Preconditioning the integral formulation of the Helmholtz equation via deflation
Master of Arts
In this thesis we propose methods for preconditioning Krylov subspace methods for solving the integral equation formulation of the Helmholtz partial differential equation for modeling scattered waves. An advantage of using an integral formulation is that only the scattering obstacle is discretized and the outgoing boundary conditions are automatically satisfied. Furthermore, convergence is dictated by the wave number kappa with only a mild dependence on the discretization. However such methods are increasingly computationally expensive for increasing values of kappa. This cost is due to GMRES iteration counts that increase like O(kappa2), for a linear system that is dense with dimension N = O(kappa 4). GMRES is slow due to a small subset of the spectrum that is well separated, a part of which approaches the origin as kappa increases. The troublesome subset corresponds to low frequency eigenfunctions which can be approximated using coarse meshes. We propose a preconditioner based on deflating this subset of the spectrum which we evaluate by interpolating coarse mesh approximations of the spectrum. We show that for discretizations of less than one node per wavelength, we can effectively precondition the full problem over a sufficiently resolved mesh.
Mathematics; Physics; Electromagnetics