% (should never be an issue)
Iterative methods have recently become more attractive than direct ones to solve
-very large sparse linear systems\cite{Saad2003}. They are more efficient in a
+very large sparse linear systems~\cite{Saad2003}. They are more efficient in a
parallel context, supporting thousands of cores, and they require less memory
and arithmetic operations than direct methods~\cite{bahicontascoutu}. This is
why new iterative methods are frequently proposed or adapted by researchers, and
%%%*********************************************************
\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 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}.
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 $||Ax_k-b||_2$.
+$error$, 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
Let us summarize the most important parameters of TSIRM:
\begin{itemize}
-\item $\epsilon_{tsirm}$: the threshold to stop the TSIRM method;
+\item $\epsilon_{tsirm}$: the threshold that stops the TSIRM method;
\item $max\_iter_{kryl}$: the maximum number of iterations for the Krylov method;
\item $s$: the number of outer iterations before applying the minimization step;
\item $max\_iter_{ls}$: the maximum number of iterations for the iterative least-squares method;
The parallelization of TSIRM relies on the parallelization of all its
parts. More precisely, except the least-squares step, all the other parts are
obvious to achieve out in parallel. In order to develop a parallel version of
-our code, we have chosen to use PETSc~\cite{petsc-web-page}. For
-line~\ref{algo:matrix_mul} the matrix-matrix multiplication is implemented and
-efficient since the matrix $A$ is sparse and since the matrix $S$ contains few
+our code, we have chosen to use PETSc~\cite{petsc-web-page}. In
+line~\ref{algo:matrix_mul}, the matrix-matrix multiplication is implemented and
+efficient since the matrix $A$ is sparse and the matrix $S$ contains few
columns in practice. As explained previously, at least two methods seem to be
interesting to solve the least-squares minimization, CGLS and LSQR.
In each iteration of CGLS, there is two matrix-vector multiplications and some
-classical operations: dot product, norm, multiplication and addition on
+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.
which concludes the induction and the proof.
\end{proof}
-%We can remark that, at each iterate, the residue of the TSIRM algorithm is lower
-%than the one of the GMRES method.
+Remark that a similar proposition can be formulated at each time
+the given solver satisfies an inequality of the form $||r_n|| \leqslant \mu^n ||r_0||$,
+with $|\mu|<1$. Furthermore, it is \emph{a priori} possible in some particular cases
+regarding $A$,
+that the proposed TSIRM converges while the GMRES($m$) does not.
%%%*********************************************************
%%%*********************************************************
\label{tab:01}
\end{center}
\end{table}
-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:
+Chosen parameters are detailed below.
+We have stopped the GMRES every 30
+iterations (\emph{i.e.}, $max\_iter_{kryl}=30$), which is the default
+setting of GMRES restart parameter. $s$, for its part, has been set to 8. CGLS
+ minimizes the least-squares problem with 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.
+$\epsilon_{tsirm}=1e-10$. These experiments have been performed on an Intel(R)
+Core(TM) i7-3630QM CPU @ 2.40GHz with the 3.5.1 version 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 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
+Experiments comparing
+a GMRES variant with TSIRM in the resolution of linear systems are given in Table~\ref{tab:02}.
+The second column describes whether GMRES or FGMRES has been used for linear systems solving.
+Different preconditioners have been used according to the matrices. With TSIRM, the same
+solver and the same preconditioner are used. This table shows that TSIRM can
+drastically reduce the number of iterations needed 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
+fact this also depends on two parameters: the number of iterations before stopping GMRES
and the number of iterations to perform the minimization.
\hline
\end{tabular}
-\caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.}
+\caption{Comparison between sequential standalone (F)GMRES and TSIRM with (F)GMRES (time in seconds).}
\label{tab:02}
\end{center}
\end{table}
In order to perform larger experiments, we have tested some example applications
-of PETSc. Those applications are available in the \emph{ksp} part which is
+of PETSc. These 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
+\item ex15 is an example that solves in parallel an operator using a finite
difference scheme. The diagonal is equal to 4 and 4 extra-diagonals
representing the neighbors in each directions are equal to -1. This example is
used in many physical phenomena, for example, heat and fluid flow, wave
propagation, etc.
-\item ex54 is another example based on 2D problem discretized with quadrilateral
- finite elements. For this example, the user can define the scaling of material
+\item ex54 is another example based on a 2D problem discretized with quadrilateral
+ finite elements. In this example, the user can define the scaling of material
coefficient in embedded circle called $\alpha$.
\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
+to read the codes available in the PETSc sources. These problems have been
chosen because they are scalable with many cores.
In the following larger experiments are described on two large scale
-architectures: Curie and Juqueen. Both these architectures are supercomputer
+architectures: Curie and Juqueen. Both these architectures are supercomputers
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
+Supercomputing Center in Germany. They belong 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.
+processors at 2.7 GHz with 2Gb memory by core. The Juqueen architecture,
+for its part, is
+composed by IBM PowerPC A2 at 1.6 GHz with 1Gb memory per core. Both those
+architectures are equipped with a dedicated high speed network.
In many situations, using preconditioners is essential in order to find the
solution of a linear system. There are many preconditioners available in PETSc.
-For parallel applications all the preconditioners based on matrix factorization
+However, 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
+preconditioners, but as it is not the subject of this paper, we will not
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}.
+multigrid (mg) and successive over-relaxation (sor). For further details on the
+preconditioners in PETSc, reader is referred to~\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 ($\epsilon_{tsirm}=1e-3$, $max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$), time is expressed in seconds.}
+\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioners (mg and sor) having 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*}
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.
+the legend of this table.
In Table~\ref{tab:04}, some experiments with example ex54 on the Curie
architecture are reported. For this application, we fixed $\alpha=0.6$. As it
-can be seen in that Table, the size of the problem has a strong influence on the
+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 necessary to reach the convergence. So Table~\ref{tab:04}
significant. Both can be good but it seems not possible to know in advance which
one will be the best.
-Table~\ref{tab:05} show a strong scaling experiment with the exemple ex54 on the
-Curie architecture. So in this case, the number of unknownws is fixed to
+Table~\ref{tab:05} shows a strong scaling experiment with the exemple ex54 on the
+Curie architecture. So in this case, the number of unknowns 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. Efficiency
+been tested. Here again we can see that TSIRM is faster than 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
+is greater than 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
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 second is more of less the same for FGMRES and TSIRM
+number of iterations per second is more or 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-squares steps are not
taken into account with TSIRM.
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, we also obtained similar gains 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
- 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
+ feature. More precisely, the solver must support to be stopped and restarted
+ without decrease its convergence. That is why with GMRES we stop it when it is
+ naturally restarted (\emph{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
+ iterations for example, the normal CG will be far more efficient. Some
+ restarted CG or CG variant versions exist and may be interesting to study in
future works.
\end{itemize}
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
-A novel two-stage iterative algorithm has been proposed in this article,
-in order to accelerate the convergence Krylov iterative methods.
+A novel two-stage iterative algorithm TSIRM has been proposed in this article,
+in order to accelerate the convergence of Krylov iterative methods.
Our TSIRM proposal acts as a merger between Krylov based solvers and
a least-squares minimization step.
The convergence of the method has been proven in some situations, while
For future work, the authors' intention is to investigate other kinds of
-matrices, problems, and inner solvers. The influence of all parameters must be
+matrices, problems, and inner solvers. In particular, the possibility
+to obtain a convergence of TSIRM in situations where the GMRES is divergent will be
+investigated. The influence of all parameters must be
tested too, while other methods to minimize the residuals must be regarded. The
-number of outer iterations to minimize should become adaptative to improve the
+number of outer iterations to minimize should become adaptive to improve the
overall performances of the proposal. Finally, this solver will be implemented
-inside PETSc. This would be very interesting because it would allow us to test
+inside PETSc, which would be of interest as it would allows us to test
all the non-linear examples and compare our algorithm with the other algorithm
implemented in PETSc.
