On Taylor Model Based Integration of Odes

Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods are able to compute guaranteed error bounds for the flow of an ODE, where traditional methods provide only approximations to a solution....

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2007-01, Vol.45 (1), p.236-262
Main Authors: Neher, M., Jackson, K. R., Nedialkov, N. S.
Format: Article
Language:English
Subjects:
Arithmetic
Cauchy problem
Exact sciences and technology
Infinity
Mathematical analysis
Mathematical intervals
Mathematical vectors
Mathematics
Measure and integration
Modeling
Musical intervals
Numerical analysis
Numerical analysis. Scientific computation
Odes
Ordinary differential equations
Parallelepipeds
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Polynomials
Sciences and techniques of general use
Software
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Summary:Interval methods for verified integration of initial value problems (IVPs) for ODEs have been used for more than 40 years. For many classes of IVPs, these methods are able to compute guaranteed error bounds for the flow of an ODE, where traditional methods provide only approximations to a solution. Overestimation, however, is a potential drawback of verified methods. For some problems, the computed error bounds become overly pessimistic, or the integration even breaks down. The dependency problem and the wrapping effect are particular sources of overestimations in interval computations. Berz and his coworkers have developed Taylor model methods, which extend interval arithmetic with symbolic computations. The latter is an effective tool for reducing both the dependency problem and the wrapping effect. By construction, Taylor model methods appear particularly suitable for integrating nonlinear ODEs. We analyze Taylor model based integration of ODEs and compare Taylor model methods with traditional enclosure methods for IVPs for ODEs.
ISSN:0036-1429
1095-7170
DOI:10.1137/050638448