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\section{Related works}
\label{sec:02}
-Krylov subspace iteration methods have increasingly become useful and successful
-techniques for solving linear, nonlinear systems and eigenvalue problems,
-especially since the increase development of the
+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 of the popularity of
-these methods is their generality, simplicity and efficiency to solve systems 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 is developed by Saad and al.~\cite{Saad86} as a
+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, it
+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. It should be
-noticed that full GMRES is too expensive in the case of large matrices since the
+most $n$ iterations, where $n$ is the size of the sparse matrix.
+Full GMRES is however too much 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,
+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 value of $m$~\cite{Huang89}. Therefore in
+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.
-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 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.
+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 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}.
examples, we also obtained similar gain between GMRES and TSIRM but those
examples are not scalable with many cores. In general, we had some problems
with more than $4,096$ cores.
-\item We have tested many iterative solvers available in PETSc. In fast, it is
+\item We have tested many iterative solvers available in PETSc. In fact, it is
possible to use most of them with TSIRM. From our point of view, the condition
to use a solver inside TSIRM is that the solver must have a restart
- feature. More precisely, the solver must support to be stoped and restarted
+ feature. More precisely, the solver must support to be stopped and restarted
without decrease its converge. That is why with GMRES we stop it when it is
- naturraly restarted (i.e. with $m$ the restart parameter). The Conjugate
+ naturally restarted (i.e. with $m$ the restart parameter). The Conjugate
Gradient (CG) and all its variants do not have ``restarted'' version in PETSc,
so they are not efficient. They will converge with TSIRM but not quickly
because if we compare a normal CG with a CG for which we stop it each 16
