# Trace-Penalty Minimization for Large-Scale Eigenspace Computation

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## Abstract

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.

## Keywords

Eigenvalue computation Exact quadratic penalty approach Gradient methods## Mathematics Subject Classification

15A18 65F15 65K05 90C06## Notes

### Acknowledgments

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.

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