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arXiv · 2209.14740

The Helmholtz equation with uncertainties in the wavenumber

Roland Pulch·Olivier Sète
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Abstract

We investigate the Helmholtz equation with suitable boundary conditions and uncertainties in the wavenumber. Thus the wavenumber is modeled as a random variable or a random field. We discretize the Helmholtz equation using finite differences in space, which leads to a linear system of algebraic equations including random variables. A stochastic Galerkin method yields a deterministic linear system of algebraic equations. This linear system is high-dimensional, sparse and complex symmetric but, in general, not hermitian. We therefore solve this system iteratively with GMRES and propose two preconditioners: a complex shifted Laplace preconditioner and a mean value preconditioner. Both preconditioners reduce the number of iteration steps as well as the computation time in our numerical experiments.

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Roland Pulch, Olivier Sète. 2022-09-29. The Helmholtz equation with uncertainties in the wavenumber. https://doi.org/10.1007/s10915-024-02450-3

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