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364
HighOrder Collocation Methods for Differential Equations with Random Inputs
 SIAM Journal on Scientific Computing
"... Abstract. Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling met ..."
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Cited by 180 (9 self)
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Abstract. Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling methods. However, when the governing equations take complicated forms, numerical implementations of stochastic Galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations. On the other hand, the traditional sampling methods, e.g., Monte Carlo methods, are straightforward to implement, but they do not offer convergence as fast as stochastic Galerkin methods. In this paper, a highorder stochastic collocation approach is proposed. Similar to stochastic Galerkin methods, the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence. However, the numerical implementation of stochastic collocation is trivial, as it requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods. The computational cost of the collocation methods depends on the choice of the collocation points, and we present several feasible constructions. One particular choice, based on sparse grids, depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs. Numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods. Key words. collocation methods, stochastic inputs, differential equations, uncertainty quantification
Galerkin Methods for Linear and Nonlinear Elliptic Stochastic Partial Differential Equations
, 2003
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Preconditioning stochastic Galerkin saddle point systems
 SIAM J. Matrix Anal. Appl
"... Abstract. Mixed finite element discretizations of deterministic secondorder elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic secondorder elliptic PDE ..."
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Cited by 110 (4 self)
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Abstract. Mixed finite element discretizations of deterministic secondorder elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic secondorder elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are blockdense and the cost of a matrix vector product is nontrivial. We implement a stochastic Galerkin discretization for the steadystate diffusion problem written as a mixed firstorder system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of unbounded random parameters. We study the resulting saddle point systems and investigate the efficiency of blockdiagonal preconditioners of Schur complement and augmented type, for use with minres. By introducing socalled Kronecker product preconditioners we improve the robustness of cheap, meanbased preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.
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 ..."
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Cited by 99 (5 self)
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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.
Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos
, 2002
"... We present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations suject to uncertain inputs. In particular, we focus on the solution of the Poisson equation with random diffusivity, forcing and boundary conditions. The stochastic input and so ..."
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Cited by 91 (16 self)
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We present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations suject to uncertain inputs. In particular, we focus on the solution of the Poisson equation with random diffusivity, forcing and boundary conditions. The stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the Askey scheme, as a generalization of the original polynomial chaos idea of Wiener (1938). A Galerkin projection in random space is applied to derive the equations in the weak form. The resulting set of deterministic equations for each random mode is solved iteratively by a block GaussSeidel iteration technique. Both discrete and continuous random distributions are considered, and convergence is verified in model problems and against Monte Carlo simulations.
An adaptive multielement generalized polynomial chaos method for stochastic differential equations
 J. COMPUT. PHYS
, 2005
"... We formulate a MultiElement generalized Polynomial Chaos (MEgPC) method to deal with longterm integration and discontinuities in stochastic differential equations. We first present this method for Legendrechaos corresponding to uniform random inputs, and subsequently we generalize it to other ra ..."
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Cited by 79 (11 self)
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We formulate a MultiElement generalized Polynomial Chaos (MEgPC) method to deal with longterm integration and discontinuities in stochastic differential equations. We first present this method for Legendrechaos corresponding to uniform random inputs, and subsequently we generalize it to other random inputs. The main idea of MEgPC is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value. In each subdomain or random element, we then employ a generalized Polynomial Chaos expansion. We develop a criterion to perform such a decomposition adaptively, and demonstrate its effectiveness for ODEs, including the KraichnanOrszag threemode problem, as well as advectiondiffusion problems. The new method is similar to spectral element method for deterministic problems but with hp discretization of the random space.
An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data
, 2007
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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 ..."
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Cited by 65 (2 self)
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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 ..."
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Cited by 63 (4 self)
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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