CERFACS CERFACS Scientific Activity Report Jan. 2010 – Dec. 2011

PARALLEL ALGORITHMS PROJECT
We are currently working on this method and some variations as the block implementation or its use to
compute the harmonic Ritz vectors (G.W. Stewart, SIAM J. Matrix Anal. Appl., 2002).
This work has been presented in conferences (e.g., P. Salas, L. Giraud, J. Muller, G. Staffelbach, T. Poinsot,
”Stability of azimuthal modes in annular combustion chambers”, INCA, November 2011) and a paper has
been accepted for its publication in the journal of Combustion and Flame (CERFACS Technical Report,
TR/CFD/11/35, J.-F. Parmentier, P. Salas, P. Wolf, G. Staffelbach, F. Nicoud and T. Poinsot, ”A simple
analytical model to study and control azimuthal instabilities in annular combustion chambers”).
5.10 Minimizing the backward error in the energy norm with
conjugate gradients.
S. Gratton : INPT-IRIT, UNIVERSITY OF TOULOUSE AND ENSEEIHT, France ;
P. Jiránek : TECHNICAL UNIVERSITY OF LIBEREC AND CERFACS, Czech Republic and France ;
X. Vasseur : CERFACS, France
In [ALG57], we derive backward error formulas for a linear system of equations in general norms induced
by given symmetric positive definite matrices and consider a special case of a backward error induced by
the energy norm when the system matrix is symmetric positive definite. We study the convergence of the
conjugate gradient method (CG) with respect to this energy backward error. For that purpose we construct a
hypothetical variant of CG called CGBACK which constructs the approximations that actually minimize the
energy backward error over the associated Krylov subspaces and can therefore be considered as an analog
of the GMBACK/MINPERT algorithms of Kasenally and Simoncini. We show that the optimal CGBACK
approximation is a scalar multiple of the current CG approximation with the coefficient depending only
on the weighting parameters appearing in the definition of the backward error and on the relative energy
norm of the error of the current CG iterate. In addition when CG makes a moderate progress in terms of the
energy norm of the error then the energy backward errors of the subsequent CG approximations start to be
very close to the optimal energy backward errors of CGBACK approximations. In this way we deduce that
CG approximations almost minimize the energy backward error.
5.11 Adaptive version of simpler GMRES.
P. Jiránek : TECHNICAL UNIVERSITY OF LIBEREC AND CERFACS, Czech Republic and France ;
M. Rozložník : ACADEMY OF SCIENCES OF THE CZECH REPUBLIC, Czech Republic
In [ALG29], we propose a stable variant of Simpler GMRES by Walker and Zhou (1994). It is based on
the adaptive choice of the Krylov subspace basis at given iteration step using the intermediate residual
norm decrease criterion. The new direction vector is chosen as in the original implementation of Simpler
GMRES or it is equal the normalized residual vector as in the GCR method. We show that such adaptive
strategy leads to a well-conditioned basis of the Krylov subspace and we support our theoretical results with
illustrative numerical examples.
CERFACS ACTIVITY REPORT 19