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Formally Verified Computation of Enclosures of Solutions of Ordinary Differential Equations

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NASA Formal Methods (NFM 2014)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 8430))

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Abstract

Ordinary differential equations (ODEs) are ubiquitous when modeling continuous dynamics. Classical numerical methods compute approximations of the solution, however without any guarantees on the quality of the approximation. Nevertheless, methods have been developed that are supposed to compute enclosures of the solution.

In this paper, we demonstrate that enclosures of the solution can be verified with a high level of rigor: We implement a functional algorithm that computes enclosures of solutions of ODEs in the interactive theorem prover Isabelle/HOL, where we formally verify (and have mechanically checked) the safety of the enclosures against the existing theory of ODEs in Isabelle/HOL.

Our algorithm works with dyadic rational numbers with statically fixed precision and is based on the well-known Euler method. We abstract discretization and round-off errors in the domain of affine forms. Code can be extracted from the verified algorithm and experiments indicate that the extracted code exhibits reasonable efficiency.

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Author information

Authors and Affiliations

  1. Institut für Informatik, Technische Universität München, Germany

    Fabian Immler

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  1. Fabian Immler
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Editors and Affiliations

  1. NASA, 2101 NASA Parkway, M/C ER4, 77058, Houston, TX, USA

    Julia M. Badger

  2. Intelligent Systems Division, NASA Ames Research Center, 94035, Moffett Field, CA, USA

    Kristin Yvonne Rozier

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Immler, F. (2014). Formally Verified Computation of Enclosures of Solutions of Ordinary Differential Equations. In: Badger, J.M., Rozier, K.Y. (eds) NASA Formal Methods. NFM 2014. Lecture Notes in Computer Science, vol 8430. Springer, Cham. https://doi.org/10.1007/978-3-319-06200-6_9

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  • DOI: https://doi.org/10.1007/978-3-319-06200-6_9

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06199-3

  • Online ISBN: 978-3-319-06200-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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