Miyaji, Tomoyuki; Pilarczyk, PawelISTA; Gameiro, Marcio; Kokubu, Hiroshi; Mischaikow, Konstantin
We study the usefulness of two most prominent publicly available rigorous ODE integrators: one provided by the CAPD group (capd.ii.uj.edu.pl), the other based on the COSY Infinity project (cosyinfinity.org). Both integrators are capable of handling entire sets of initial conditions and provide tight rigorous outer enclosures of the images under a time-T map. We conduct extensive benchmark computations using the well-known Lorenz system, and compare the computation time against the final accuracy achieved. We also discuss the effect of a few technical parameters, such as the order of the numerical integration method, the value of T, and the phase space resolution. We conclude that COSY may provide more precise results due to its ability of avoiding the variable dependency problem. However, the overall cost of computations conducted using CAPD is typically lower, especially when intervals of parameters are involved. Moreover, access to COSY is limited (registration required) and the rigorous ODE integrators are not publicly available, while CAPD is an open source free software project. Therefore, we recommend the latter integrator for this kind of computations. Nevertheless, proper choice of the various integration parameters turns out to be of even greater importance than the choice of the integrator itself. © 2016 IMACS. Published by Elsevier B.V. All rights reserved.
Applied Numerical Mathematics
MG was partially supported by FAPESP grants 2013/07460-7 and 2010/00875-9, and by CNPq grants 305860/2013-5 and 306453/2009-6, Brazil. The work of HK was partially supported by Grant-in-Aid for Scientific Research (Nos.24654022, 25287029), Ministry of Education, Science, Technology, Culture and Sports, Japan. KM was supported by NSF grants NSF-DMS-0835621, 0915019, 1125174, 1248071, and contracts from AFOSR and DARPA. TM was supported by Grant-in-Aid for JSPS Fellows No. 245312. A part of the research of TM and HK was also supported by JST, CREST. Research conducted by PP has received funding from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE – Programa Operacional Factores de Competitividade (POFC) and from the Portuguese national funds through Fundação para a Ciência e a Tecnologia (FCT) in the framework of the research project FCOMP-01-0124-FEDER-010645 (Ref. FCT PTDC/MAT/098871/2008); from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement No. 622033; and from the same sources as HK. The authors express their gratitude to the Department of Mathematics of Kyoto University for making their server available for conducting the computations described in the paper, and to the reviewers for helpful comments that contributed towards increasing the quality of the paper.
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Miyaji T, Pilarczyk P, Gameiro M, Kokubu H, Mischaikow K. A study of rigorous ODE integrators for multi scale set oriented computations. Applied Numerical Mathematics. 2016;107:34-47. doi:10.1016/j.apnum.2016.04.005
Miyaji, T., Pilarczyk, P., Gameiro, M., Kokubu, H., & Mischaikow, K. (2016). A study of rigorous ODE integrators for multi scale set oriented computations. Applied Numerical Mathematics. Elsevier. https://doi.org/10.1016/j.apnum.2016.04.005
Miyaji, Tomoyuki, Pawel Pilarczyk, Marcio Gameiro, Hiroshi Kokubu, and Konstantin Mischaikow. “A Study of Rigorous ODE Integrators for Multi Scale Set Oriented Computations.” Applied Numerical Mathematics. Elsevier, 2016. https://doi.org/10.1016/j.apnum.2016.04.005.
T. Miyaji, P. Pilarczyk, M. Gameiro, H. Kokubu, and K. Mischaikow, “A study of rigorous ODE integrators for multi scale set oriented computations,” Applied Numerical Mathematics, vol. 107. Elsevier, pp. 34–47, 2016.
Miyaji T, Pilarczyk P, Gameiro M, Kokubu H, Mischaikow K. 2016. A study of rigorous ODE integrators for multi scale set oriented computations. Applied Numerical Mathematics. 107, 34–47.
Miyaji, Tomoyuki, et al. “A Study of Rigorous ODE Integrators for Multi Scale Set Oriented Computations.” Applied Numerical Mathematics, vol. 107, Elsevier, 2016, pp. 34–47, doi:10.1016/j.apnum.2016.04.005.