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

我们发展了一些有效的迭代方法来求解计算科学中产生的大型稀疏移位线性方程组。在此基础上，提出了求解复杂对称情形的平移COCR方法，以及求解(复杂)非对称情形的平移GMRES方法的变种。值得注意的是，对于大规模电子结构计算中出现的问题，移位COCR方法比COCR方法快约26倍。作为相关工作，得到了如下结果：(1)求解复杂对称线性方程组的COCR方法的改进；(2)IDR方法的改进；(3)非对称线性方程组的GMRES(M)方法的改进；(4)具有特殊矩阵的线性方程组的快速求解器；(5)具有复对称矩阵的广义移位线性方程组的高效迭代方法。

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.