Publication: How AD can help solve differential-algebraic equations
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How AD can help solve differential-algebraic equations

- Article in a journal -
 

Author(s)
John D. Pryce , Nedialko S. Nedialkov , Guangning Tan , Xiao Li

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
A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of the so-called index reduction or regularization, to prepare them for numerical solution. This is often done with the help of a computer algebra system. We show in two significant cases that it can be done efficiently by pure algorithmic differentiation. The first is the Dummy Derivatives method; here we give a mainly theoretical description, with tutorial examples. The second is the solution of a mechanical system directly from its Lagrangian formulation. Here, we outline the theory and show several non-trivial examples of using the ‘Lagrangian facility’ of the Nedialkov–Pryce initial-value solver DAETS, namely a spring-mass-multi-pendulum system; a prescribed-trajectory control problem; and long-time integration of a model of the outer planets of the solar system, taken from the DETEST testing package for ODE solvers.

Cross-References
Christianson2018Sio

BibTeX
@ARTICLE{
         Pryce2018HAc,
       crossref = "Christianson2018Sio",
       author = "John D. Pryce and Nedialko S. Nedialkov and Guangning Tan and Xiao Li",
       title = "How {AD} can help solve differential-algebraic equations",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "729--749",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2018.1428605",
       url = "https://doi.org/10.1080/10556788.2018.1428605",
       eprint = "https://doi.org/10.1080/10556788.2018.1428605",
       abstract = "A characteristic feature of differential-algebraic equations is that one needs to
         find derivatives of some of their equations with respect to time, as part of the so-called index
         reduction or regularization, to prepare them for numerical solution. This is often done with the
         help of a computer algebra system. We show in two significant cases that it can be done efficiently
         by pure algorithmic differentiation. The first is the Dummy Derivatives method; here we give a
         mainly theoretical description, with tutorial examples. The second is the solution of a mechanical
         system directly from its Lagrangian formulation. Here, we outline the theory and show several
         non-trivial examples of using the ‘Lagrangian facility’ of the
         Nedialkov–Pryce initial-value solver DAETS, namely a spring-mass-multi-pendulum system; a
         prescribed-trajectory control problem; and long-time integration of a model of the outer planets of
         the solar system, taken from the DETEST testing package for ODE solvers.",
       booktitle = "Special issue of Optimization Methods \& Software: Advances in
         Algorithmic Differentiation",
       editor = "Bruce Christianson and Shaun A. Forth and Andreas Griewank"
}


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