【作者】 荆燕飞；

【导师】 黄廷祝；

【作者基本信息】 电子科技大学 ， 应用数学， 2010， 博士

【摘要】 本文研究内容涉及数值计算方法中的两个方面.其一,主要发展了求解线性方程组的新的子空间投影法及其在计算电磁学、计算化学等科学与工程计算领域中的应用研究.其二,以开发基于物理问题的高效预条件子为出发点,对弱奇异积分处理技术及应用也做了研究.本文包括新算法的设计和算法的应用研究.算法设计方面主要创新成果包括:提出了一种新的求解对称正定线性方程组的算法–“二维双连续投影法(2D-DSPM)”,建立了带位移线性系统重开始的加权位移全正交法(RWS-FOM),开发了一类基于Lanczos双共轭A-标准正交过程的Krylov子空间投影法(BiCOR/CORS/BiCORSTAB).而推广应用方面的主要贡献包括:推广应用Lanczos双共轭A-标准正交法求解在计算电磁学和计算化学模型问题中产生的线性方程组以及将最新提出的处理弱奇异积分的机械求积技术推广应用于无限长金属柱体的散射计算问题.具体来说,本文研究内容组织为方法篇和应用篇,其中,下面前三部分归于方法研究篇,后三部分属于应用研究篇,具体内容如下:2D-DSPM方法.首先从科学计算领域广泛使用的投影技术角度重新解析了学者Ujevic′于2006年提出的一种新颖的数值迭代算法.然后结合投影技术和线性子空间理论,给出了一种新的求解对称正定线性方程组的算法–“二维双连续投影法(2D-DSPM)”.我们从理论上深入分析探讨了2D-DSPM方法较Ujevic′迭代算法的优越性,并且通过数值实验表明, 2D-DSPM方法在大部分实际问题求解时极其有效,收敛速度较Ujevic′方法快1～2倍.这方面研究工作可以说为定常迭代法和非定常迭代法之间搭建了一个桥梁,使我们进一步更加深刻地认识到了这两类方法之间的关系.目前为止,此研究工作已得到进一步推广和应用,例如3D-OPM、1V-DSMR和mD-DSPM方法的产生以及2D-DSPM方法在矩阵方程和边界层问题方面的推广应用.RWS-FOM方法.基于Essai于1998年给出的Arnoldi加权过程,提出了一种重开始的加权位移全正交法(RWS-FOM)用于求解带位移线性系统,并且对于该算法中涉及的权重选择策略进行了深入的探讨和研究.数值实验表明了RWS-FOM方法比重开始的位移全正交法(RS-FOM)在重开始次数方面所具有的加速性能.而且,RWS-FOM方法在一定程度上可以求解使用RS-FOM方法不收敛的带位移线性系统.BiCOR/CORS/BiCORSTAB方法.这部分研究内容是对用于求解复非Hermitian线性方程组的Krylov子空间方法作出的进一步发展与创新.在Sogabe和Zhang的COCR方法基础上,首先给出了一种Lanczos双共轭A-标准正交过程,然后在此过程基础上,提出了三种Krylov子空间投影法.前两种方法数值上改进和推广了Sogabe的相关方法.第三种方法是一个新算法,并且,第三种算法在某些情形下比前两种方法数值上更加稳定,收敛行为更加平滑.在大量数值比较实验的基础上,发现所提出的这三种算法无论从残差平滑性上,还是收敛速度上,都比经典的复双共轭梯度法(CBiCG)及其两个衍生方法(CGS, BiCGSTAB)以及Sogabe所提出的相关方法表现出色.特别是共轭A-正交残量平方法(CORS),某些情形下就收敛速度方面甚至可以与BiCGSTAB方法相媲美.目前为止,此研究工作已推广应用于计算电磁学、计算化学等科学与工程计算领域,例如应用Lanczos双共轭A-标准正交法求解通过矩量法(MoM)离散Maxwell方程组所得的稠密复非Hermitian线性系统以及求解由NERSC的Sherry Li提出的计算化学模型问题中产生的两个复非对称线性方程组的问题.Maxwell方程组和计算化学模型问题中迭代法应用研究.这两部分分别将我们提出的Lanczos双共轭A-标准正交法推广应用于求解在计算电磁学和计算化学模型问题中产生的大型线性方程组.其中,第一部分是对通过矩量法(MoM)离散Maxwell方程组所得的稠密复非Hermitian线性系统的迭代法应用研究.通过在一组表征计算实际目标雷达散射截面(RCS)的模型问题上进行的数值实验,展示了这类方法较其他流行的Krylov子空间方法所具有的竞争优势,特别是在存储量有一定限制要求的情况下.结果有助于评价Krylov子空间迭代法用于求解大尺寸电磁散射问题的潜力,同时也充实了该领域迭代法的求解技术.第二部分研究运用迭代法结合简单对角预条件技术求解由NERSC的Sherry Li提出的计算化学模型问题中产生的两个复非对称线性方程组的问题,进一步反映了BiCOR/CORS/BiCORSTAB方法与其他常用的迭代法相比所具有的竞争性.同时指出基于物理问题的特定预条件子对于Krylov子空间方法的成功运用起着关键性的作用.计算电磁学中弱奇异积分处理技术应用研究.最后一部分将最新提出的处理弱奇异积分的机械求积技术进行修正后应用于计算电磁学中的散射计算问题.其核心思想是高效计算阻抗矩阵,从而得到高精度的表面电流分布及雷达散射截面(RCS).通过计算无限长金属圆柱和椭圆柱的相关数据,所修正方法和计算电磁学界的经典矩量法相比在计算结果精度和求解速度方面具有明显的优越性.而且,所修正的机械求积方法对于电大问题也只需少量点就可以得到较好的表面电流分布及RCS.此工作将数学界的最新理论成果应用于电磁计算研究.另一方面,在应用的过程中又会产生新的数学问题,比如误差分析等,为今后的研究探明方向.同时,这方面的研究工作有助于深入理解阻抗矩阵的形成过程,为构造基于物理问题的高效预条件子奠定了坚实的算法理论基础.另外,采用迭代法求解此研究工作中产生的系数矩阵为复满阵的线性代数方程组的数值实验再一次验证了我们提出的Lanczos双共轭A-标准正交法所具有的有效性和实用性.更多还原

【Abstract】 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.更多还原