Study Of Solutions To Large Linear Systems With Applications In Computational Electromagnetics

Computational mathematics and scientific computing are involved in many appli-cations, such as aeromechanics, modern biomedicine, oil reservoir, stealth technology. Usually, we are required to solve one or a series of systems of linear equations. With the increment of the size of the underlying problem, the size of the corresponding linear system increases dramatically. The order of the coefficient matrix often turns out to be millions or even hundreds of millions. The solution to the linear system is the key of the solution to the entire problem. Lack of efficient solutions to large-scale linear systems becomes a bottleneck in solving certain practical problems. Therefore, study of effec-tive solutions to very large linear systems turns out to be one of the focuses in modern scientific computing. This dissertation attempts to develop and construct algorithms of high performance for solutions of linear systems in depth, especially those arising from computational electromagnetics.Effective preconditioners involving incomplete factorization for symmetric positive definite matrices are developed. What we do in the common procedure of the precon-ditioned iterative methods is that we firstly reorder the original coefficient matrix, and then decompose the reordered matrix in an incomplete manner, and finally proceed with the preconditioned iterative methods. We investigate an effective preconditioning tech-nique, which interleaves the incomplete Cholesky (IC) factorization with an approximate minimum degree ordering. An IC factorization algorithm derived from the IKJ-version Gaussian elimination is proposed and some details on its implementation are presented. Then we discuss how to compute the degrees of the unnumbered nodes both exactly and approximately. When used in conjunction with the conjugate gradient algorithm, the new preconditioners usually lead to fast convergence. Numerical experiments show that the preconditioners generated by the interleaving of symbolic ordering and numerical IC fac-torization are better than those generated by the IC factorization without ordering or with purely symbolic ordering ahead of the factorization.A class of lopsided Hermitian/skew-Hermitian (LHSS) methods and a class of asym-metric Hermitian/skew-Hermitian (AHSS) methods to solve non-Hermitian and positive-definite systems of linear equations are established. Convergence properties and optimal parameter selections of these methods are studied. In each iteration of both LHSS and AHSS, two subsystems should be solved. If these subsystems are solved exactly with direct solvers, the HSS iterations are meaningless, because the cost for solving the sub-systems is equal to that for solving the original system by direct solvers. Consequently, we solve them inexactly by iterative solvers and develop ILHSS and IAHSS methods. The presented numerical methods illustrate the effectiveness of the proposed methods.Indefinite preconditioners for solving symmetric indefinite matrices are studied. These preconditioners are developed through a modified incomplete Cholesky (MIC) fac-torization with the pivoting strategy employed in the RBBK algorithm. Both the coef-ficient matrix and the preconditioner are indefinite, so the preconditioned matrix is ex-pected to have more favorable properties for iterative solution. When combined with the SQMR algorithm, this kind of preconditioners works very well according to the presented numerical examples.Preconditioning techniques for solving unsymmetric systems of linear equations are investigated. We study the incomplete orthogonal factorization based on the Givens ro-tations. An FQIGO preconditioning technique is developed, and its efficiency is demon-strated by the presented numerical tests. Then we have an investigation on how the order-ing affects the QR factorization and propose a reordering scheme to reduce the number of Givens rotations in the QR factorization.Finally, we study preconditioning techniques for the solution to the linear systems arising from electromagnetic computation. A kind of MIC preconditioners is proposed to solve the complex symmetric linear systems when FEM is used to handle scattering problems. Then block preconditioners are proposed when hybrid FEM/MoM is employed to deal with electromagnetic radiation and scattering problems.