Abstract
AbstractWe 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.
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,General Engineering,Theoretical Computer Science,Software,Applied Mathematics,Computational Mathematics,Numerical Analysis