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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, 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.
+
+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
+noticed 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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-\section{Two-stage iteration with least-squares residuals minimization algorithm}
+\section{TSIRM: Two-stage iteration with least-squares residuals minimization algorithm}
\label{sec:03}
A two-stage algorithm is proposed to solve large sparse linear systems of the
form $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is a sparse and square
\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}
we suggest to set this parameter equal to the restart number in the GMRES-like
method. Moreover, a tolerance threshold must be specified for the solver. In
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 $||Ax^k-b||_2$.
+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 $||Ax_k-b||_2$.
- Line~\ref{algo:store},
-$S_{k \mod s}=x^k$ 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
-and the threshold to stop the method.
+ Line~\ref{algo:store}, $S_{k \mod s}=x_k$ 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 ($max\_iter_{ls}$) and the threshold to stop
+ the method ($\epsilon_{ls}$).
Let us summarize the most important parameters of TSIRM:
\begin{itemize}
columns in practice. As explained previously, at least two methods seem to be
interesting to solve the least-squares minimization, CGLS and LSQR.
-In the following we remind the CGLS algorithm. The LSQR method follows more or
-less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
+In Algorithm~\ref{algo:02} we remind the CGLS algorithm. The LSQR method follows
+more or less the same principle but it takes more place, so we briefly explain
+the parallelization of CGLS which is similar to LSQR.
\begin{algorithm}[t]
\caption{CGLS}
In each iteration of CGLS, there is two matrix-vector multiplications and some
-classical operations: dot product, norm, multiplication and addition on vectors. All
-these operations are easy to implement in PETSc or similar environment.
-
+classical operations: dot product, norm, multiplication and addition on
+vectors. All these operations are easy to implement in PETSc or similar
+environment. It should be noticed that LSQR follows the same principle, it is a
+little bit longer but it performs more or less the same operations.
%%%*********************************************************
\label{sec:05}
-In order to see the behavior of the proposal when considering only one processor, a first
-comparison with GMRES or FGMRES and the new algorithm detailed previously has been experimented.
-Matrices that have been used with their characteristics (names, fields, rows, and nonzero coefficients) are detailed in
-Table~\ref{tab:01}. These latter, which are real-world applications matrices,
-have been extracted
- from the Davis collection, University of
+In order to see the behavior of our approach when considering only one processor,
+a first comparison with GMRES or FGMRES and the new algorithm detailed
+previously has been experimented. Matrices that have been used with their
+characteristics (names, fields, rows, and nonzero coefficients) are detailed in
+Table~\ref{tab:01}. These latter, which are real-world applications matrices,
+have been extracted from the Davis collection, University of
Florida~\cite{Dav97}.
\begin{table}[htbp]
\label{tab:01}
\end{center}
\end{table}
-Chosen parameters are detailed below.
-%The following parameters have been chosen for our experiments.
-As by default
-the restart of GMRES is performed every 30 iterations, we have chosen to stop
-the GMRES every 30 iterations (\emph{i.e.} $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is
-chosen to minimize the least-squares problem with the following parameters:
+Chosen parameters are detailed below. As by default the restart of GMRES is
+performed every 30 iterations, we have chosen to stop the GMRES every 30
+iterations (\emph{i.e.} $max\_iter_{kryl}=30$). $s$ is set to 8. CGLS is chosen
+to minimize the least-squares problem with the following parameters:
$\epsilon_{ls}=1e-40$ and $max\_iter_{ls}=20$. The external precision is set to
$\epsilon_{tsirm}=1e-10$. Those experiments have been performed on a Intel(R)
Core(TM) i7-3630QM CPU @ 2.40GHz with the version 3.5.1 of PETSc.
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 GMRES or FGMRES
+(Flexible GMRES)~\cite{Saad:1993} is used to solve the linear system. According
+to the matrices, different preconditioners are 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 two parameters: the number of iterations to stop GMRES
+and the number of iterations to perform the minimization.
\begin{table}[htbp]
In order to perform larger experiments, we have tested some example applications
-of PETSc. Those applications are available in the ksp part which is suited for
-scalable linear equations solvers:
+of PETSc. Those applications are available in the \emph{ksp} part which is
+suited for scalable linear equations solvers:
\begin{itemize}
\item ex15 is an example which solves in parallel an operator using a finite
difference scheme. The diagonal is equal to 4 and 4 extra-diagonals
\end{itemize}
For more technical details on these applications, interested readers are invited
to read the codes available in the PETSc sources. Those problems have been
-chosen because they are scalable with many cores which is not the case of other
-problems that we have tested.
+chosen because they are scalable with many cores.
In the following larger experiments are described on two large scale
-architectures: Curie and Juqeen. Both these architectures are supercomputer
-composed of 80,640 cores for Curie and 458,752 cores for Juqueen. Those machines
-are respectively hosted by GENCI in France and Jülich Supercomputing Centre in
-Germany. They belongs with other similar architectures of the PRACE initiative (
-Partnership for Advanced Computing in Europe) which aims at proposing high
-performance supercomputing architecture to enhance research in Europe. The Curie
-architecture is composed of Intel E5-2680 processors at 2.7 GHz with 2Gb memory
-by core. The Juqueen architecture is composed of IBM PowerPC A2 at 1.6 GHz with
-1Gb memory per core. Both those architecture are equiped with a dedicated high
-speed network.
+architectures: Curie and Juqueen. Both these architectures are supercomputer
+respectively composed of 80,640 cores for Curie and 458,752 cores for
+Juqueen. Those machines are respectively hosted by GENCI in France and Jülich
+Supercomputing Centre in Germany. They belongs with other similar architectures
+of the PRACE initiative (Partnership for Advanced Computing in Europe) which
+aims at proposing high performance supercomputing architecture to enhance
+research in Europe. The Curie architecture is composed of Intel E5-2680
+processors at 2.7 GHz with 2Gb memory by core. The Juqueen architecture is
+composed of IBM PowerPC A2 at 1.6 GHz with 1Gb memory per core. Both those
+architecture are equiped with a dedicated high speed network.
In many situations, using preconditioners is essential in order to find the
For parallel applications all the preconditioners based on matrix factorization
are not available. In our experiments, we have tested different kinds of
preconditioners, however as it is not the subject of this paper, we will not
-present results with many preconditioners. In practise, we have chosen to use a
+present results with many preconditioners. In practice, we have chosen to use a
multigrid (mg) and successive over-relaxation (sor). For more details on the
preconditioner in PETSc please consult~\cite{petsc-web-page}.
\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*}
Table~\ref{tab:03} shows the execution times and the number of iterations of
example ex15 of PETSc on the Juqueen architecture. Different numbers of cores
-are studied ranging from 2,048 up-to 16,383 with the two preconditioners {\it mg} and {\it sor}. For those experiments, the number of components (or unknowns of the
-problems) per core is fixed to 25,000, also called weak scaling. This
-number can seem relatively small. In fact, for some applications that need a lot
-of memory, the number of components per processor requires sometimes to be
-small.
-
-
-
-In Table~\ref{tab:03}, we can notice that TSIRM is always faster than FGMRES. The last
-column shows the ratio between FGMRES and the best version of TSIRM according to
-the minimization procedure: CGLS or LSQR. Even if we have computed the worst
-case between CGLS and LSQR, it is clear that TSIRM is always faster than
-FGMRES. For this example, the multigrid preconditioner is faster than SOR. The
-gain between TSIRM and FGMRES is more or less similar for the two
+are studied ranging from 2,048 up-to 16,383 with the two preconditioners {\it
+ mg} and {\it sor}. For those experiments, the number of components (or
+unknowns of the problems) per core is fixed to 25,000, also called weak
+scaling. This number can seem relatively small. In fact, for some applications
+that need a lot of memory, the number of components per processor requires
+sometimes to be small. Other parameters for this application are described in
+the legend of this Table.
+
+
+
+In Table~\ref{tab:03}, we can notice that TSIRM is always faster than
+FGMRES. The last column shows the ratio between FGMRES and the best version of
+TSIRM according to the minimization procedure: CGLS or LSQR. Even if we have
+computed the worst case between CGLS and LSQR, it is clear that TSIRM is always
+faster than FGMRES. For this example, the multigrid preconditioner is faster
+than SOR. The gain between TSIRM and FGMRES is more or less similar for the two
preconditioners. Looking at the number of iterations to reach the convergence,
it is obvious that TSIRM allows the reduction of the number of iterations. It
should be noticed that for TSIRM, in those experiments, only the iterations of
the Krylov solver are taken into account. Iterations of CGLS or LSQR were not
-recorded but they are time-consuming. In general each $max\_iter_{kryl}*s$ which
-corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.
+recorded but they are time-consuming. In general each $max\_iter_{kryl}*s$
+iterations which corresponds to 30*12, there are $max\_iter_{ls}$ iterations for
+the least-squares method which corresponds to 15.
\begin{figure}[htbp]
\centering
\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*}
can be seen in that Table, the size of the problem has a strong influence on the
number of iterations to reach the convergence. That is why we have preferred to
change the threshold. If we set it to $1e-3$ as with the previous application,
-only one iteration is necessray to reach the convergence. So Table~\ref{tab:04}
-shows the results of differents executions with differents number of cores and
-differents thresholds. As with the previous example, we can observe that TSIRM
-is faster than FGMRES. The ratio greatly depends on the number of iterations for
+only one iteration is necessary to reach the convergence. So Table~\ref{tab:04}
+shows the results of different executions with different number of cores and
+different thresholds. As with the previous example, we can observe that TSIRM is
+faster than FGMRES. The ratio greatly depends on the number of iterations for
FMGRES to reach the threshold. The greater the number of iterations to reach the
convergence is, the better the ratio between our algorithm and FMGRES is. This
experiment is also a weak scaling with approximately $25,000$ components per
Curie architecture. So in this case, the number of unknownws is fixed to
$204,919,225$ and the number of cores ranges from $512$ to $8192$ with the power
of two. The threshold is fixed to $5e-5$ and only the $mg$ preconditioner has
-been tested. Here again we can see that TSIRM is faster that FGMRES. Efficiecy
-of each algorithms is reported. It can be noticed that FGMRES is more efficient
-than TSIRM except with $8,192$ cores and that its efficiency is greater that one
-whereas the efficiency of TSIRM is lower than one. Nevertheless, the ratio of
-TSIRM with any version of the least-squares method is always faster. With
-$8,192$ cores when the number of iterations is far more important for FGMRES, we
-can see that it is only slightly more important for TSIRM.
+been tested. Here again we can see that TSIRM is faster that FGMRES. Efficiency
+of each algorithm is reported. It can be noticed that the efficiency of FGMRES
+is better than the TSIRM one except with $8,192$ cores and that its efficiency
+is greater that one whereas the efficiency of TSIRM is lower than
+one. Nevertheless, the ratio of TSIRM with any version of the least-squares
+method is always faster. With $8,192$ cores when the number of iterations is
+far more important for FGMRES, we can see that it is only slightly more
+important for TSIRM.
In Figure~\ref{fig:02} we report the number of iterations per second for
-experiments reported in Table~\ref{tab:05}. This Figure highlights that the
-number of iterations per seconds is more of less the same for FGMRES and TSIRM
+experiments reported in Table~\ref{tab:05}. This Figure highlights that the
+number of iterations per second is more of less the same for FGMRES and TSIRM
with a little advantage for FGMRES. It can be explained by the fact that, as we
-have previously explained, that the iterations of the least-sqaure steps are not
+have previously explained, that the iterations of the least-squares steps are not
taken into account with TSIRM.
\begin{table*}[htbp]
\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 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 have 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 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 stopped and restarted
+ without decrease its converge. That is why with GMRES we stop it when it is
+ 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
+ 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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