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Adaptivity and Computational Complexity in the Numerical Solution of ODEs

Ilie, Silvana LU ; Söderlind, Gustaf LU and Corless, Robert M. (2008) In Journal of Complexity 24(3). p.341-361
Abstract
In this paper we analyze the problem of adaptivity for numerical methods for solving ODEs, both IVPs and BVPs, with a view to generating optimal grids for local error control. The grids are generated by introducing an auxiliary independent variable au and finding a grid deformation map, t=Theta(au), that maps an equidistant grid au_j to a non-equidistant grid in the original independent variable, {t_j}. The optimal deformation Theta is determined by a variational approach. Finally, we investigate the cost of the solution procedure and compare it to the cost of using equidistant grids. We show that if the principal error function is non-constant, an adaptive method is always more efficient than a nonadaptive method.
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author
; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Information-based complexity, Adaptive step size control, Adaptive numerical methods, Ordinary differential equations, Initial value problems, Boundary value problems, Hölder mean
in
Journal of Complexity
volume
24
issue
3
pages
341 - 361
publisher
Elsevier
external identifiers
  • wos:000257629700002
  • scopus:44649201223
ISSN
0885-064X
DOI
10.1016/j.jco.2007.11.004
language
English
LU publication?
yes
additional info
The information about affiliations in this record was updated in December 2015. The record was previously connected to the following departments: Numerical Analysis (011015004)
id
1801a6e3-5b45-4995-9fd8-b2388712b43a (old id 633992)
alternative location
http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WHX-4RNR6WK-1-1&_cdi=6862&_user=745831&_orig=search&_coverDate=06%2F30%2F2008&_sk=999759996&view=c&wchp=dGLbVtb-zSkzV&md5=a9644e63811b786b931ede8214e2a517&ie=/sdarticle.pdf
date added to LUP
2016-04-01 12:06:37
date last changed
2024-10-08 22:03:30
@article{1801a6e3-5b45-4995-9fd8-b2388712b43a,
  abstract     = {{In this paper we analyze the problem of adaptivity for numerical methods for solving ODEs, both IVPs and BVPs, with a view to generating optimal grids for local error control. The grids are generated by introducing an auxiliary independent variable au and finding a grid deformation map, t=Theta(au), that maps an equidistant grid au_j to a non-equidistant grid in the original independent variable, {t_j}. The optimal deformation Theta is determined by a variational approach. Finally, we investigate the cost of the solution procedure and compare it to the cost of using equidistant grids. We show that if the principal error function is non-constant, an adaptive method is always more efficient than a nonadaptive method.}},
  author       = {{Ilie, Silvana and Söderlind, Gustaf and Corless, Robert M.}},
  issn         = {{0885-064X}},
  keywords     = {{Information-based complexity; Adaptive step size control; Adaptive numerical methods; Ordinary differential equations; Initial value problems; Boundary value problems; Hölder mean}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{341--361}},
  publisher    = {{Elsevier}},
  series       = {{Journal of Complexity}},
  title        = {{Adaptivity and Computational Complexity in the Numerical Solution of ODEs}},
  url          = {{http://dx.doi.org/10.1016/j.jco.2007.11.004}},
  doi          = {{10.1016/j.jco.2007.11.004}},
  volume       = {{24}},
  year         = {{2008}},
}