Adaptive Projection Subspace Dimension for the Thick-Restart Lanczos Method

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The Thick-Restart Lanczos (TRLan) method is an effective method for solving large-scale Hermitian eigenvalue problems. However, its performance strongly depends on the dimension of the projection subspace. In this paper, we propose an objective function to quantify the effectiveness of a chosen subspace dimension, and then introduce an adaptive scheme to dynamically adjust the dimension at each restart. An open-source software package, nu-TRLan, which implements the TRLan method with this adaptive projection subspace dimension is available in the public domain. The numerical results of synthetic eigenvalue problems are presented to demonstrate that nu-TRLan achieves speedups of between 0.9 and 5.1 ... continued below

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Yamazaki, Ichitaro; Bai, Zhaojun; Simon, Horst; Wang, Lin-Wang & Wu, K. October 1, 2008.

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The Thick-Restart Lanczos (TRLan) method is an effective method for solving large-scale Hermitian eigenvalue problems. However, its performance strongly depends on the dimension of the projection subspace. In this paper, we propose an objective function to quantify the effectiveness of a chosen subspace dimension, and then introduce an adaptive scheme to dynamically adjust the dimension at each restart. An open-source software package, nu-TRLan, which implements the TRLan method with this adaptive projection subspace dimension is available in the public domain. The numerical results of synthetic eigenvalue problems are presented to demonstrate that nu-TRLan achieves speedups of between 0.9 and 5.1 over the static method using a default subspace dimension. To demonstrate the effectiveness of nu-TRLan in a real application, we apply it to the electronic structure calculations of quantum dots. We show that nu-TRLan can achieve speedups of greater than 1.69 over the state-of-the-art eigensolver for this application, which is based on the Conjugate Gradient method with a powerful preconditioner.

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  • Journal Name: ACM Transactions on Mathematical Software

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  • Report No.: LBNL-1059E
  • Grant Number: DE-AC02-05CH11231
  • Office of Scientific & Technical Information Report Number: 941539
  • Archival Resource Key: ark:/67531/metadc899585

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  • October 1, 2008

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  • Sept. 27, 2016, 1:39 a.m.

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  • Nov. 8, 2016, 12:12 p.m.

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Yamazaki, Ichitaro; Bai, Zhaojun; Simon, Horst; Wang, Lin-Wang & Wu, K. Adaptive Projection Subspace Dimension for the Thick-Restart Lanczos Method, article, October 1, 2008; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc899585/: accessed September 24, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.