www.Autodiff.org - Publication: Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation

Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation

- Article in a journal -

Author(s)
Andreas Griewank , Richard Hasenfelder , Manuel Radons , Lutz Lehmann , Tom Streubel

Published in
Special issue of Optimization Methods & Software: Advances in Algorithmic Differentiation Optimization Methods & Software

Editor(s)
Bruce Christianson, Shaun A. Forth, Andreas Griewank

Year
2018

Publisher
Taylor & Francis

Abstract
In this article we analyse a generalized trapezoidal rule for initial value problems with piecewise smooth right-hand side based on a generalization of algorithmic differentiation. When applied to such a problem, the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a nondifferentiability of F. The advantage of the proposed generalized trapezoidal rule is threefold: Firstly, we can achieve a higher convergence order than with the classical method. Moreover, the method is energy preserving for piecewise linear Hamiltonian systems. Finally, in analogy to the classical case we derive a third-order interpolation polynomial for the numerical trajectory. In the smooth case, the generalized rule reduces to the classical one. Hence, it is a proper extension of the classical theory. An error estimator is given and numerical results are presented.

Cross-References
Christianson2018Sio

AD Theory and Techniques
Piecewise Linear

BibTeX
@ARTICLE{
         Griewank2018ILd,
       crossref = "Christianson2018Sio",
       author = "Andreas Griewank and Richard Hasenfelder and Manuel Radons and Lutz Lehmann and Tom
         Streubel",
       title = "Integrating {L}ipschitzian dynamical systems using piecewise algorithmic
         differentiation",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "1089--1107",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2017.1378653",
       url = "https://doi.org/10.1080/10556788.2017.1378653",
       eprint = "https://doi.org/10.1080/10556788.2017.1378653",
       abstract = "In this article we analyse a generalized trapezoidal rule for initial value
         problems with piecewise smooth right-hand side based on a generalization of algorithmic
         differentiation. When applied to such a problem, the classical trapezoidal rule suffers from a loss
         of accuracy if the solution trajectory intersects a nondifferentiability of F. The advantage of the
         proposed generalized trapezoidal rule is threefold: Firstly, we can achieve a higher convergence
         order than with the classical method. Moreover, the method is energy preserving for piecewise linear
         Hamiltonian systems. Finally, in analogy to the classical case we derive a third-order interpolation
         polynomial for the numerical trajectory. In the smooth case, the generalized rule reduces to the
         classical one. Hence, it is a proper extension of the classical theory. An error estimator is given
         and numerical results are presented.",
       booktitle = "Special issue of Optimization Methods \& Software: Advances in
         Algorithmic Differentiation",
       editor = "Bruce Christianson and Shaun A. Forth and Andreas Griewank",
       ad_theotech = "Piecewise Linear"
}

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