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111
Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise
, 2009
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Preconditioning stochastic Galerkin saddle point systems
 SIAM J. MATRIX ANAL. APPL
, 2009
"... 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 ..."
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Cited by 110 (4 self)
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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.
Deterministic Multilevel Algorithms for INFINITEDIMENSIONAL INTEGRATION ON R^N
 PREPRINT 40, DFGSPP 1324
, 2010
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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 97 (3 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.
Fast HighDimensional Approximation with Sparse Occupancy Trees
, 2010
"... Abstract This paper is concerned with scattered data approximation in high dimensions: Given a data set X ⊂ R d of N data points x i along with values y i ∈ R d , i = 1, . . . , N , and viewing the y i as values y i = f (x i ) of some unknown function f , we wish to return for any query point x ∈ R ..."
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Cited by 94 (9 self)
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Abstract This paper is concerned with scattered data approximation in high dimensions: Given a data set X ⊂ R d of N data points x i along with values y i ∈ R d , i = 1, . . . , N , and viewing the y i as values y i = f (x i ) of some unknown function f , we wish to return for any query point x ∈ R d an approximationf (x) to y = f (x). Here the spatial dimension d should be thought of as large. We wish to emphasize that we do not seek a representation off in terms of a fixed set of trial functions but definef through recovery schemes which, in the first place, are designed to be fast and to deal efficiently with large data sets. For this purpose we propose new methods based on what we call sparse occupancy trees and piecewise linear schemes based on simplex subdivisions.