A Generalized Eigensolver based on Smoothed Aggregation (GES-SA) for Initializing Smoothed Aggregation Multigrid (SA)

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Consider the linear system Ax = b, where A is a large, sparse, real, symmetric, and positive definite matrix and b is a known vector. Solving this system for unknown vector x using a smoothed aggregation multigrid (SA) algorithm requires a characterization of the algebraically smooth error, meaning error that is poorly attenuated by the algorithm's relaxation process. For relaxation processes that are typically used in practice, algebraically smooth error corresponds to the near-nullspace of A. Therefore, having a good approximation to a minimal eigenvector is useful to characterize the algebraically smooth error when forming a linear SA solver. This ... continued below

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Brezina, M; Manteuffel, T; McCormick, S; Ruge, J; Sanders, G & Vassilevski, P S May 31, 2007.

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Consider the linear system Ax = b, where A is a large, sparse, real, symmetric, and positive definite matrix and b is a known vector. Solving this system for unknown vector x using a smoothed aggregation multigrid (SA) algorithm requires a characterization of the algebraically smooth error, meaning error that is poorly attenuated by the algorithm's relaxation process. For relaxation processes that are typically used in practice, algebraically smooth error corresponds to the near-nullspace of A. Therefore, having a good approximation to a minimal eigenvector is useful to characterize the algebraically smooth error when forming a linear SA solver. This paper discusses the details of a generalized eigensolver based on smoothed aggregation (GES-SA) that is designed to produce an approximation to a minimal eigenvector of A. GES-SA might be very useful as a standalone eigensolver for applications that desire an approximate minimal eigenvector, but the primary aim here is for GES-SA to produce an initial algebraically smooth component that may be used to either create a black-box SA solver or initiate the adaptive SA ({alpha}SA) process.

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PDF-file: 23 pages; size: 2.7 Mbytes

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  • Journal Name: Numerical Linear Algebra with Applications, vol. 15, no. 2-3, January 15, 2008, pp. 249-269; Journal Volume: 15; Journal Issue: 2-3

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  • Report No.: UCRL-JRNL-231442
  • Grant Number: W-7405-ENG-48
  • Office of Scientific & Technical Information Report Number: 940855
  • Archival Resource Key: ark:/67531/metadc901621

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  • May 31, 2007

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  • Sept. 27, 2016, 1:39 a.m.

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  • Nov. 28, 2016, 6:30 p.m.

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Brezina, M; Manteuffel, T; McCormick, S; Ruge, J; Sanders, G & Vassilevski, P S. A Generalized Eigensolver based on Smoothed Aggregation (GES-SA) for Initializing Smoothed Aggregation Multigrid (SA), article, May 31, 2007; Livermore, California. (digital.library.unt.edu/ark:/67531/metadc901621/: accessed October 16, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.