approximations on L^2 spaces is proposed and analyzed.
Book ChapterDOI
A quasi-optimal sparse grids procedure for groundwater flows
TL;DR: This work proposes an explicit a-priori/a-posteriori procedure for the construction of a quasi-optimal sparse grids method using an estimate of the decay of the Hermite coefficients of the solution and an efficient nested quadrature rule with respect to the Gaussian weight.
Journal ArticleDOI
A Leray regularized ensemble-proper orthogonal decomposition method for parameterized convection-dominated flows
TL;DR: The Leray ensemble-POD model as mentioned in this paper employs the POD spatial filter to smooth (regularize) the convection term in the Navier-Stokes equations and greatly diminishes the numerical inaccuracies produced by the ensemble-pOD method in the numerical simulation of convection-dominated flows.
Multi-level Monte Carlo finite element method for elliptic PDE's with stochastic coefficients
TL;DR: In this paper, the authors proposed the Multi-Level Monte Carlo (MLMC) method for stochastic partial differential equations (SDPE), which has the same convergence rate as the standard Monte Carlo method.
Journal ArticleDOI
A generalized multi-resolution expansion for uncertainty propagation with application to cardiovascular modeling.
TL;DR: A previously proposed multi-resolution approach to uncertainty propagation is generalized to develop a method that improves computational efficiency, can handle arbitrarily distributed random inputs and non-smooth stochastic responses, and naturally facilitates adaptivity, i.e., the expansion coefficients encode information on solution refinement.
References
More filters
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.