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Convergence Proofs for Numerical IVP Software

Lamba, Harbir and Stuart, Andrew (2000) Convergence Proofs for Numerical IVP Software. In: Dynamics of Algorithms. IMA Volumes in Mathematics and its Applications. No.118. Springer , New York, NY, pp. 107-125. ISBN 9781461270737. https://resolver.caltech.edu/CaltechAUTHORS:20170613-142949630

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Abstract

The study of the running times of algorithms in computer science can be broken down into two broad types: worst-case and average-case analyses. For many problems this distinction is very important as the orders of magnitude (in terms of some measure of the problem size) of the running times may differ significantly in each case, providing useful information about the merits of the algorithm. Historically average-case analyses were first done with respect to a measure on the input data; to counter the argument that it is often difficult to find a natural measure on the data, randomised algorithms were then developed. In this paper similar questions are studied for adaptive software used to integrate initial value problems for ODEs. In worst case these algorithms may fail completely giving O (1) errors. We consider the probability of failure for generic vector fields with random initial data chosen from a ball and perform average-case and worst-case analyses.We then perform a different average-case analysis where, having fixed the initial data, it is the algorithm that is chosen at random from some suitable class.This last analysis suggests a modified deterministic algorithm which cannot fail for generic vector fields.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://dx.doi.org/10.1007/978-1-4612-1274-4_6DOIArticle
https://link.springer.com/chapter/10.1007%2F978-1-4612-1274-4_6PublisherArticle
Additional Information:© 2000 Springer Science+Business Media New York. Supported by NSF Grant DMS-95-04879.
Funders:
Funding AgencyGrant Number
NSFDMS-95-04879
Subject Keywords:Error control; adaptivity; random algorithms; convergence; tolerance proportionality
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartC7
Series Name:IMA Volumes in Mathematics and its Applications
Issue or Number:118
Classification Code:AMS(MOS) subject classifications: 34C35; 65L07; 65L20; 65L50
Record Number:CaltechAUTHORS:20170613-142949630
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170613-142949630
Official Citation:Lamba H., Stuart A. (2000) Convergence Proofs for Numerical IVP Software. In: de la Llave R., Petzold L.R., Lorenz J. (eds) Dynamics of Algorithms. The IMA Volumes in Mathematics and its Applications, vol 118. Springer, New York, NY
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78174
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:13 Jun 2017 21:40
Last Modified:03 Oct 2019 18:06

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