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Automatic Differentiation:

A Structure-Exploiting Forward Mode with almost Optimal Complexity for Kantorovic Trees

Michael Ulbrich, and Stefan Ulbrich Pfeil

Technische Universität München

Zentrum Mathematik

Lehrstuhl für Angewandte Mathematik und Mathematische Statistik

January 1996

Technical Report


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Abstract:

A structure-exploiting forward mode is discussed that achieves almost optimal complexity for functions given by Kantorovic trees. It is based on approriate representations of the gradient and the Hessian. After a brief exposition of the forward and reverse mode of automatic differentiation for derivatives up to second order and compact proofs of their complexities, the new forward mode is presented and analyzed. It is shown that in the case of functions f: R^n -> R with a tree as Kantorovic graph the algorithm is only O(ln(n)) times as expensive as the reverse mode. Except for the fact that the new method is a very efficient implementation of the forward mode, it can be used to significantly reduce the length of characterizing sequences before applying the memory expensive reverse mode. For the Hessian all discussed algorithms are shown to be efficiently parallelizable. Some numerical examples confirm the advantages of the new forward mode.

Keywords:

automatic differentiation, characterizing sequence, code list, forward mode, reverse mode, Kantorovic graph, Kantorovic tree, time complexity, parallelization.


Michael Ulbrich , 1996-05-21

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