用于格点QCD计算中多右端项和多平移的CG方法
A CG Method for Multiple Right Hand Sides and Multiple Shifts in Lattice QCD Calculations
AI总结：
本文提出一种基于块Lanczos过程的块Krylov子空间方法，同时利用平移和多个右端项，高效求解格点QCD中出现的线性系统。
摘要：
我们考虑求解仅与单位矩阵相差一个平移的线性系统以及具有多个不同右端项的线性系统的任务。过去已经开发了Krylov子空间方法，这些方法利用了对多个右端项（例如，收缩型方法和块方法）或多个平移（例如，平移CG）的需求，并取得了一定成功。在本文中，我们提出了一种基于块Lanczos过程的块Krylov子空间方法，该方法同时利用了平移和多个右端项这两个特征。例如，在理性混合蒙特卡洛算法中的格点QCD模拟中会出现这种情况。我们提供数值证据表明，我们的方法优于对每个系统单独应用其他迭代方法，并且在某些情况下也优于平移或块Krylov子空间方法。
英文摘要：
We consider the task of computing solutions of linear systems that only differ by a shift with the identity matrix as well as linear systems with several different right hand sides. In the past Krylov subspace methods have been developed which exploit either the need for solutions to multiple right hand sides (e.g. deflation type methods and block methods) or multiple shifts (e.g. shifted CG) with some success. In this paper we present a block Krylov subspace method which, based on a block Lanczos process, exploits both features - shifts and multiple right hand sides - at once. Such situations arise, for example, in lattice QCD simulations within the Rational Hybrid Monte Carlo algorithm. We give numerical evidence that our method is superior to applying other iterative methods to each of the systems individually as well as, in some cases, to shifted or block Krylov subspace methods.
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