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.
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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