- J. Erhel and F. Guyomarc'h.

An Augmented Conjugate Gradient Method for solving consecutive symmetric positive definite systems.

SIAM Journal on Matrix Analysis and Applications , Volume 21, no. 4, pp. 1279-1299, 2000.

- Y. Saad, M. Yeung, J. Erhel, and F. Guyomarc'h.

A deflated version of the Conjugate Gradient Algorithm.

SIAM Journal on Scientific Computing, Volume 21, no. 5 pp.1909-1926, 2000.

- F. Guyomarc'h

Méthodes de Krylov : régularisation de la solution et accélération de la convergence.

Ph-D thesis, University of Rennes 1, 2000.

- M-O. Bristeau, J. Erhel.

Augmented Conjugate Gradient. Application in an iterative process for the solution of scattering problems.

Numerical Algorithms, 18, pp. 71-90, 1998.

- M.O. Bristeau and P. Féat and J. Erhel and R. Glowinski and J. Périaux.

Solving the Helmholtz equation at high wave numbers on a parallel computer with a shared virtual memory.

International Journal of Supercomputer Applications and High Performance Computing, Volume 9, pp.18-28, 1995.

Applications to groundwater flow

- de Dreuzy, J.-R.; Pichot, G.; Poirriez, B. and Erhel, J.

Synthetic benchmark for modeling flow in 3D fractured media

Computers & Geosciences, 50, 59-71, 2013

- Pichot, G.; Poirriez, B.; Erhel, J. & de Dreuzy, J.-R.

A Mortar BDD method for solving flow in stochastic discrete fracture networks

Domain Decomposition Methods in Science and Engineering XXI

Jocelyne Erhel; Martin Gander; Laurence Halpern; Géraldine Pichot; Taoufik Sassi & Olof Widlund, ed

LNCSE, Springer, 2014

- Poirriez, B.

Etude et mise en oeuvre d'une méthode de sous-domaines pour la modélisation de l'écoulement dans des réseaux de fractures en 3D

Ph-D thesis, University of Rennes 1, 2011

DEFGMRES computes and updates a preconditioner at each restart by using an estimation of an invariant subspace. It can be used with any other preconditioner. It is parallel but involves some communications. Deflation avoids stagnation when the restarting parameter is small.

AUGMRES computes an augmented basis of the Krylov subspace by estimating an invariant subspace. Like DEFGMRES, it can be used with any preconditioner, it avoids stagnation and is parallel.

MSGMRES computes a parallel Multiplicative Schwarz preconditioner. It is based on an algebraic matrix partition and on an explicit formulation of the preconditioning matrix. The matrix vector product and the preconditioning are pipelined through parallel processes.

- GPREMS implements a combination of the three algorithms PARGMRES,
MSGMRES and AUGMRES: it
is parallel and preconditioned by Multiplicative Schwarz. The basis can
be augmented by deflating approximate eigenvalues.

- DGMRES implements DEFGMRES in the Petsc library. It can be combined with GMRES and any preconditioner included in Petsc. DGMRES is included in Petsc library.
- AGMRES implements a combination of the algorithms PARGMRES and
AUGMRES in the Petsc library. It is included in the repository of Petsc.

Algorithms are applied to 3D industrial problems stemming from Computational Fluid Dynamics applications aiming at reducing fuel consumption.

Matrices are provided by the industrial partners of the LIBRAERO and CINEMAS2 projects (see below).

- D. Nuentsa Wakam and F. Pacull

Memory efficient hybrid algebraic solvers for linear systems arising from compressible flows

Computers and Fluids, 2013, Volume 80, 158-167 - J. Erhel

Some Properties of Krylov Projection Methods for Large Linear Systems.

P. Ivanyi and B.H.V. Topping (ed), Computational Technology Reviews,

Saxe-Coburg Publications, 2011, Volume 3, 41-70 - B. Philippe and L. Reichel

on the generation of Krylov subspace bases

Applied Numerical Mathematics (APNUM), 2012, Volume 62, 1171-1186 - N. Gmati, B. Philippe,

Comments on the GMRES convergence for preconditioned systems

*proceedings of the international conference on Large-Scale Scientific Computations,***2007,**40-51, LNCS

- D. Nuentsa-Wakam and J. Erhel.

Parallelism and robustness in GMRES with the Newton basis and the deflated restarting.

ETNA, 40, 381-406, 2013.

- D. Nuentsa-Wakam and J. Erhel and W. Gropp.

Parallel adaptive deflated GMRES.

Domain Decomposition Methods in Science and Engineering XX (DD20), R. Bank, M. Holst, O. Widlund and J. Xu editors,

pp. 631-638, LNCSE, Springer, 2013. - D. Nuentsa-Wakam and J. Erhel and É. Canot
**.**Robustness in hybrid algebraic solvers for large linear systems.

Proceedings of Parallel CFD 2011, 2011, online. - D. Nuentsa Wakam

Parallélisme et robustesse dans les solveurs hybrides pour grands systèmes linéaires: Application à l'optimisation en dynamique des fluides. Ph-D thesis, University of Rennes 1, 2011

- K. Burrage, J. Erhel, B. Pohl and A. Williams.

A deflation technique for linear systems of equations.

*SIAM Journal on Science and Computation*, Volume 19, Number 4, pp. 1245-1260, 1998. - K. Burrage, J. Erhel.

On the performance of various adaptive preconditioned GMRES strategies.

Numerical Linear Algebra with Applications, Volume 5, pp. 101-121, 1998. - J. Erhel and K. Burrage and B. Pohl.

Restarted GMRES preconditioned by deflation.

Journal of Computation and Applied Mathematics, Volume 69, pp.303-318, 1996. - K. Burrage and A. Williams and J. Erhel and B. Pohl.

The implementation of a Generalized Cross Validation algorithm using deflation techniques for linear systems.

Applied Numerical Mathematics, Volume 19, pp.17-31, 1995.

- G.-A.
Atenekeng Kahou and D. Nuentsa-Wakam.

Parallel GMRES with a multiplicative Schwarz preconditioner.

ARIMA, 2011, Volume 14, 81-99

- D. Nuentsa-Wakam, J. Erhel, É. Canot and G.
Atenekeng-Kahou

A comparative study of some distributed linear solvers on systems arising from fluid dynamics simulations

*Parallel Computing: from Multicores and GPU's to Petascale (Proceedings of PARCO'2009*), 2010**,**51-58

B. Chapman and F. Desprez and G. Joubert and A. Lichnewsky and F. Peters and T. Priol*(ed.)*

IOS Press - D. Nuentsa-Wakam and J. Erhel and É. Canot
Parallélisme à deux niveaux dans {GMRES} avec un préconditionneur Schwarz multiplicatif

proceedings of CARI'2010

**,****2010,**189-196**,**INRIA - G.-A. Atenekeng Kahou, L.
Grigori, and M. Sosonkina.

A Partitioning Algorithm for Block-Diagonal Matrices with Overlap.

*Parallel Computing,***2008,***Volume 34, Issues 6-8*, 332-344 - G.-A.
Atenekeng Kahou, E. Kamgnia, and B. Philippe.

An explicit formulation of the multiplicative Schwarz preconditionner.

*Applied Numerical Mathematics,***2007***, Volume 57*, 1197-1213*(also INRIA research report RR-5685).* - G.-A.
Atenekeng Kahou, E. Kamgnia, and B. Philippe.

A Combinatorial tool for GMRES preconditioned by Multiplicative Schwarz.

*Proceedings of PPAM'2007, CTPSM07 Workshop,***2007.**

- D. Imberti and J. Erhel

Vary the s in Your s-step GMRES

preprint, 2016. https://hal.inria.fr/hal-01299652 - Sidje, R.B.

Alternatives for Parallel Krylov Basis Computation.

Numerical Linear Algebra with Applications, Vol. 4(4), 305-331, 1997. - J. Erhel.

A parallel GMRES version for general sparse matrices.

Electronic Transactions on Numerical Analysis, Volume 3, pp.160-176, 1995. - J. Erhel.

A Parallel Preconditioned GMRES Algorithm for Sparse Matrices.

*Lectures in Applied Mathematics, The Mathematics of Numerical Analysis, 1996, pp. 345-355.* - R-B. Sidje.

Algorithmes parallèles pour le calcul d'exponentielles de matrices de grandes tailles.

Ph-D thesis, University of Rennes 1, 1994. - B. Philippe and R. B. Sidje.

Parallel Krylov Subspace Basis Computation.

proceedings of the 2nd Colloque Africain sur la Recherche en Informatique (CARI), 1994, 421-440.

J. Tankoano Ed., ORSTOM Editions.

This topic started in the Aladin team (participants B. Philippe and
J. Erhel) with the Ph-D thesis of R. Sidje. The thesis was defended in
1994. It was extended with several internships to define PARGMRES.

The work on Multiplicative Schwarz was done in the Sage team (participants B. Philippe and L. Grigori) with the Ph-D thesis of G-A. Atenekeng Kahou, in collaboration with E. Kamgnia, Cameroon and M. Sosonkina, USA. The thesis was defended in 2008.

Some convergence aspects of GMRES were studied (participant B.
Philippe), in collaboration with N. Gmati, Tunisia and with L. Reichel, USA.

The research continues in the Fluminance team ((participant J. Erhel) with the post-doctoral position of D. Imberti.

Work on parallel and deflated GMRES was funded by the ANR RNTL with the project LIBRAERO (2008-2011).

It was also funded by Région Rhône-Alpes, in the framework of LUTB, with
the project CINEMAS2 (2007-2011).

It is partly funded by the INRIA Illinois Joint Lab on Petascale Computing (2010-2011).

D. Nuentsa Wakam spent a month at UIUC (USA), thanks to a grant from the university of Rennes 1 (2011).

The visits of L. Reichel (Kent Univ, USA) in the team were funded by the university of Rennes 1 (2010).

The exchanges between INRIA and Univ. of Queensland (Australia) were partly funded by the australian government (1994-1997).