基于电磁散射的多右端向量线性方程组的块Krylov子空间方法

Solution for large-scale systems of linear equations with multiple right-hand sides is one of the core issues in modern computational electromagnetism, and is also one of the key points in efficient parallel computing of computational electromagnetism using computers with multicore architectures. Associated with the Boundary Element Method in numerical solution of the Maxwell equations for electromagnetic scattering in the frequency domain, the project aims at studying block Krylov subspace methods and preconditioning techniques of high-performance for large linear systems with multiple right-hand sides, and to illustrating convergence and divergence principles of block subspace methods, finally to designing efficient algorithms suitable for a sequence of linear systems with several right-hand sides given simultaneously or in sequence. Firstly we have in-depth investigation of the linear correlation among the right-hand sides given simultaneously as well as the block residuals generated at each iteration step in order to effectively address the issue of singularity of the corresponding block Krylov matrix. In addition, we employ the deflating restarted technique of small eigenvalues in magnitude to accelerate the convergence speed of the developed subspace methods. Secondly, we investigate the relationships of sequences of Krylov subspaces corresponding to the right-hand sides available in sequence. We exploit spectral information of the Krylov subspaces with the previous right-hand sides to build augmented Krylov subspaces for latter right-hand sides, and then perform incremental update of the preconditioner to establish efficient preconditioners. Finally, the proposed algorithms will be investigated and applied into practical problems arising from electromagnetic scattering computing taking into account aspects of practicality and stability. And development of the corresponding software package will further enrich block Krylov subspace solution techniques in the field of electromagnetic scattering.

大规模多右端向量线性方程组求解问题是现代计算电磁学核心问题之一，是利用多核计算机系统实现计算电磁学并行高效计算的关键所在。本项目拟结合离散电磁散射频域中麦克斯韦方程组的边界元方法，进行多右端向量大型线性系统的高性能块Krylov子空间方法与预条件技术研究，阐明块子空间方法敛散原理，设计适合多右端向量同时给定和分时给定两种情形的高效算法，包括：深入考察同时给定多右端向量及迭代过程中各残量的线性相关性，有效解决块Krylov子空间矩阵的奇异问题，并采用小特征值抑制技术加速子空间法方法收敛速度；考察分时给定多右端向量产生的Krylov子空间序列关系，运用之前右端向量的子空间谱信息为后续右端向量设计增广子空间，通过更新预处理子的谱信息从而建立高效预处理子。最后，结合电磁散射计算实际问题，考察上述方法的实用性和稳定性，开发相应程序软件包，进一步充实电磁散射领域块Krylov子空间求解技术。

大规模多右端向量线性方程组求解问题是现代计算电磁学核心问题之一，是利用多核计算机系统实现计算电磁学并行高效计算的关键所在。本项目以电磁散射问题为背景，以高效求解多右端向量大型线性系统为目标，采用理论分析、算法设计与数值实验相结合的方法，旨在进行算法创新，构造高性能块Krylov子空间方法，进一步充实电磁散射领域块Krylov子空间求解技。本项目按计划顺利进行，目前已发表的SCI期刊论文6篇，均标注该课题基金资助。该课题在新型高效块Krylov子空间方法方面取得的重要进展和研究成果概述如下：.(1) 同时给定多右端向量线性系统求解方法研究：.(a). 新型高效块Krylov子空间方法—IB-BGMRES-DR方法；.(b). IB-BGMRES-DR的变型算法——alpha-IB-BGMRES-DR;.(c). 块Krylov子空间方法—DBGCROT(m, k)方法。.(2) 分时给定多右端向量线性系统求解方法：.(a). 循环块残量极小化方法——IB-BFGCRO-DR方法.(b). 多右端向量移位线性系统求解方法——BGMRES-DR-Sh方法.(3) IB-BGMRES-DR方法被法国INRIA Hi-Box project采用解决电磁学和声学的波传播问题。