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92
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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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.
An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data
, 2007
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On Elliptic Partial Differential Equations with Random Coefficients
, 2011
"... The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distrib ..."
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Cited by 71 (2 self)
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The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors.
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
Blockdiagonal preconditioning for spectral stochastic finite element systems
, 2007
"... Deterministic models of fluid flow and the transport of chemicals in flows in heterogeneous porous media incorporate PDEs whose material parameters are assumed to be known exactly. To tackle more realistic stochastic flow problems, it is fitting to represent the permeability coefficients as random ..."
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Cited by 33 (9 self)
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Deterministic models of fluid flow and the transport of chemicals in flows in heterogeneous porous media incorporate PDEs whose material parameters are assumed to be known exactly. To tackle more realistic stochastic flow problems, it is fitting to represent the permeability coefficients as random fields with prescribed statistics. Traditionally, large numbers of deterministic problems are solved in a Monte Carlo framework and the solutions averaged to obtain statistical properties of the solution variables. Alternatively, socalled stochastic finite element methods (SFEMs) discretise the probabilistic dimension of the PDE directly leading to a single structured linear system. The latter approach is becoming extremely popular but its computational cost is still perceived to be problematic as this system is orders of magnitude larger than for the corresponding deterministic problem. A simple blockdiagonal preconditioning strategy, incorporating only the mean component of the random field coefficient and based on incomplete factorisations has been employed in the literature, and observed to be robust, for problems of moderate variance, but without theoretical analysis. We solve the stochastic Darcy flow problem in primal formulation via the spectral SFEM and focus on its efficient iterative solution. To achieve optimal computational complexity, we base our blockdiagonal preconditioner on algebraic multigrid. In addition, we provide new theoretical eigenvalue bounds for the preconditioned system matrix and highlight the dependence of the iteration counts on all the SFEM parameters.