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\section{Related works}
\label{sec:02}
-%Wherever Times is specified, Times Roman or Times New Roman may be used. If neither is available on your system, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc.
+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 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
+method, 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.
+
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\Input $A$ (sparse matrix), $b$ (right-hand side)
\Output $x$ (solution vector)\vspace{0.2cm}
\State Set the initial guess $x_0$
- \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsirm}$)} \label{algo:conv}
+ \For {$k=1,2,3,\ldots$ until convergence ($error<\epsilon_{tsirm}$)} \label{algo:conv}
\State $[x_k,error]=Solve(A,b,x_{k-1},max\_iter_{kryl})$ \label{algo:solve}
- \State $S_{k \mod s}=x_k$ \label{algo:store} \Comment{update column (k mod s) of S}
- \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
+ \State $S_{k \mod s}=x_k$ \label{algo:store} \Comment{update column ($k \mod s$) of $S$}
+ \If {$k \mod s=0$ {\bf and} $error>\epsilon_{kryl}$}
\State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
\State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
\State $x_k=S\alpha$ \Comment{compute new solution}
practice, this threshold must be much smaller than the convergence threshold of
the TSIRM algorithm (\emph{i.e.}, $\epsilon_{tsirm}$). We also consider that
after the call of the $Solve$ function, we obtain the vector $x_k$ and the error
-which is defined by $||Ax^k-b||_2$.
+which is defined by $||Ax_k-b||_2$.
Line~\ref{algo:store},
-$S_{k \mod s}=x^k$ consists in copying the solution $x_k$ into the column $k
+$S_{k \mod s}=x_k$ consists in copying the solution $x_k$ into the column $k
\mod s$ of $S$. After the minimization, the matrix $S$ is reused with the new
values of the residuals. To solve the minimization problem, an iterative method
is used. Two parameters are required for that: the maximum number of iterations
In Table~\ref{tab:02}, some experiments comparing the solving of the linear
-systems obtained with the previous matrices with a GMRES variant and with out 2
-stage algorithm are given. In the second column, it can be noticed that either
-GRMES or FGMRES (Flexible GMRES)~\cite{Saad:1993} is used to solve the linear
-system. According to the matrices, different preconditioner is used. With
-TSIRM, the same solver and the same preconditionner are used. This Table shows
-that TSIRM can drastically reduce the number of iterations to reach the
-convergence when the number of iterations for the normal GMRES is more or less
-greater than 500. In fact this also depends on tow parameters: the number of
-iterations to stop GMRES and the number of iterations to perform the
-minimization.
+systems obtained with the previous matrices with a GMRES variant and with TSIRM
+are given. In the second column, it can be noticed that either GRMES or FGMRES
+(Flexible GMRES)~\cite{Saad:1993} is used to solve the linear system. According
+to the matrices, different preconditioner is used. With TSIRM, the same solver
+and the same preconditionner are used. This Table shows that TSIRM can
+drastically reduce the number of iterations to reach the convergence when the
+number of iterations for the normal GMRES is more or less greater than 500. In
+fact this also depends on tow parameters: the number of iterations to stop GMRES
+and the number of iterations to perform the minimization.
\begin{table}[htbp]
\hline
\end{tabular}
-\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (mg and sor) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12), time is expressed in seconds.}
+\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (mg and sor) with 25,000 components per core on Juqueen ($\epsilon_{tsirm}=1e-3$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$), time is expressed in seconds.}
\label{tab:03}
\end{center}
\end{table*}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
\hline
- nb. cores & threshold & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\
+ nb. cores & $\epsilon_{tsirm}$ & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\
\cline{3-8}
& & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline
2,048 & 8e-5 & 108.88 & 16,560 & 23.06 & 3,630 & 22.79 & 3,630 & 4.77 \\
\hline
\end{tabular}
-\caption{Comparison of FGMRES and TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 components per core on Curie (restart=30, s=12), time is expressed in seconds.}
+\caption{Comparison of FGMRES and TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 components per core on Curie ($max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$), time is expressed in seconds.}
\label{tab:04}
\end{center}
\end{table*}
\hline
\end{tabular}
-\caption{Comparison of FGMRES and TSIRM with FGMRES for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores (restart=30, s=12, threshold 5e-5), time is expressed in seconds.}
+\caption{Comparison of FGMRES and TSIRM with FGMRES for ex54 of Petsc (both with the MG preconditioner) with 204,919,225 components on Curie with different number of cores ($\epsilon_{tsirm}=5e-5$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$), time is expressed in seconds.}
\label{tab:05}
\end{center}
\end{table*}
\end{figure}
-Concerning the experiments some other remarks are interesting. We can tested
-other examples of PETSc (ex29, ex45, ex49). For all these 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.
+Concerning the experiments some other remarks are interesting.
+\begin{itemize}
+\item We can tested other examples of PETSc (ex29, ex45, ex49). For all these
+ 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
+ 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
+ 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
+ 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
+ iterations for example, the normal CG will be for more efficient. Some
+ restarted CG or CG variant versions exist and may be interested to study in
+ future works.
+\end{itemize}
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