Iterative Methods For Linear Systems And Preconditioning Techniques With Applications In Electromagnetic Scattering Computing

This thesis contributes to the development of numerical methods in two aspects. Onthe one hand, new subspace projection methods for solving systems of linear equationshave been further developed with application investigations in fields of, for instance, com-putational electromagnetics and computational chemistry. On the other hand, motivatedby recent development of efficient physics-based preconditioners, this thesis also has aninvestigation of techniques for dealing with weakly singular integrations as well as theirapplications in scattering problems.The research work herein involves designs of novel algorithms and application in-vestigations of pertinent algorithms. The main innovations with respect to algorithmdesign include: proposal of a new algorithm to solve symmetric positive definite lin-ear equations with the algorithm name as"two-dimensional double successive projectionmethod"(referred to as 2D-DSPM); establishment of the restarted weighted full orthogo-nalization method for shifted linear systems (referred to as RWS-FOM); development ofa novel class of Krylov subspace projection methods based on Lanczos Biconjugate A-Orthonormalizaion Procedure (referred to as BiCOR/CORS/BiCORSTAB). Some majorcontributions in applications comprise: extensive applications of Lanczos Biconjugate A-Orthonormalization methods for Method of Moments discretization of Maxwell's equa-tions in computational electromagnetics as well as for linear systems arising in quantummechanics in computational chemistry model problems; adaptive application of recentlyproposed mechanical quadrature methods for an efficient solution of weakly singular in-tegral equations arising in two-dimensional electromagnetic scattering problems. Specif-ically, the contents of this thesis are organized into method study parts and applicationstudy parts, among which, the following first three parts are attributed to the first kind ofparts while the latter successive three parts belong to the second kind. They are illustratedin detail as follows:2D-DSPM method. A different interpretation of Ujevic′'s new iterative method isfirst given in terms of projection techniques widely used in scientific computing. Andthen a different new approach termed as"two-dimensional double successive projectionmethod"(referred to as 2D-DSPM) is obtained with the combination of projection tech- niques and theory of linear spaces. The proposed method is both theoretically and numer-ically proven in-depth to be better than (at least the same as) Ujevic′'s. As the presentednumerical examples show, in most cases, the convergence rate is more than one and ahalf that of Ujevic′. Such research work may build up a bridge between stationary andnonstationary iterative methods, further strengthening our profound understanding of therelationships between these two kinds of methods. So far, 2D-DSPM has been furthergeneralized and employed, such as presentations of 3D-OPM, 1V-DSMR and mD-DSPMmethods and applications of 2D-DSPM in areas of matrix equation and boundary layerproblems on heterogeneous cluster systems.RWS-FOM method. According to the weighted Arnoldi process proposed by Essaiin 1998, this part presents a hybrid algorithm, termed as restarted weighted full orthogo-nalization method (referred to as RWS-FOM) for shifted linear systems, which in effect,brings together the best of the restarted shifted full orthogonalization method (referred toas RS-FOM) on the one hand and the weighted Arnoldi process on the other. The issueon the strategy for the choice of the weights is investigated detailedly. Our method in-deed may provide accelerating convergence rate with respect to the number of restarts ata little extra expense, shown by the numerical experiments. In some circumstances whereour hybrid algorithm needs less enough number of restarts to converge than the RS-FOMmethod, it can amortize the extra cost in the weighted Arnoldi process and consequentlyit can reduce the CPU computing time. Moreover, our algorithm is able to solve certainshifted systems which the RS-FOM method cannot handle sometimes.BiCOR/CORS/BiCORSTAB methods. The contents in this part are to further de-velop and innovate Krylov subspace methods for complex non-Hermitian systems of lin-ear equations. Based on the recent Conjugate A-Orthogonal Conjugate Residual (COCR)method of Sogabe and Zhang, a version of the Biconjugate A-Orthonormalizaion Proce-dure is described first. Based on this new procedure, three Lanczos-type Krylov subspaceprojection methods are explored. The first two can be respectively considered as math-ematically equivalent but numerically improved popularizing versions of the BiCR andCRS methods for complex systems presented in Sogabe's Ph.D. Thesis. And the last oneis somewhat new and is a stabilized and more smoothly converging variant of the firsttwo in some circumstances. Based on extensive numerical experiments, it is found thatthe presented algorithms can obtain smoother and, hopefully, faster convergence behav- ior in comparison with the CBiCG method as well as its two corresponding variants aswell as the counterparts of Sogabe. In particular, the Conjugate A-Orthogonal ResidualSquared method (CORS) gains a lot and sometimes may be amazingly competitive withthe BiCGSTAB method. So far, the methods developed in this work have been employedin scientific and engineering computing such as computational electromagnetics and com-putational chemistry. For instance, the Lanczos Biconjugate A-Orthonormalization meth-ods have been already used for the solution of dense complex non-Hermitian linear sys-tems arising from the Method of Moments discretization of Maxwell's equations as wellas for the solution of two complex-valued nonsymmetric systems of linear equations aris-ing from a computational chemistry model problem proposed by Sherry Li of NERSC.Comparative study of iterative solutions to linear systems arising from Maxwellequations and computational chemistry model problems. These two parts respectivelyapply the Lanczos Biconjugate A-Orthonormalization methods developed to solve large-scale linear systems arising from model problems in computational electromagnetics andcomputational chemistry. Specifically, the first part considers the Lanczos BiconjugateA-Orthonormalization methods for the solution of dense complex non-Hermitian linearsystems arising from the Method of Moments discretization of Maxwell's equations. Nu-merical experiments are reported on a set of model problems representative of realisticradar-cross section calculations to show their competitiveness with other popular Krylovsolvers, especially when memory is a concern. The results presented in this study willcontribute to assess the potential of iterative Krylov methods for solving electromag-netic scattering problems from large structures, enriching the database of this technol-ogy. While the second part is mainly focused on iterative solutions with simple diagonalpreconditioning to two complex-valued nonsymmetric systems of linear equations arisingfrom a computational chemistry model problem proposed by Sherry Li of NERSC, show-ing again the competitiveness of the Lanczos Biconjugate A-Orthonormalization meth-ods to other classical and popular iterative methods. By the way, experiment results alsoindicate that application specific preconditioners may be mandatory and required for ac-celerating convergence.Investigation of techniques for coping with weakly singular integral in computationalelectromagnetics as well as their applications. The focus of the last part is to exploreand adapt a novel mechanical quadrature method (MQM) to deal with weakly singular integral equations arising in two-dimensional electromagnetic scattering problems. Thepresented MQM method takes the essential strategy of splitting the integrands into appro-priate parts to obtain the solutions of high accuracy to two–dimensional TM–polarized in-duced currents and scattered fields of infinite two–dimensional cylinders under impressedelectric field illumination. It is numerically demonstrated that the MQM method is ob-viously superior to the classical method of moments (MoM) in both terms of accuracyand calculation speed. Moreover, the adapted MQM method can get nice surface currentdistribution and radar cross section (RCS) with a few discretization points for electri-cally large problems. This work applies the latest theoretical results in the mathematicalcommunity into research of electromagnetic computation. On the other hand, new math-ematical problems, such as error analysis, will be encountered during the applicationprocess, directing further research. Moreover, this research effort helps us to understandthe formation of the impedance matrix so as to provide a solid algorithmic and theoreticalfoundation for constructing physics-based preconditioners for linear systems. In addition,experiments on iterative solution to the generated dense complex nonsymmetric linearsystems in this study will again contribute to assess the validity and practicality of ourLanczos Biconjugate A-Orthonormalization methods.