Publication: Pattern graph for sparse Hessian matrix determination
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Pattern graph for sparse Hessian matrix determination

- Article in a journal -
 

Author(s)
Shahadat Hossain , Nasrin Hakim Mithila

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
In a recent work, we have proposed the pattern graph as a unifying framework for methods that exploit sparsity by matrix compression: row compression, column compression or a combination of the two in sparse Jacobian matrix determination. Utilization of structural similarity between the matrix and its graph has been found to be beneficial. In this paper, we show that an important structural property, symmetry, can be exploited in the formulation of sparse Hessian matrix calculations using the pattern graph model. Using the notion of ‘direct cover’, we present a new general direct method for the determination of sparse Hessian matrices with fixed sparsity pattern and a multicolouring interpretation of it on the pattern graph associated with the matrix. A heuristic procedure for finding direct covers is sketched and some preliminary numerical test results are provided.

Cross-References
Christianson2018Sio

BibTeX
@ARTICLE{
         Hossain2018Pgf,
       crossref = "Christianson2018Sio",
       author = "Shahadat Hossain and Nasrin Hakim Mithila",
       title = "Pattern graph for sparse {H}essian matrix determination",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "1250--1263",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2018.1458849",
       url = "https://doi.org/10.1080/10556788.2018.1458849",
       eprint = "https://doi.org/10.1080/10556788.2018.1458849",
       abstract = "In a recent work, we have proposed the pattern graph as a unifying framework for
         methods that exploit sparsity by matrix compression: row compression, column compression or a
         combination of the two in sparse Jacobian matrix determination. Utilization of structural similarity
         between the matrix and its graph has been found to be beneficial. In this paper, we show that an
         important structural property, symmetry, can be exploited in the formulation of sparse Hessian
         matrix calculations using the pattern graph model. Using the notion of ‘direct
         cover’, we present a new general direct method for the determination of sparse Hessian
         matrices with fixed sparsity pattern and a multicolouring interpretation of it on the pattern graph
         associated with the matrix. A heuristic procedure for finding direct covers is sketched and some
         preliminary numerical test results are provided.",
       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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