Fast Iterative Techniques For Electromagnetic Scattering Analysis

In this work, we consider the iterative solution of large dense complex linear systems that arise from integral equations in electromagnetic scattering applications. The goal of this study is the development of efficient acceleration strategies and robust preconditioning techniques for the Krylov subspace iterative methods.We first adapt to the dense situation the preconditioners initially developed for sparse linear systems. We compare their respective numerical scalabilities and propose an efficient perturbation technique for constructing robust incomplete factorization preconditoners. We also develop a robust sparse approximate inverse preconditioner in the fast multipole method framework.We then investigate several acceleration strategies based on generalized minimal residual method and compare their numerical behaviours respectively. This inspires us to improve the performance of an augmentation strategy with more accurate eigenvector information, resulting in the enhanced generalized minimal residual method. The eigenvector information can be evaluated by an eigensolver in a preprocessing phase or by a deflated generalized minimal residual method at run time.The performance of most existing preconditioners suffers heavily from the lack of global coupling in the sparsified matrix on which they have been constructed. To tack this problem we propose a multi-step hybrid preconditioning technique. It uses a spectral preconditioner in a multi-step manner to continuously update an existing preconditioner so as to recover the global information.Inherited from the basic idea from the multigrid methods that are fully defined by the choice of the smoother and the selection of the coarse space, we develop a robust algebraic multigrid method. We use a preconditioned Krylov iterative method to implement the smoother and the coarse space is defined by the span of the eigenvectors associated with the smallest eigenvalues close to the origin.Finally, we develop two spectral block iterative methods for the solution of a set of systems involving the same coefficient matrix but different right-hand sides. Both of these methods exploit spectral information of the coefficient system to improve the performance of traditional block GMRES iterative method.