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International Journal for Uncertainty Quantification
Block and multilevel preconditioning for stochastic Galerkin problems with lognormally distributed parameters and tensor product polynomials
Czech Technical University in Prague
The stochastic Galerkin method is a popular numerical method for solution of differential equations with randomly distributed data. We focus on isotropic elliptic problems with lognormally distributed coefficients. We study the block-diagonal preconditioning and the algebraic multilevel preconditioning based on the block splitting according to some hierarchy of approximation spaces for the stochastic part of the solution. We introduce upper bounds for the resulting condition numbers, and we derive a tool for obtaining sharp guaranteed upper bounds for the strengthened Cauchy-Bunyakowsky-Schwarz constant, which can serve as an indicator of the efficiency of some of these preconditioning methods. The presented multilevel approach yields a tool for efficient guaranteed two-sided a~posteriori estimates of algebraic errors and for adaptive algorithms as well.
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