Browsing by Author "Sifuentes, Josef"
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Item Preconditioned iterative methods for inhomogeneous acoustic scattering applications(2010) Sifuentes, Josef; Embree, MarkThis thesis develops and analyzes efficient iterative methods for solving discretizations of the Lippmann--Schwinger integral equation for inhomogeneous acoustic scattering. Analysis and numerical illustrations of the spectral properties of the scattering problem demonstrate that a significant portion of the spectrum is approximated well on coarse grids. To exploit this, I develop a novel restarted GMRES method with adaptive deflation preconditioning based on spectral approximations on multiple grids. Much of the literature in this field is based on exact deflation, which is not feasible for most practical computations. This thesis provides an analytical framework for general approximate deflation methods and suggests a way to rigorously study a host of inexactly-applied preconditioners. Approximate deflation algorithms are implemented for scattering through thin inhomogeneities in photonic band gap problems. I also develop a short term recurrence for solving the one dimensional version of the problem that exploits the observation that the integral operator is a low rank perturbation of a self-adjoint operator. This method is based on strategies for solving Schur complement problems, and provides an alternative to a recent short term recurrence algorithm for matrices with such structure that we show to be numerically unstable for this application. The restarted GMRES method with adaptive deflation preconditioning over multiple grids, as well as the short term recurrence method for operators with low rank skew-adjoint parts, are very effective for reducing both the computational time and computer memory required to solve acoustic scattering problems. Furthermore, the methods are sufficiently general to be applicable to a wide class of problems.Item Preconditioning the integral formulation of the Helmholtz equation via deflation(2006) Sifuentes, Josef; Embree, MarkIn 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.