Implicity Restarted GMRES and Arnoldi Methods for Nonsymmetric Systems of Equations

Morgan 9522612在这个项目中，将研究求解大型非对称线性方程组的重新启动GMRES方法的改进。 我们的目标是开发一种方法，这是有竞争力的短复发Lanczos方法，如QMR，但保持理想的性能GMRES。 这些性质包括从子空间中提取最小剩余解，一般稳定收敛和数值稳定性。 还期望不显著提高每次迭代的成本和存储要求。 通过在子空间中引入近似特征向量，可以提高GMRES的收敛性。 一旦这些特征值基本上从矩阵的谱中收缩，从谱中去除甚至仅仅几个小的特征值可以显著改善收敛。 在这个项目中，提出了一种新的和有效的方法，包括特征向量。 Sorensen的隐式QR Arnoldi方法的特征值相结合的内部特征值版本的Arnoldi。 令人惊讶的结果是，近似特征向量成为一个新的Krylov子空间的一部分。 这个Krylov子空间可以与GMRES一起使用。 在矩阵向量乘积很昂贵的情况下，每次迭代的开销甚至小于常规GMRES，并且所需的迭代次数有时会大大减少。 将研究由此产生的隐式重新启动GMRES方法，并与其他方法进行比较。 ***

Morgan 9522612 In this project, improvements of the restarted GMRES method for solving large nonsymmetric systems of linear equations will be investigated. The goal is to develop a method that is competitive with short recurrence Lanczos methods such as QMR, but maintains the desirable properties of GMRES. These properties include extraction of the minimum residual solution from the subspace, generally steady convergence, and numerical stability. It is also desired to not significantly raise cost per iteration and storage requirements. The convergence of GMRES can be improved by including approximate eigenvectors in the subspace. Once these eigenvalues are essentially deflated from the spectrum of the matrix, removing even just a few small eigenvalues from the spectrum can significantly improve the convergence. In this project a new and efficient way of including eigenvectors is proposed. Sorensen's implicit-QR Arnoldi method for eigenvalues is combined with an interior eigenvalue version of Arnoldi. The surprising result is that the approximate eigenvectors become part of a new Krylov subspace. This Krylov subspace can be used with GMRES. The expense per iteration is even less than that of regular GMRES for the case where matrix-vector products are expensive, and the number of iterations needed is sometimes greatly reduced. The resulting implicitly restarted GMRES method will be studied and compared with other methods. ***