多右端的收缩Hermitian Lanczos方法
Deflated Hermitian Lanczos Methods for Multiple Right-Hand Sides
摘要：
本文提出了一种求解厄米特线性方程组的收缩重启Lanczos算法，并给出了该算法在计算特征值和特征向量时的多个右手边的应用。对于第一右手边，计算具有小特征值的特征向量，同时求解线性系统。给出了该算法的两个版本。第一种称为Lan-DR，基于Lanczos算法的共轭梯度（CG）实现。这个版本将是最佳的埃尔米特正定的情况下。第二个版本称为MinRes-DR，基于Lanczos算法的最小残差（MinRes）实现。这个版本是最佳的不定厄米特系统的CG算法是不稳定的。对于额外的右手边，我们在计算的特征向量上投影以加速收敛。用于后续右侧的算法分别称为D-CG和D-MinRes。在给出一些介绍性的例子后，我们给出了Wilson费米子在Kappa临界的情况下的测试。与未修改的CG和MinRes相比，观察到收敛速度大大加快。
英文摘要：
A deflated and restarted Lanczos algorithm to solve hermitian linear systems, and at the same time compute eigenvalues and eigenvectors for application to multiple right-hand sides, is described. For the first right-hand side, eigenvectors with small eigenvalues are computed while simultaneously solving the linear system. Two versions of this algorithm are given. The first is called Lan-DR and is based on conjugate gradient (CG) implementation of the Lanczos algorithm. This version will be optimal for the hermitian positive definite case. The second version is called MinRes-DR and is based on the minimum residual (MinRes) implementation of Lanczos algorithm. This version is optimal for indefinite hermitian systems where the CG algorithm is subject to instabilities. For additional right-hand sides, we project over the calculated eigenvectors to speed up convergence. The algorithms used for subsequent right-hand sides are called D-CG and D-MinRes respectively. After some introductory examples are given, we show tests for the case of Wilson fermions at kappa critical. A considerable speed up in the convergence is observed compared to unmodified CG and MinRes.
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