

Preconditioning Jacobian Systems by Superimposing Diagonal Blocks
Part of a collection
  

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 35, 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 Jacobianvector products and transposed Jacobianvector products in a matrixfree fashion. Standard preconditioning techniques, however, typically require access to individual nonzero elements of the coefficient matrix. These operations are computationally expensive in a matrixfree 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 semimatrixfree 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 35, 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 = "101115",
doi = "10.1007/9783030504175_8",
url = "https://doi.org/10.1007/9783030504175_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 Jacobianvector products and transposed Jacobianvector products in a
matrixfree fashion. Standard preconditioning techniques, however, typically require access to
individual nonzero elements of the coefficient matrix. These operations are computationally
expensive in a matrixfree 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 semimatrixfree 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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