Publication: Preconditioning Jacobian Systems by Superimposing Diagonal Blocks
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Preconditioning Jacobian Systems by Superimposing Diagonal Blocks

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Author(s)
M. A. Rostami , H. M. Bücker

Published in
Computational Science -- ICCS 2020, Proceedings of the 20th International Conference on Computational Science, Amsterdam, The Netherlands, June 3--5, 2020. Part II

Editor(s)
V. V. Krzhizhanovskaya, G. Závodszky, M. H. Lees, J. J. Dongarra, P. M. A. Sloot, S. Brissos, J. Teixeira

Year
2020

Publisher
Springer International Publishing

Abstract
Preconditioning constitutes an important building block for the solution of large sparse systems of linear equations. If the coefficient matrix is the Jacobian of some mathematical function given in the form of a computer program, automatic differentiation enables the efficient and accurate evaluation of Jacobian-vector products and transposed Jacobian-vector products in a matrix-free fashion. Standard preconditioning techniques, however, typically require access to individual nonzero elements of the coefficient matrix. These operations are computationally expensive in a matrix-free approach where the coefficient matrix is not explicitly assembled. We propose a novel preconditioning technique that is designed to be used in combination with automatic differentiation. A key element of this technique is the formulation and solution of a graph coloring problem that encodes the rules of partial Jacobian computation that determines only a proper subset of the nonzero elements of the Jacobian matrix. The feasibility of this semi-matrix-free approach is demonstrated on a set of numerical experiments using the automatic differentiation tool ADiMat.

AD Tools
ADiMat

AD Theory and Techniques
Sparsity

BibTeX
@INPROCEEDINGS{
         Rostami2020PJS,
       author = "M. A. Rostami and H. M. B{\"u}cker",
       title = "Preconditioning {J}acobian Systems by Superimposing Diagonal Blocks",
       booktitle = "Computational Science -- ICCS~2020, Proceedings of the 20th International
         Conference on Computational Science, Amsterdam, The Netherlands, June 3--5, 2020. Part~II",
       editor = "V. V. Krzhizhanovskaya and G. Z\'{a}vodszky and M. H. Lees and J. J.
         Dongarra and P. M. A. Sloot and S. Brissos and J. Teixeira",
       volume = "12138",
       series = "Lecture Notes in Computer Science",
       publisher = "Springer International Publishing",
       pages = "101--115",
       doi = "10.1007/978-3-030-50417-5_8",
       url = "https://doi.org/10.1007/978-3-030-50417-5_8",
       address = "Cham, Switzerland",
       abstract = "Preconditioning constitutes an important building block for the solution of large
         sparse systems of linear equations. If the coefficient matrix is the Jacobian of some mathematical
         function given in the form of a computer program, automatic differentiation enables the efficient
         and accurate evaluation of Jacobian-vector products and transposed Jacobian-vector products in a
         matrix-free fashion. Standard preconditioning techniques, however, typically require access to
         individual nonzero elements of the coefficient matrix. These operations are computationally
         expensive in a matrix-free approach where the coefficient matrix is not explicitly assembled. We
         propose a novel preconditioning technique that is designed to be used in combination with automatic
         differentiation. A key element of this technique is the formulation and solution of a graph coloring
         problem that encodes the rules of partial Jacobian computation that determines only a proper subset
         of the nonzero elements of the Jacobian matrix. The feasibility of this semi-matrix-free approach is
         demonstrated on a set of numerical experiments using the automatic differentiation tool ADiMat.",
       year = "2020",
       ad_tools = "ADiMat",
       ad_theotech = "Sparsity"
}


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