2022/09/29 by Roland Pulch, Olivier Sète, Pulch, Roland +1 · 1 citation
Computer Science · Decision Sciences · Physics and Astronomy · #35R60 #65C20 #65N30 #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2209.14740
openalex publication_date 2022/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.