Restarting Conjugate Gradient Methods for Nonsymmetric Linear Equations

对于大型非对称线性方程组，重新启动是降低共轭梯度法存储和正交化成本的一种方法。但是，重新启动会减慢收敛速度。在共轭梯度法的Arnoldi和GMRES版本中，可以求解相关的特征值问题。本研究利用特征值问题保留了重新启动过程中丢失的部分信息，从而提高了线性方程求解方法的收敛性。这是通过向用于线性方程问题的子空间中添加一些近似特征向量来完成的。初步实验表明，该方法是非常有效的。需要测试和理论研究来确定该方法在不同类型的矩阵上的工作效果。还需要更多的研究来确定最佳实施方案。

Restarting is one way to reduce storage and orthogonalization costs of conjugate gradient methods for large systems of nonsymmetric linear equations. However, restarting slows down the convergence. In the Arnoldi and GMRES versions of the conjugate gradient method, the associated eigenvalue problem can be solved. This investigation uses the eigenvalue problem to retain some of the information lost in the restart and thus improve the convergence of the linear equations solution method. This is done by adding a few approximate eigenvectors to the subspace that is used for the linear equations problem. Some preliminary experimentation show that the method can be very effective. Testing and theoretical research are needed to determine how well the method works with different types of matrices. More research is also needed to determine the best implementation.