Subspace Methods with Local Refinements for Eigenvalue Computation Using Low-Rank Tensor-Train Format
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Computing a few eigenpairs from large-scale symmetric eigenvalue problems is far beyond the tractability of classic eigensolvers when the storage of the eigenvectors in the classical way is impossible. We consider a tractable case in which both the coefficient matrix and its eigenvectors can be represented in the low-rank tensor train formats. We propose a subspace optimization method combined with some suitable truncation steps to the given low-rank Tensor Train formats. Its performance can be further improved if the alternating minimization method is used to refine the intermediate solutions locally. Preliminary numerical experiments show that our algorithm is competitive to the state-of-the-art methods on problems arising from the discretization of the stationary Schrödinger equation.
KeywordsHigh-dimensional eigenvalue problem Tensor-train format Alternating least square method Subspace optimization method
We thank D. Kressner, M. Steinlechner and A. Uschmajew for sharing online their matlab codes on EVAMEn and the TT/MPS tensor toolbox TTeMPS. The authors would like to thank the associate editor Prof. Wotao Yin and two anonymous referees for their detailed and valuable comments and suggestions.
- 4.Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31, 2029–2054 (2009/10)Google Scholar
- 6.Hackbusch, W.: Entwicklungen nach exponentialsummen, Technical Report 4, Max Planck Institute for Mathematics in the Sciences, 2005, MPI MIS Leipzig, Revised version (2010)Google Scholar
- 13.Kressner, D., Tobler, C.: htucker, a MATLAB toolbox for tensors in hierarchical Tucker format, Technical Report (2012)Google Scholar
- 20.Wen, Z., Zhang, Y.: Block algorithms with augmented rayleigh-ritz projections for large-scale eigenpair computation, arXiv:1507.06078 (2015)