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Termination criteria for inexact fixed-point schemes

Birken, Philipp LU (2015) In Numerical Linear Algebra with Applications 22(4). p.702-716
Abstract
We analyze inexact fixed-point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed-point iteration. Important applications are the Picard iteration and partitioned fluid-structure interaction. For the analysis, the iteration is modeled as a perturbed fixed-point iteration, and existing analysis is extended to the nested case x=F(S(x)). We prove that if the iteration converges, it converges to the exact solution irrespective of the tolerance in the inner systems, provided that a nonstandard relative termination criterion is employed, whereas standard relative and absolute criteria do not have... (More)
We analyze inexact fixed-point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed-point iteration. Important applications are the Picard iteration and partitioned fluid-structure interaction. For the analysis, the iteration is modeled as a perturbed fixed-point iteration, and existing analysis is extended to the nested case x=F(S(x)). We prove that if the iteration converges, it converges to the exact solution irrespective of the tolerance in the inner systems, provided that a nonstandard relative termination criterion is employed, whereas standard relative and absolute criteria do not have this property.

Numerical results demonstrate the effectiveness of the approach with the nonstandard termination criterion. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
fixed-point iteration, Picard iteration, transmission problem, Dirichlet–Neumann iteration, termination criteria
in
Numerical Linear Algebra with Applications
volume
22
issue
4
pages
702 - 716
publisher
Wiley-Blackwell
external identifiers
  • wos:000357833700007
  • scopus:84935757636
ISSN
1070-5325
DOI
10.1002/nla.1982
language
English
LU publication?
yes
id
f26bdfeb-5fa8-4c67-aaf2-20ed9198902c (old id 5277077)
date added to LUP
2015-04-27 10:11:59
date last changed
2017-10-22 03:21:21
@article{f26bdfeb-5fa8-4c67-aaf2-20ed9198902c,
  abstract     = {We analyze inexact fixed-point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed-point iteration. Important applications are the Picard iteration and partitioned fluid-structure interaction. For the analysis, the iteration is modeled as a perturbed fixed-point iteration, and existing analysis is extended to the nested case x=F(S(x)). We prove that if the iteration converges, it converges to the exact solution irrespective of the tolerance in the inner systems, provided that a nonstandard relative termination criterion is employed, whereas standard relative and absolute criteria do not have this property.<br/><br>
Numerical results demonstrate the effectiveness of the approach with the nonstandard termination criterion.},
  author       = {Birken, Philipp},
  issn         = {1070-5325},
  keyword      = {fixed-point iteration,Picard iteration,transmission problem,Dirichlet–Neumann iteration,termination criteria},
  language     = {eng},
  number       = {4},
  pages        = {702--716},
  publisher    = {Wiley-Blackwell},
  series       = {Numerical Linear Algebra with Applications},
  title        = {Termination criteria for inexact fixed-point schemes},
  url          = {http://dx.doi.org/10.1002/nla.1982},
  volume       = {22},
  year         = {2015},
}