Publication: Combinatorial Optimization of Stencil-based Jacobian Computations
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Combinatorial Optimization of Stencil-based Jacobian Computations

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Author(s)
H. M. Bücker , M. Lülfesmann

Published in
Proceedings of the 2010 International Conference on Computational and Mathematical Methods in Science and Engineering, Almería, Andalucía, Spain, June 26--30, 2010

Editor(s)
J. V. Aguiar

Year
2010

Abstract
The computation of all nonzero entries of a sparse Jacobian matrix using either divided differencing or the forward mode of automatic differentiation is considered. Throughout this article, we assume that the sparsity of the Jacobian stems from a stencil-based computation of the underlying function which is typical for numerical applications in computational science and engineering involving partial differential equations. The minimization of the time needed to compute all nonzero Jacobian entries is formulated as a combinatorial optimization problem. We present three different, yet equivalent, representations of that problem and discuss each of its advantages and disadvantages. Broadly speaking, the three representations belong to the areas of linear algebra, grid discretization, and graph theory.

AD Theory and Techniques
graph coloring, Sparsity

BibTeX
@INPROCEEDINGS{
         Bucker2010COo,
       author = "H. M. B{\"u}cker and M. L{\"u}lfesmann",
       title = "Combinatorial Optimization of Stencil-based {J}acobian Computations",
       booktitle = "Proceedings of the 2010 International Conference on Computational and Mathematical
         Methods in Science and Engineering, Almer\'{i}a, Andaluc\'{i}a, Spain,
         June~26--30, 2010",
       editor = "J. V. Aguiar",
       pages = "284--295",
       isbn = "978-84-613-5510-5",
       abstract = "The computation of all nonzero entries of a sparse Jacobian matrix using either
         divided differencing or the forward mode of automatic differentiation is considered. Throughout this
         article, we assume that the sparsity of the Jacobian stems from a stencil-based computation of the
         underlying function which is typical for numerical applications in computational science and
         engineering involving partial differential equations. The minimization of the time needed to compute
         all nonzero Jacobian entries is formulated as a combinatorial optimization problem. We present three
         different, yet equivalent, representations of that problem and discuss each of its advantages and
         disadvantages. Broadly speaking, the three representations belong to the areas of linear algebra,
         grid discretization, and graph theory.",
       year = "2010",
       volume = "1",
       ad_theotech = "graph coloring, Sparsity"
}


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