Multilevel solution strategies for the Stochastic Galerkin Method

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2015-08

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Abstract

When using partial differential equations (PDEs) to model physical problems, the exact values of coefficients are often unknown. To obtain more realistic models, the coefficients are modeled by a set of random variables which induce variability in the solution of the physical model. We seek numerical solutions of elliptic PDEs with random coefficients using the stochastic Galerkin method. There are two main computational challenges associated with this method that will be addressed in this work. First, the efficient solution of a large system of linear equations is required. In addition, the method suffers from the curse of dimensionality where the computational effort increases greatly as the stochastic dimension increases. It is the purpose of this dissertation to investigate and improve efficiency of the stochastic Galerkin method by addressing these two computational challenges using multilevel approaches.

A smoothed aggregation algebraic multigrid preconditioner is proposed to enhance the convergence of the iterative solution of the system of linear equations arising from stochastic Galerkin discretizations. Numerical results are provided to demonstrate the convergence properties of the proposed preconditioner.

We next present and analyze a multilevel solution strategy to alleviate some of the prohibitive computational cost of the stochastic Galerkin method. The idea is to reduce computational complexity by balancing errors across a sequence of spatial approximations. Analysis of the proposed multilevel method and numerical results are presented that compare the multilevel method to the traditional, single level stochastic Galerkin method.

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Keywords

Stochastic Galerkin Method, Preconditioning, Multigrid, Uncertainty Quantification, Polynomial Chaos, Fast Solvers, Multilevel

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