A Fast Hierarchically Preconditioned Eigensolver Based on Multiresolution Matrix Decomposition

In this paper we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse symmetric positive matrix under the multiresolution operator compression framework. We exploit the well-conditioned property of every decomposition component by integrating the multiresolution framework into the implicitly restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned. Theoretical analysis and numerical illustration are also reported to illustrate the efficiency and effectiveness of this algorithm. ; © 2019 Society for Industrial and Applied Mathematics. Received by the editors April 16, 2018; accepted for publication (in revised form) December 3, 2018; published electronically January 30, 2019. This research was supported in part by the NSF grants DMS-1318377 and DMS-1613861. ; Published - 18m1180827.pdf Submitted - 1804.03415.pdf