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Convergence Theory

  • James LottesEmail author
Chapter
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Part of the Springer Theses book series (Springer Theses)

Abstract

This chapter presents a convergence theory for a fairly general two-level algorithm applied to a nonsymmetric operator with positive real part. The final convergence bounds are separated into a “smoothing property” and two separate “approximation properties”, one for the prolongation and one for the restriction, a novel feature of the theory, and one that depends critically on the absolute value presented in the previous chapter. Also shown is how, in the symmetric case, the theory reduces to reproduce and in some cases slightly generalize existing results from the literature.

Keywords

Symmetric Case Positive Real Part Ideal Weight Hierarchical Decomposition Prolongation Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Eijkhout, V., Vassilevski, P.: The role of the strengthened Cauchy-Buniakowskii-Schwarz inequality in multilevel methods. SIAM Rev. 33(3), 405–419 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Falgout, R.D., Vassilevski, P.S., Zikatanov, L.T.: On two-grid convergence estimates. Numer. Lin. Algebr. Appl. 12(5–6), 471–494 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Griebel, M.: Multilevel algorithms considered as iterative methods on semidefinite systems. SIAM J. Sci. Comput. 15(3), 547–565 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Heidelberg (1966)CrossRefzbMATHGoogle Scholar
  5. 5.
    Notay, Y.: Algebraic analysis of two-grid methods: the nonsymmetric case. Numer. Lin. Algebr. Appl. 17, 73–96 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Google Inc.Mountain ViewUSA

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