Publication: A Low Rank Approach to Automatic Differentiation
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A Low Rank Approach to Automatic Differentiation

- incollection -
 

Author(s)
Hany S. Abdel-Khalik , Paul D. Hovland , Andrew Lyons , Tracy E. Stover , Jean Utke

Published in
Advances in Automatic Differentiation

Editor(s)
Christian H. Bischof, H. Martin Bücker, Paul D. Hovland, Uwe Naumann, J. Utke

Year
2008

Publisher
Springer

Abstract
This manuscript introduces a new approach for increasing the efficiency of automatic differentiation (ad) computations for estimating the first order derivatives comprising the Jacobian matrix of a complex large-scale computational model. The objective is to approximate the entire Jacobian matrix with minimized computational and storage resources. This is achieved by finding low rank approximations to a Jacobian matrix via the Efficient Subspace Method (ESM). Low rank Jacobian matrices arise in many of today's important scientific and engineering problems, e.g. nuclear reactor calculations, weather climate modeling, geophysical applications, etc. A low rank approximation replaces the original Jacobian matrix J (whose size is dictated by the size of the input and output data streams) with matrices of much smaller dimensions (determined by the numerical rank of the Jacobian matrix). This process reveals the rank of the Jacobian matrix and can be obtained by ESM via a series of r randomized matrix-vector products of the form: Jq, and J^T ω which can be evaluated by the ad forward and reverse modes, respectively.

Cross-References
Bischof2008AiA

AD Tools
OpenAD

AD Theory and Techniques
Jacobian-vector product

BibTeX
@INCOLLECTION{
         Abdel-Khalik2008ALR,
       title = "A Low Rank Approach to Automatic Differentiation",
       doi = "10.1007/978-3-540-68942-3_6",
       author = "Hany S. Abdel-Khalik and Paul D. Hovland and Andrew Lyons and Tracy E. Stover and
         Jean Utke",
       abstract = "This manuscript introduces a new approach for increasing the efficiency of
         automatic differentiation (AD) computations for estimating the first order derivatives comprising
         the Jacobian matrix of a complex large-scale computational model. The objective is to approximate
         the entire Jacobian matrix with minimized computational and storage resources. This is achieved by
         finding low rank approximations to a Jacobian matrix via the Efficient Subspace Method (ESM). Low
         rank Jacobian matrices arise in many of today's important scientific and engineering problems,
         e.g. nuclear reactor calculations, weather climate modeling, geophysical applications, etc. A low
         rank approximation replaces the original Jacobian matrix $J$ (whose size is dictated by the size of
         the input and output data streams) with matrices of much smaller dimensions (determined by the
         numerical rank of the Jacobian matrix). This process reveals the rank of the Jacobian matrix and can
         be obtained by ESM via a series of r randomized matrix-vector products of the form: $Jq$, and $J^{T}
         \omega$ which can be evaluated by the AD forward and reverse modes, respectively.",
       crossref = "Bischof2008AiA",
       pages = "55--65",
       booktitle = "Advances in Automatic Differentiation",
       publisher = "Springer",
       editor = "Christian H. Bischof and H. Martin B{\"u}cker and Paul D. Hovland and Uwe
         Naumann and J. Utke",
       isbn = "978-3-540-68935-5",
       issn = "1439-7358",
       year = "2008",
       ad_theotech = "Jacobian-vector product",
       ad_tools = "OpenAD"
}


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