Publication: Differentiating through conjugate gradient
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Differentiating through conjugate gradient

- Article in a journal -
 

Author(s)
Bruce Christianson

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
We show that, although the conjugate gradient (CG) algorithm has a singularity at the solution, it is possible to differentiate forward through the algorithm automatically by re-declaring all the variables as truncated Taylor series, the type of active variable widely used in automatic differentiation (ad) tools such as ADOL-C. If exact arithmetic is used, this approach gives a complete sequence of correct directional derivatives of the solution, to arbitrary order, in a single cycle of at most n iterations, where n is the number of dimensions. In the inexact case, the approach emphasizes the need for a means by which the programmer can communicate certain conditions involving derivative values directly to an ad tool.

Cross-References
Christianson2018Sio

BibTeX
@ARTICLE{
         Christianson2018Dtc,
       crossref = "Christianson2018Sio",
       author = "Bruce Christianson",
       title = "Differentiating through conjugate gradient",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "988--994",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2018.1425862",
       url = "https://doi.org/10.1080/10556788.2018.1425862",
       eprint = "https://doi.org/10.1080/10556788.2018.1425862",
       abstract = "We show that, although the conjugate gradient (CG) algorithm has a singularity at
         the solution, it is possible to differentiate forward through the algorithm automatically by
         re-declaring all the variables as truncated Taylor series, the type of active variable widely used
         in automatic differentiation (AD) tools such as ADOL-C. If exact arithmetic is used, this approach
         gives a complete sequence of correct directional derivatives of the solution, to arbitrary order, in
         a single cycle of at most n iterations, where n is the number of dimensions. In the inexact case,
         the approach emphasizes the need for a means by which the programmer can communicate certain
         conditions involving derivative values directly to an AD tool.",
       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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