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Journal ArticleDOI

An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

01 May 2008-SIAM Journal on Numerical Analysis (Society for Industrial and Applied Mathematics)-Vol. 46, Iss: 5, pp 2411-2442
TL;DR: 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.

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Citations
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Journal ArticleDOI
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.
Abstract: In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuscka, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

1,597 citations


Cites background or methods from "An Anisotropic Sparse Grid Stochast..."

  • ...Their systematic choice can be based either on a priori or a posteriori information and is motivated by the regularity of the solution and the error estimates derived in [45]....

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  • ...It is also shown in [45] that, under the same condition, the constant Ĉ(r, N) is uniformly bounded with respect to N ....

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  • ...Anisotropic sparse stochastic collocation [45] combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective in reducing the curse of dimensionality for problems depending on random variables which weigh approximately equally in the solution, and the second is appropriate for solving highly anisotropic problems with relatively low effective dimension, as in the case where input random variables are Karhunen–Loève truncations of “smooth” random fields....

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  • ...The actual construction of α can be based on either a priori or a posteriori information, as suggested in [45]....

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  • ...The second result we state is taken from [45] and analyzes the rate of convergence for the anisotropic Smolyak formula....

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ReportDOI
01 May 2010
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.
Abstract: The DAKOTA (Design Analysis Kit for Optimization and Terascale Applications) toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. DAKOTA contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic finite element methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the DAKOTA toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a reference manual for the commands specification for the DAKOTA software, providing input overviews, option descriptions, and example specifications. DAKOTA Version 5.0 Reference Manual generated on May 7, 2010

757 citations

Journal ArticleDOI
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.
Abstract: The quantification of uncertainty in computational fluid dynamics (CFD) predictions is both a significant challenge and an important goal. Probabilistic uncertainty quantification (UQ) methods have been used to propagate uncertainty from model inputs to outputs when input uncertainties are large and have been characterized probabilistically. Polynomial chaos (PC) methods have found increased use in probabilistic UQ over the past decade. 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. Many CFD applications are considered, including flow in porous media, incompressible and compressible flows, and thermofluid and reacting flows. The review examines each application area, focusing on the demonstrated use of PC UQ and the associated challenges. Cross-cutting challenges with time unsteadiness and long time horizons are also discussed.

731 citations

Journal Article
Dongbin Xiu1
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.
Abstract: This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations. The focus is on efficient high-order methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework of gPC is reviewed, along with its Galerkin and collocation approaches for solving stochastic equations. Properties of these methods are summarized by using results from literature. This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multi-dimensional random spaces. AMS subject classifications: 41A10, 60H35, 65C30, 65C50

665 citations


Cites methods from "An Anisotropic Sparse Grid Stochast..."

  • ...The current work includes adaptive choice of polynomial basis [19, 79], adaptive element selection in multi-element gPC [80], and adaptive sparse grid collocation [20, 78]....

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  • ...Althoughwith the fast growth of computing power and some newly developed adaptive algorithms, in both stochastic Galerkin and collocation methods [19, 20, 78–80], the difficulty is alleviated to some degree....

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Journal ArticleDOI
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.

484 citations

References
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Book
01 Jan 1978
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.
Abstract: Preface 1. Elliptic boundary value problems 2. Introduction to the finite element method 3. Conforming finite element methods for second-order problems 4. Other finite element methods for second-order problems 5. Application of the finite element method to some nonlinear problems 6. Finite element methods for the plate problem 7. A mixed finite element method 8. Finite element methods for shells Epilogue Bibliography Glossary of symbols Index.

8,407 citations

Book
01 Apr 2002
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.
Abstract: From the Publisher: This book is particularly useful to graduate students, researchers, and engineers using finite element methods. The reader should have knowledge of analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces. Other than these basics, the book is mathematically self-contained. About the Author Philippe G. Ciarlet is a Professor at the Laboratoire d'Analyse Numerique at the Universite Pierre et Marie Curie in Paris. He is also a member of the French Academy of Sciences. He is the author of more than a dozen books on a variety of topics and is a frequent invited lecturer at meetings and universities throughout the world. Professor Ciarlet has served approximately 75 visiting professorships since 1973, and he is a member of the editorial boards of more than 20 journals.

8,052 citations


Additional excerpts

  • ...[7, 5])....

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Book
14 Feb 2013
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.
Abstract: Preface(2nd ed.).- Preface(1st ed.).- Basic Concepts.- Sobolev Spaces.- Variational Formulation of Elliptic Boundary Value Problems.- The Construction of a Finite Element of Space.- Polynomial Approximation Theory in Sobolev Spaces.- n-Dimensional Variational Problems.- Finite Element Multigrid Methods.- Additive Schwarz Preconditioners.- Max-norm Estimates.- Adaptive Meshes.- Variational Crimes.- Applications to Planar Elasticity.- Mixed Methods.- Iterative Techniques for Mixed Methods.- Applications of Operator-Interpolation Theory.- References.- Index.

7,158 citations


Additional excerpts

  • ...That is, s = 1 and C(s;φ) = ‖φ‖H2(D); see, for example, [5]....

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Book
01 Jan 1963
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.
Abstract: 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. 1 Sample spaces and events To treat probability rigorously, we define a sample space S whose elements are the possible outcomes of some process or experiment. For example, the sample space might be the outcomes of the roll of a die, or flips of a coin. To each element x of the sample space, we assign a probability, which will be a non-negative number between 0 and 1, which we will denote by p(x). We require that x∈S p(x) = 1, so the total probability of the elements of our sample space is 1. What this means intuitively is that when we perform our process, exactly one of the things in our sample space will happen. Example. The sample space could be S = {a, b, c}, and the probabilities could be p(a) = 1/2, p(b) = 1/3, p(c) = 1/6. If all elements of our sample space have equal probabilities, we call this the uniform probability distribution on our sample space. For example, if our sample space was the outcomes of a die roll, the sample space could be denoted S = {x 1 , x 2 ,. .. , x 6 }, where the event x i correspond to rolling i. The uniform distribution, in which every outcome x i has probability 1/6 describes the situation for a fair die. Similarly, if we consider tossing a fair coin, the outcomes would be H (heads) and T (tails), each with probability 1/2. In this situation we have the uniform probability distribution on the sample space S = {H, T }. We define an event A to be a subset of the sample space. For example, in the roll of a die, if the event A was rolling an even number, then A = {x 2 , x 4 , x 6 }. The probability of an event A, denoted by P(A), is the sum of the probabilities of the corresponding elements in the sample space. For rolling an even number, we have P(A) = p(x 2) + p(x 4) + p(x 6) = 1 2 Given an event A of our sample space, there is a complementary event which consists of all points in our sample space that are not …

6,236 citations


"An Anisotropic Sparse Grid Stochast..." refers background in this paper

  • ...4 (Karhunen-Loève expansion) We recall that any second order random field g(ω, x), with continuous covariance function cov[g] : D ×D → R, can be represented as an infinite sum of random variables, by means, for instance, of a Karhunen-Loève expansion [21]....

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Book
29 Mar 1977

6,171 citations