Trace-Penalty Minimization for Large-Scale Eigenspace Computation
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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.
KeywordsEigenvalue computation Exact quadratic penalty approach Gradient methods
Mathematics Subject Classification15A18 65F15 65K05 90C06
The computational results were obtained at the National Energy Research Scientific Computing Center (NERSC), which is supported by the Director, Office of Advanced Scientific Computing Research of the U.S. Department of Energy under contract number DE-AC02-05CH11232. Z. Wen would like to thank Prof. Michael Ulbrich for hosting his visit at Technische Universität München. X. Liu would like to thank Prof. Yuhong Dai for discussing nonlinear programming techniques for eigenvalue computation. C. Yang would like to thank Dr. Eugene Vencharynski for helping test EigPen, especially the preconditioned version. The authors are grateful to Prof. Chi-Wang Shu, the associate editor and the anonymous referees for their detailed and valuable comments and suggestions.
- 1.Anderson, E., Bai, Z., Dongarra, J., Greenbaum, A., McKenney, A., Du Croz, J., Hammerling, S., Demmel, J., Bischof, C., Sorensen, D.: Lapack: a portable linear algebra library for high-performance computers, in Proceedings of the 1990 ACM/IEEE conference on Supercomputing, Supercomputing ’90, IEEE Computer Society Press, pp. 2–11 (1990)Google Scholar
- 9.Kronik, L., Makmal, A., Tiago, M., Alemany, M.M.G., Huang, X., Saad, Y., Chelikowsky, J.R.: PARSEC - the pseudopotential algorithm for real-space electronic structure calculations: recent advances and novel applications to nanostructures. Phys. Status Solidi. (b) 243, 1063–1079 (2006)CrossRefGoogle Scholar
- 10.Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK users’ guide: Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods, vol. 6 of software, environments, and tools, society for industrial and applied mathematics (SIAM), Philadelphia, PA, (1998)Google Scholar
- 11.Nocedal, Jorge, Wright, Stephen J.: Numerical Optimization, Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, New York (2006)Google Scholar
- 16.Sun, Wenyu, Yuan, Yaxiang: Optimization Theory and Methods: Nonlinear Programming. Springer, New York (2006)Google Scholar