Trace-Penalty Minimization for Large-scale Eigenspace Computation
Zaiwen Wen, Chao Yang, Xin Liu, Yin Zhang⋆ · 2013
The Rayleigh-Ritz (RR) procedure, including orthogonalization, constitutes a major bottleneck in computing relatively high-dimensional eigenspaces of large sparse matrices.Although operations involved 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 matrixmatrix 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.