

Enabling Implicit Time Integration for Compressible Flows by Partial Coloring: A Case Study of a Semimatrixfree Preconditioning Technique
Part of a collection
  

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 1012

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 matrixvector products. Techniques of automatic differentiation are capable of evaluating these Jacobianvector products efficiently and accurately in a matrixfree 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 Semimatrixfree Preconditioning Technique",
booktitle = "2016 Proceedings of the Seventh SIAM Workshop on Combinatorial Scientific
Computing, Albuquerque, New Mexico, USA, October~1012",
editor = "A. H. Gebremedhin and E. G. Boman and B. Ucar",
pages = "2332",
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 matrixvector products. Techniques of automatic
differentiation are capable of evaluating these Jacobianvector products efficiently and accurately
in a matrixfree 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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