-Krylov subspace iteration methods have increasingly become useful and successful techniques for solving linear and nonlinear systems and eigenvalue problems, especially since the increase development of the preconditioners~\cite{Saad2003,Meijerink77}. One reason of the popularity of these methods is their generality, simplicity and efficiency to solve systems of equations arising from very large and complex problems. %A Krylov method is based on a projection process onto a Krylov subspace spanned by vectors and it forms a sequence of approximations by minimizing the residual over the subspace formed~\cite{}.
-
-GMRES is one of the most widely used Krylov iterative method for solving sparse and large linear systems. It is developed by Saad and al.~\cite{Saad86} as a generalized method to deal with unsymmetric and non-Hermitian problems, and indefinite symmetric problems too. In its original version called full GMRES, it minimizes the residual over the current Krylov subspace until convergence in at most $n$ iterations, where $n$ is the size of the sparse matrix. It should be noted that full GMRES is too expensive in the case of large matrices since the required orthogonalization process per iteration grows quadratically with the number of iterations. For that reason, in practice GMRES is restarted after each $m\ll n$ iterations to avoid the storage of a large orthonormal basis. However, the convergence behavior of the restarted GMRES, called GMRES($m$), in many cases depends quite critically on the value of $m$~\cite{Huang89}. Therefore in most cases, a preconditioning technique is applied to the restarted GMRES method in order to improve its convergence.
-
-In order to enhance the robustness of Krylov iterative solvers, some techniques have been proposed allowing the use of different preconditioners, if necessary, within the iteration instead of restarting. Those techniques may lead to considerable savings in CPU time and memory requirements. Van der Vorst in~\cite{Vorst94} has proposed variants of the GMRES algorithm in which a different preconditioner is applied in each iteration, so-called GMRESR family of nested methods. In fact, the GMRES method is effectively preconditioned with other iterative schemes (or GMRES itself), where the iterations of the GMRES method are called outer iterations while the iterations of the preconditioning process referred to as inner iterations. Saad in~\cite{Saad:1993} has proposed FGMRES which is another variant of the GMRES algorithm using a variable preconditioner. In FGMRES the search directions are preconditioned whereas in GMRESR the residuals are preconditioned. However in practice the good preconditioners are based on direct methods, as ILU preconditioners, which are not easy to parallelize and suffer from the scalability problems on large clusters of thousands of cores.
-
-% two-stage, communication avoiding
+Krylov subspace iteration methods have increasingly become key
+techniques for solving linear and nonlinear systems, or eigenvalue problems,
+especially since the increasing development of
+preconditioners~\cite{Saad2003,Meijerink77}. One reason for the popularity of
+these methods is their generality, simplicity, and efficiency to solve systems of
+equations arising from very large and complex problems.
+
+GMRES is one of the most widely used Krylov iterative method for solving sparse
+and large linear systems. It has been developed by Saad \emph{et
+ al.}~\cite{Saad86} as a generalized method to deal with unsymmetric and
+non-Hermitian problems, and indefinite symmetric problems too. In its original
+version called full GMRES, this algorithm minimizes the residual over the
+current Krylov subspace until convergence in at most $n$ iterations, where $n$
+is the size of the sparse matrix. Full GMRES is however too expensive in the
+case of large matrices, since the required orthogonalization process per
+iteration grows quadratically with the number of iterations. For that reason,
+GMRES is restarted in practice after each $m\ll n$ iterations, to avoid the
+storage of a large orthonormal basis. However, the convergence behavior of the
+restarted GMRES, called GMRES($m$), in many cases depends quite critically on
+the $m$ value~\cite{Huang89}. Therefore in most cases, a preconditioning
+technique is applied to the restarted GMRES method in order to improve its
+convergence.
+
+To enhance the robustness of Krylov iterative solvers, some techniques have been
+proposed allowing the use of different preconditioners, if necessary, within the
+iteration itself instead of restarting. Those techniques may lead to
+considerable savings in CPU time and memory requirements. Van der Vorst
+in~\cite{Vorst94} has for instance proposed variants of the GMRES algorithm in
+which a different preconditioner is applied in each iteration, leading to the
+so-called GMRESR family of nested methods. In fact, the GMRES method is
+effectively preconditioned with other iterative schemes (or GMRES itself), where
+the iterations of the GMRES method are called outer iterations while the
+iterations of the preconditioning process is referred to as inner iterations.
+Saad in~\cite{Saad:1993} has proposed Flexible GMRES (FGMRES) which is another
+variant of the GMRES algorithm using a variable preconditioner. In FGMRES the
+search directions are preconditioned whereas in GMRESR the residuals are
+preconditioned. However, in practice, good preconditioners are those based on
+direct methods, as ILU preconditioners, which are not easy to parallelize and
+suffer from the scalability problems on large clusters of thousands of cores.
+
+Recently, communication-avoiding methods have been developed to reduce the
+communication overheads in Krylov subspace iterative solvers. On modern computer
+architectures, communications between processors are much slower than
+floating-point arithmetic operations on a given
+processor. Communication-avoiding techniques reduce either communications
+between processors or data movements between levels of the memory hierarchy, by
+reformulating the communication-bound kernels (more frequently SpMV kernels) and
+the orthogonalization operations within the Krylov iterative solver. Different
+works have studied the communication-avoiding techniques for the GMRES method,
+so-called CA-GMRES, on multicore processors and multi-GPU
+machines~\cite{Mohiyuddin2009,Hoemmen2010,Yamazaki2014}.
+
+Compared to all these works and to all the other works on Krylov iterative
+methods, the originality of our work is to build a second iteration over a
+Krylov iterative method and to minimize the residuals with a least-squares
+method after a given number of outer iterations.
