Publication: Enabling Implicit Time Integration for Compressible Flows by Partial Coloring: A Case Study of a Semi-matrix-free Preconditioning Technique
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Enabling Implicit Time Integration for Compressible Flows by Partial Coloring: A Case Study of a Semi-matrix-free Preconditioning Technique

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Area
Aerodynamics

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

Published in
2016 Proceedings of the Seventh SIAM Workshop on Combinatorial Scientific Computing, Albuquerque, New Mexico, USA, October 10--12

Editor(s)
A. H. Gebremedhin, E. G. Boman, B. Ucar

Year
2016

Publisher
SIAM

Abstract
Numerical techniques involving linearizations of nonlinear functions require the repeated solution of systems of linear equations whose coefficient matrix is the Jacobian of that nonlinear function. If the Jacobian is large and sparse, iterative methods offer the advantage that they involve the Jacobian solely in the form of matrix-vector products. Techniques of automatic differentiation are capable of evaluating these Jacobian-vector products efficiently and accurately in a matrix-free fashion. So, the numerical technique does not need to store the Jacobian explicitly. When the solution of the linear system is preconditioned, however, there is currently a considerable gap between automatic differentiation and preconditioning because the latter typically requires to explicitly store the Jacobian in a sparse data format. In an attempt to bridge this gap, we introduce an approach based on block diagonal preconditioning that brings together known computational building blocks in a novel way. The crucial methodological ingredient to that approach is the formulation and solution of a partial coloring problem in which colors are assigned to only a subset of the vertices of the underlying graph. Numerical experiments are reported that demonstrate the feasibility of this approach.

AD Theory and Techniques
Sparsity

BibTeX
@INPROCEEDINGS{
         Bucker2016EIT,
       author = "H. M. B{\"u}cker and M. L{\"u}lfesmann and M. A. Rostami",
       title = "Enabling Implicit Time Integration for Compressible Flows by Partial Coloring: {A}
         Case Study of a Semi-matrix-free Preconditioning Technique",
       booktitle = "2016 Proceedings of the Seventh SIAM Workshop on Combinatorial Scientific
         Computing, Albuquerque, New Mexico, USA, October~10--12",
       editor = "A. H. Gebremedhin and E. G. Boman and B. Ucar",
       pages = "23--32",
       address = "Philadelphia, PA, USA",
       publisher = "SIAM",
       doi = "10.1137/1.9781611974690.ch3",
       abstract = "Numerical techniques involving linearizations of nonlinear functions require the
         repeated solution of systems of linear equations whose coefficient matrix is the Jacobian of that
         nonlinear function. If the Jacobian is large and sparse, iterative methods offer the advantage that
         they involve the Jacobian solely in the form of matrix-vector products. Techniques of automatic
         differentiation are capable of evaluating these Jacobian-vector products efficiently and accurately
         in a matrix-free fashion. So, the numerical technique does not need to store the Jacobian
         explicitly. When the solution of the linear system is preconditioned, however, there is currently a
         considerable gap between automatic differentiation and preconditioning because the latter typically
         requires to explicitly store the Jacobian in a sparse data format. In an attempt to bridge this gap,
         we introduce an approach based on block diagonal preconditioning that brings together known
         computational building blocks in a novel way. The crucial methodological ingredient to that approach
         is the formulation and solution of a partial coloring problem in which colors are assigned to only a
         subset of the vertices of the underlying graph. Numerical experiments are reported that demonstrate
         the feasibility of this approach.",
       year = "2016",
       ad_area = "Aerodynamics",
       ad_theotech = "Sparsity"
}


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