Results 1  10
of
188
ON THE CONVERGENCE OF GENERALIZED POLYNOMIAL CHAOS EXPANSIONS
"... A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal w ..."
Abstract

Cited by 97 (3 self)
 Add to MetaCart
(Show Context)
A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to nonGaussian probability measures. We present conditions on such measures which imply meansquare convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.
Fast Numerical Methods for Stochastic Computations: A Review
, 2009
"... This paper presents a review of the current stateoftheart of numerical methods for stochastic computations. The focus is on efficient highorder methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework ..."
Abstract

Cited by 76 (5 self)
 Add to MetaCart
This paper presents a review of the current stateoftheart of numerical methods for stochastic computations. The focus is on efficient highorder 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 multidimensional random spaces.
Efficient collocational approach for parametric uncertainty analysis
 Commun. Comput. Phys
, 2007
"... Abstract. A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as co ..."
Abstract

Cited by 65 (6 self)
 Add to MetaCart
(Show Context)
Abstract. A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. A highorder stochastic collocation method is employed to solve the solution statistics, and more importantly, to reconstruct the polynomial expansion. While retaining the high accuracy by polynomial expansion, the resulting “pseudospectral ” type algorithm is straightforward to implement as it requires only repetitive deterministic simulations. An estimate on error bounded is presented, along with numerical examples for problems with relatively complicated forms of governing equations. Key words: Collocation methods; pseudospectral methods; stochastic inputs; random differential equations; uncertainty quantification. 1
The MultiElement Probabilistic Collocation Method: Error Analysis and Applications
 J Comp Physics
"... Stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters. In this work, we present and examine the multielement probabilistic collocation method (MEPCM), which is a generalized form of the probabilistic collocation me ..."
Abstract

Cited by 36 (4 self)
 Add to MetaCart
(Show Context)
Stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters. In this work, we present and examine the multielement probabilistic collocation method (MEPCM), which is a generalized form of the probabilistic collocation method. In the MEPCM, the parametric space is discretized and a collocation/cubature grid is prescribed on each element. Both full and sparse tensor product grids based on Gauss and ClenshawCurtis quadrature rules are considered. We prove analytically and observe in numerical tests that as the parameter space mesh is refined, the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness. In addition, the L2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided. Numerical examples demonstrating adaptive MEPCM are shown, including lowregularity problems and longtime integration. We test the MEPCM on twodimensional Navier Stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions. While the convergence rate of MEPCM deteriorates in 50 dimensions, the error in the mean and variance is two orders of magnitude lower than the error obtained with the Monte Carlo method using only a small number of samples (e.g., 100). The computational cost of MEPCM is found to be favorable when compared to the cost of other methods including stochastic Galerkin, Monte Carlo and quasirandom sequence methods. 1
A stochastic collocation approach to Bayesian inference in inverse problems
 Communications in computational physics 6
, 2009
"... Abstract. We present an efficient numerical strategy for the Bayesian solution of inverse problems. Stochastic collocation methods, based on generalized polynomial chaos (gPC), are used to construct a polynomial approximation of the forward solution over the support of the prior distribution. This a ..."
Abstract

Cited by 32 (5 self)
 Add to MetaCart
(Show Context)
Abstract. We present an efficient numerical strategy for the Bayesian solution of inverse problems. Stochastic collocation methods, based on generalized polynomial chaos (gPC), are used to construct a polynomial approximation of the forward solution over the support of the prior distribution. This approximation then defines a surrogate posterior probability density that can be evaluated repeatedly at minimal computational cost. The ability to simulate a large number of samples from the posterior distribution results in very accurate estimates of the inverse solution and its associated uncertainty. Combined with high accuracy of the gPCbased forward solver, the new algorithm can provide great efficiency in practical applications. A rigorous error analysis of the algorithm is conducted, where we establish convergence of the approximate posterior to the true posterior and obtain an estimate of the convergence rate. It is proved that fast (exponential) convergence of the gPC forward solution yields similarly fast (exponential) convergence of the posterior. The numerical strategy and the predicted convergence rates are then demonstrated on nonlinear inverse problems of
EFFICIENT SOLVERS FOR A LINEAR STOCHASTIC GALERKIN MIXED FORMULATION OF DIFFUSION PROBLEMS WITH RANDOM DATA
, 2007
"... We introduce a stochastic Galerkin mixed formulation of the steadystate diffusion equation and focus on the efficient iterative solution of the saddlepoint systems obtained by combining standard finite element discretisations with two distinct types of stochastic basis functions. Socalled mean ..."
Abstract

Cited by 25 (11 self)
 Add to MetaCart
(Show Context)
We introduce a stochastic Galerkin mixed formulation of the steadystate diffusion equation and focus on the efficient iterative solution of the saddlepoint systems obtained by combining standard finite element discretisations with two distinct types of stochastic basis functions. Socalled meanbased preconditioners, based on fast solvers for scalar diffusion problems, are introduced for use with the minimum residual method. We derive eigenvalue bounds for the preconditioned system matrices and report on the efficiency of the chosen preconditioning schemes with respect to all the discretisation parameters.
Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients
, 2008
"... ..."
EFFICIENT STOCHASTIC GALERKIN METHODS FOR RANDOM DIFFUSION EQUATIONS
"... Abstract. We discuss in this paper efficient solvers for stochastic diffusion equations in random media. We employ generalized polynomial chaos (gPC) expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by Galerkin projec ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We discuss in this paper efficient solvers for stochastic diffusion equations in random media. We employ generalized polynomial chaos (gPC) expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by Galerkin projection. Although the resulting system of diffusion equations are coupled, we show that one can construct fast numerical methods to solve them in a decoupled fashion. The methods are based on separation of the diagonal terms and offdiagonal terms in the matrix of the Galerkin system. We examine properties of this matrix and show that the proposed method is unconditionally stable for unsteady problems and convergent for steady problems with a convergent rate independent of discretization parameters. Numerical examples are provided, for both steady and unsteady random diffusions, to support the analysis. Key words. Generalized polynomial chaos, stochastic Galerkin, random diffusion, uncertainty quantification 1. Introduction. We
Adaptive Smolyak pseudospectral approximation
 SIAM Journal on Scientific Computing
, 2012
"... Abstract. Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for nonintrusive pseudospectral approximation, based on Smolyak’s algorithm with generalized sparse grids. We rigorously analyze and extend the ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
(Show Context)
Abstract. Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for nonintrusive pseudospectral approximation, based on Smolyak’s algorithm with generalized sparse grids. We rigorously analyze and extend the nonadaptive method proposed in [6], and compare it to a common alternative approach for using sparse grids to construct polynomial approximations, direct quadrature. Analysis of direct quadrature shows that O(1) errors are an intrinsic property of some configurations of the method, as a consequence of internal aliasing. We provide precise conditions, based on the chosen polynomial basis and quadrature rules, under which this aliasing error occurs. We then establish theoretical results on the accuracy of Smolyak pseudospectral approximation, and show that the Smolyak approximation avoids internal aliasing and makes far more effective use of sparse function evaluations. These results are applicable to broad choices of quadrature rule and generalized sparse grids. Exploiting this flexibility, we introduce a greedy heuristic for adaptive refinement of the pseudospectral approximation. We numerically demonstrate convergence of the algorithm on the Genz test functions, and illustrate the accuracy and efficiency of the adaptive approach on a realistic chemical kinetics problem. Key words. Smolyak algorithms, sparse grids, orthogonal polynomials, pseudospectral approximation, approximation theory, uncertainty quantification AMS subject classifications. 41A10, 41A63, 47A80, 65D15, 65D32 1. Introduction. A