Constructive Approximation

, Volume 44, Issue 1, pp 121–139 | Cite as

Schwarz Iterative Methods: Infinite Space Splittings

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Abstract

We prove the convergence of greedy and randomized versions of Schwarz iterative methods for solving linear elliptic variational problems based on infinite space splittings of a Hilbert space. For the greedy case, we show a squared error decay rate of \(O((m+1)^{-1})\) for elements of an approximation space \(\mathscr {A}_1\) related to the underlying splitting. For the randomized case, we show an expected squared error decay rate of \(O((m+1)^{-1})\) on a class \(\mathscr {A}_{\infty }^{\pi }\subset \mathscr {A}_1\) depending on the probability distribution.

Keywords

Infinite space splitting Subspace correction Multiplicative Schwarz Block coordinate descent Greedy  Randomized Convergence rates 

Mathematics Subject Classification

65F10 65F08 65N22 65H10 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute for Numerical SimulationUniversität BonnBonnGermany
  2. 2.Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)Schloss BirlinghovenSankt AugustinGermany
  3. 3.Jacobs University BremenBremenGermany

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