A Fast Hierarchically Preconditioned Eigensolver Based on Multiresolution Matrix Decomposition

Abstract

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 components 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. (K) ex and D (K) ex to obtain V (K−1) ini and D (K−1) ini, which will then be the initial spectrum in the consecutive finer level. The efficiency of the cross-level refinement is achieved by a modified version of the orthogonal iteration with the Ritz Acceleration, where we exploit the proximity of the eigenspace across levels to accelerate the Conjugate gradient (CG) method within the refinement step. Using this refined initial spectrum, our second stage is to extend spectrum up to some prescribed control of the well-posedness using the Implicit Restarted Lanczos architecture. Recall that a shifting approach is introduced to reduce 2.2. Operator Compression. The procedures of compressing the solver A −1 with broad-banded spectrum are: (i) construct a partition of the computational basis using local information of A; (ii) construct the

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