Eigenvalues of the Hermitian Wilson-Dirac operator are of special interest in several lattice QCD simulations, e.g., for noise reduction when evaluating all-to-all propagators. In this paper we present a Davidson-type eigensolver that utilizes the structural properties of the Hermitian Wilson-Dirac operator $Q$ to compute eigenpairs of this operator corresponding to small eigenvalues. The main idea is to exploit a synergy between the (outer) eigensolver and its (inner) iterative scheme which solves shifted linear systems. This is achieved by adapting the multigrid DD-$α$AMG algorithm to a solver for shifted systems involving the Hermitian Wilson-Dirac operator. We demonstrate that updating the coarse grid operator using eigenvector information obtained in the course of the generalized Davidson method is crucial to achieve good performance when calculating many eigenpairs, as our study of the local coherence shows. We compare our method with the commonly used software-packages PARPACK and PRIMME in numerical tests, where we are able to achieve significant improvements, with speed-ups of up to one order of magnitude and a near-linear scaling with respect to the number of eigenvalues. For illustration we compare the distribution of the small eigenvalues of $Q$ on a $64\times 32^3$ lattice with what is predicted by the Banks-Casher relation in the infinite volume limit.
A first look at Structured-Multiscale Algebraic Multigrid for Lattice Field Theory
Fraunhofer Institute for Algorithms and Scientific Computing SCAI（弗劳恩霍夫算法与科学计算研究所）；Institute for Numerical Simulation, Rheinische Friedrich-Wilhelms-Universität（莱茵弗里德里希·威廉斯大学数值模拟研究所）；Jülich Supercomputing Centre, Forschungszentrum Jülich（于利希超级计算中心，于利希研究中心）；Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn（波恩莱茵弗里德里希·威廉斯大学亥姆霍兹辐射与核物理研究所）
