Abstract.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 non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.Mathematics Subject Classification. 33C45, 35R60, 40A30, 41A10, 60H35, 65N30.

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Standard multigrid algorithms have proven ine ective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's e ectiveness through theoretical analysis of a model problem and experimental results. In particular, we show that the combined use of GMRES as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as two hundred. For xed wave numbers, it displays grid-independent convergence rates and has costs proportional to number of unknowns.Abstract. Standard multigrid algorithms have proven ine ective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's e ectiveness through theoretical analysis of a model problem and experimental results. In particular, we show that the combined use of GMRES as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as two hundred. For xed wave numbers, it displays grid-independent convergence rates and has costs proportional to number of unknowns.

Abstract. We show how the Arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted Krylov subspace methods for solving linear systems of equations. The resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions. We further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments.

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We analyze the Ensemble and Polynomial Chaos Kalman filters applied to nonlinear stationary Bayesian inverse problems. In a sequential data assimilation setting such stationary problems arise in each step of either filter. We give a new interpretation of the approximations produced by these two popular filters in the Bayesian context and prove that, in the limit of large ensemble or high polynomial degree, both methods yield approximations which converge to a well-defined random variable termed the analysis random variable. We then show that this analysis variable is more closely related to a specific linear Bayes estimator than to the solution of the associated Bayesian inverse problem given by the posterior measure. This suggests limited or at least guarded use of these generalized Kalman filter methods for the purpose of uncertainty quantification.

Abstract. We investigate the structural, spectral, and sparsity properties of Stochastic Galerkin matrices as they arise in the discretization of linear differential equations with random coefficient functions. These matrices are characterized as the Galerkin representation of polynomial multiplication operators. In particular, it is shown that the global Galerkin matrix associated with complete polynomials cannot be diagonalized in the stochastically linear case.

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