Increasing the performance of the Jacobi-Davidson method by blocking

Melven Röhrig-Zöllner, Jonas Thies, Moritz Kreutzer, Andreas Alvermann, Andreas Pieper, Achim Basermann, Georg Hager, Gerhard Wellein, Holger Fehske

Research output: Contribution to journalArticleScientificpeer-review

21 Citations (Scopus)

Abstract

Block variants of the Jacobi-Davidson method for computing a few eigenpairs of a large sparse matrix are known to improve the robustness of the standard algorithm when it comes to computing multiple or clustered eigenvalues. In practice, however, they are typically avoided because the total number of matrix-vector operations increases. In this paper we present the implementation of a block Jacobi-Davidson solver. By detailed performance engineering and numerical experiments we demonstrate that the increase in operations is typically more than compensated by performance gains through better cache usage on modern CPUs, resulting in a method that is both more efficient and robust than its single vector counterpart. The steps to be taken to achieve a block speedup involve both kernel optimizations for sparse matrix and block vector operations, and algorithmic choices to allow using blocked operations in most parts of the computation. We discuss the aspect of avoiding synchronization in the algorithm and show by numerical experiments with our hybrid parallel implementation that a significant speedup through blocking can be achieved for a variety of matrices on up to 5 120 CPU cores as long as at least about 20 eigenpairs are sought.

Original languageEnglish
Pages (from-to)C697-C722
JournalSIAM Journal on Scientific Computing
Volume37
Issue number6
DOIs
Publication statusPublished - 2015
Externally publishedYes

Keywords

  • Block methods
  • High performance computing
  • Hybrid parallel implementation
  • Jacobi-Davidson
  • Multicore processors
  • Performance engineering
  • Sparse eigenvalue problems

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