Matrix-Free Polynomial-Based Nonlinear Least Squares Optimized Preconditioning and its Application to Discontinuous Galerkin Discretizations of the Euler Equations
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Authors
Carr, L.E.
Borges, C.F.
Giraldo, F.X.
Subjects
preconditioning
polynomial preconditioner
nonlinear least squares
spectral elements
Galerkin methods
Euler Equations
nonhydrostatic atmospheric model
polynomial preconditioner
nonlinear least squares
spectral elements
Galerkin methods
Euler Equations
nonhydrostatic atmospheric model
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2015-02-12
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Abstract
We introduce a preconditioner that can be both constructed and applied using
only the ability to apply the underlying operator. Such a preconditioner can be very
attractive in scenarios where one has a highly efficient parallel code for applying the operator.
Our method constructs a polynomial preconditioner using a nonlinear least squares
(NLLS) algorithm. We show that this polynomial-based NLLS-optimized (PBNO) preconditioner
significantly improves the performance of a discontinuous Galerkin (DG)
compressible Euler equation model when run in an implicit-explicit time integration
mode. The PBNO preconditioner achieves significant reduction in GMRES iteration
counts and model wall-clock time, and significantly outperforms several existing types
of generalized (linear) least squares (GLS) polynomial preconditioners. Comparisons of
the ability of the PBNO preconditioner to improve DG model performance when employing
the Stabilized Biconjugate Gradient algorithm (BICGS) and the basic Richardson
(RICH) iteration are also included. In particular, we show that higher order PBNO
preconditioning of the Richardson iteration (run in a dot product free mode) makes the
algorithm competitive with GMRES and BICGS in a serial computing environment. Because
the NLLS-based algorithm used to construct the PBNO preconditioner can handle
both positive definite and complex spectra without any need for algorithm modification,
we suggest that the PBNO preconditioner is, for certain types of problems, an attractive
alternative to existing polynomial preconditioners based on linear least-squares methods.
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Journal of Scientific Computing manuscript
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Applied Mathematics
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This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.