Journal of Scientific Computing

, Volume 66, Issue 3, pp 1175–1203

Trace-Penalty Minimization for Large-Scale Eigenspace Computation


DOI: 10.1007/s10915-015-0061-0

Cite this article as:
Wen, Z., Yang, C., Liu, X. et al. J Sci Comput (2016) 66: 1175. doi:10.1007/s10915-015-0061-0


In a block algorithm for computing relatively high-dimensional eigenspaces of large sparse symmetric matrices, the Rayleigh-Ritz (RR) procedure often constitutes a major bottleneck. Although dense eigenvalue calculations for subproblems in RR steps can be parallelized to a certain level, their parallel scalability, which is limited by some inherent sequential steps, is lower than dense matrix-matrix multiplications. The primary motivation of this paper is to develop a methodology that reduces the use of the RR procedure in exchange for matrix-matrix multiplications. We propose an unconstrained trace-penalty minimization model and establish its equivalence to the eigenvalue problem. With a suitably chosen penalty parameter, this model possesses far fewer undesirable full-rank stationary points than the classic trace minimization model. More importantly, it enables us to deploy algorithms that makes heavy use of dense matrix-matrix multiplications. Although the proposed algorithm does not necessarily reduce the total number of arithmetic operations, it leverages highly optimized operations on modern high performance computers to achieve parallel scalability. Numerical results based on a preliminary implementation, parallelized using OpenMP, show that our approach is promising.


Eigenvalue computation Exact quadratic penalty approach Gradient methods 

Mathematics Subject Classification

15A18 65F15 65K05 90C06 

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Beijing International Center for Mathematical ResearchPeking UniversityBeijingChina
  2. 2.Computational Research DivisionLawrence Berkeley National LaboratoryBerkeleyUSA
  3. 3.State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  4. 4.Department of Computational and Applied MathematicsRice UniversityHoustonUSA