Study on fast and robust iterative methods for solving large and sparse shifted linear systems arising from computational science

我们开发了一些有效的迭代方法来解决来自计算科学的大型稀疏移位线性系统。在此基础上，提出了求解复杂对称情况的位移COCR方法，以及求解（复杂）非对称情况的位移GMRES方法的变体。值得注意的是，对于大规模电子结构计算问题，位移COCR方法比COCR方法快约26倍。在相关工作中，得到了以下结果：(1)对求解复杂对称线性系统的COCR方法进行了改进；(2) IDR方法的改进；(3)求解非对称线性系统的GMRES(m)方法的改进；(4)具有特殊矩阵的线性系统的快速求解器；(5)具有复对称矩阵的广义位移线性系统的有效迭代方法。

We developed some efficient iterative methods for solving large and sparse shifted linear systems that arise from computational science. As a result, the shifted COCR method was proposed for solving complex symmetric case, and a variant of the shifted GMRES method was proposed for (complex) nonsymmetric case. Remarkably, the shifted COCR method was about 26 times faster than the COCR method for a problem arising from large scale electronic structure calculation. As related work, the following results are obtained: (1) An improvement of the COCR method for solving complex symmetric linear systems; (2) Improvements of the IDR method; (3) An improvement of the GMRES(m) method for solving nonsymmetric linear systems; (4) a fast solver for linear systems with a special matrix; (5) Efficient iterative method for generalized shifted linear systems with complex symmetric matrices.