大型稀疏多右端位移线性方程组的高性能算法研究

Solving the large sparse linear systems with multiple right-hand sides and multiple shifts has important engineering application values and practical significance，which is also a central task in scientific computing. Since the expending scale and the worsing ill-condition of the problem, many existing numerical methods fail to cope with this challenge. Therefore, this project will take the structure of equations and information haring as a starting point to study high performance algorithms for linear systems with multiple right-hand sides and shifts: (1) we make full use of the information sharing characteristics of the deflation idea to propose new short recurrence shift-deflated algorithm; (2) the eigenvalue deflation technique and the vector deflation technique are respectively introduce into the shifted IDR(s) method and the block IDR(s) method, and then two efficient IDR(s) variant algorithms are constructed; (3) in order to improve the computational speed, we study the high performance preconditioning techniques for (generalized) shifted linear systems with multiple right-hand sides and analysis the convergence of algorithms.

大型稀疏多右端位移线性方程组的求解在工程领域中有着极其重要的应用价值和实际意义，是目前科学计算领域中的热点问题之一。由于该类问题的规模不断扩大，其性态越来越复杂，使得求解难度不断加大，给现有算法带来了严峻的挑战。本项目结合方程组的结构特征，从信息共享的角度出发，对大型稀疏多右端位移方程组的高效算法主要展开以下三方面研究：(1) 充分利用收缩方法的信息共享特性，设计高效稳定的短递归位移-收缩迭代算法；(2) 分别将特征值收缩技巧与列向量收缩技巧有效的嵌入到位移IDR(s)算法和块IDR(s)算法中，构造两种高效IDR(s)变型算法；(3) 为了提高求解速度，系统研究针对多右端(广义)位移线性方程组的高效预处理迭代算法，并对其收敛性进行分析。

大型稀疏多右端(广义)位移线性方程组求解在工程领域中有着极其重要的应用价值。由于该类问题的规模不断扩大，其性态越来越复杂，使得求解难度不断加大，给现有算法带来了严峻的挑战。为了高效求解，课题组结合方程组的结构特征，从保位移不变性、信息共享的角度出发，提出了多右端(广义)位移线性方程组的短递归灵活预处理IDR(s)类变型算法，带有收缩特征值技巧的灵活预处理全局GCRO-DR算法与灵活预处理全局GMRES-DR算法，短递归SIDRstab算法和BI-DGMRES(m)算法等多种高效算法，同时完善相应算法的理论分析工作；研制出求解多右端(广义)位移线性方程组的实用性数值软件包。