Publication: Jet Space as the Geometric Arena of Automatic Differentiation
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Jet Space as the Geometric Arena of Automatic Differentiation

- incollection -
 

Author(s)
Gordon D. Pusch

Published in
Computational Differentiation: Techniques, Applications, and Tools

Editor(s)
Martin Berz, Christian H. Bischof, George F. Corliss, Andreas Griewank

Year
1996

Publisher
SIAM

Abstract
I discuss the connection between the differential geometric concept of a jet and Berz's differential algebra (DA) method of automatic differentiation to arbitrarily high order with respect to arbitrarily many variables. A DA-valued object is a representation of the target-space projection of a jet under a particular coordinate chart. Jets generalize the coordinate-independent concept of the tangent map of a function to higher orders of contact. Since the tangent map unifies the concepts of derivative, tangent vector, and the tangent linear model into a single geometrical object, the geometrical framework of higher-order differentiation and its natural connections to perturbation theory and to higher-order tangent models become clear. Jets contain additional structure beyond that of DA vectors; in particular, the jet concept meshes nicely with both the ``applicative″ and ``object oriented″ programming paradigms. The jet concept may therefore provide insights to guide new implementations of automatic differentiation. In particular, jets may help to define derivatives of composite objects such as structures; they also introduce a new means of error-checking, since operations combining jets originating from different source points are undefined.

Cross-References
Berz1996CDT

BibTeX
@INCOLLECTION{
         Pusch1996JSa,
       author = "Gordon D. Pusch",
       editor = "Martin Berz and Christian H. Bischof and George F. Corliss and Andreas Griewank",
       title = "Jet Space as the Geometric Arena of Automatic Differentiation",
       booktitle = "Computational Differentiation: Techniques, Applications, and Tools",
       pages = "53--62",
       publisher = "SIAM",
       address = "Philadelphia, PA",
       key = "Pusch1996JSa",
       crossref = "Berz1996CDT",
       abstract = "I discuss the connection between the differential geometric concept of a {\em
         jet\/} and Berz's differential algebra (DA) method of automatic differentiation to
         arbitrarily high order with respect to arbitrarily many variables. A DA-valued object is a
         representation of the {\em target-space projection\/} of a jet under a particular
         coordinate chart. Jets generalize the coordinate-independent concept of the {\em tangent
         map\/} of a function to higher orders of contact. Since the tangent map unifies the concepts of
         derivative, tangent vector, and the tangent linear model into a single geometrical object, the
         geometrical framework of higher-order differentiation and its natural connections to perturbation
         theory and to higher-order tangent models become clear. Jets contain additional structure beyond
         that of DA vectors; in particular, the jet concept meshes nicely with both the
         ``applicative'' and ``object oriented'' programming paradigms. The jet concept
         may therefore provide insights to guide new implementations of automatic differentiation. In
         particular, jets may help to define derivatives of composite objects such as structures; they also
         introduce a new means of error-checking, since operations combining jets originating from different
         source points are undefined.",
       keywords = "Jet spaces, differential geometry, coordinate-free differentiation, fiber
         bundles.",
       year = "1996"
}


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