\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja\IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}
\IEEEauthorblockA{\IEEEauthorrefmark{1} Femto-ST Institute, University of Franche-Comt\'e, France\\
Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
-\IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
-Email: lilia.ziane@inria.fr}
+\IEEEauthorblockA{\IEEEauthorrefmark{2} LTAS-Mécanique numérique non linéaire, University of Liege, Belgium\\ Email: l.zianekhodja@ulg.ac.be}
+%INRIA Bordeaux Sud-Ouest, France\\ Email: lilia.ziane@inria.fr}
}
% (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
so-called ``Krylov subspaces''. They consist in forming a basis of successive
matrix powers multiplied by an initial vector, which can be for instance the
residual. These methods use vectors orthogonality of the Krylov subspace basis
-in order to solve linear systems. The most known iterative Krylov subspace
+in order to solve linear systems. The best known iterative Krylov subspace
methods are conjugate gradient and GMRES ones (Generalized Minimal RESidual).
%%%*********************************************************
\section{Related works}
\label{sec:02}
-GMRES method is one of the most widely used iterative solvers chosen to deal with the sparsity and the large order of linear systems. It was initially developed by Saad \& al.~\cite{Saad86} to deal with non-symmetric and non-Hermitian problems, and indefinite symmetric problems too. The convergence of the restarted GMRES with preconditioning is faster and more stable than those of some other iterative solvers.
+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.
-The next two chapters explore a few methods which are considered currently to be among the
-most important iterative techniques available for solving large linear systems. These techniques
-are based on projection processes, both orthogonal and oblique, onto Krylov subspaces, which
-are subspaces spanned by vectors of the form p(A)v where p is a polynomial. In short, these
-techniques approximate A −1 b by p(A)b, where p is a “good” polynomial. This chapter covers
-methods derived from, or related to, the Arnoldi orthogonalization. The next chapter covers
-methods based on Lanczos biorthogonalization.
-
-Krylov subspace techniques have inceasingly been viewed as general purpose iterative methods, especially since the popularization of the preconditioning techniqes.
-
-Preconditioned Krylov-subspace iterations are a key ingredient in
-many modern linear solvers, including in solvers that employ support
-preconditioners.
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
-\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
+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
-nonsingular matrix, $x\in\mathbb{R}^n$ is the solution vector, and
-$b\in\mathbb{R}^n$ is the right-hand side. As explained previously,
-the algorithm is implemented as an
-inner-outer iteration solver based on iterative Krylov methods. The main
-key-points of the proposed solver are given in Algorithm~\ref{algo:01}.
-It can be summarized as follows: the
-inner solver is a Krylov based one. In order to accelerate its convergence, the
-outer solver periodically applies a least-squares minimization on the residuals computed by the inner one. %Tsolver which does not required to be changed.
+nonsingular matrix, $x\in\mathbb{R}^n$ is the solution vector, and
+$b\in\mathbb{R}^n$ is the right-hand side. As explained previously, the
+algorithm is implemented as an inner-outer iteration solver based on iterative
+Krylov methods. The main key-points of the proposed solver are given in
+Algorithm~\ref{algo:01}. It can be summarized as follows: the inner solver is a
+Krylov based one. In order to accelerate its convergence, the outer solver
+periodically applies a least-squares minimization on the residuals computed by
+the inner one.
At each outer iteration, the sparse linear system $Ax=b$ is partially solved
using only $m$ iterations of an iterative method, this latter being initialized
-with the last obtained approximation. GMRES method~\cite{Saad86}, or any of its
-variants, can potentially be used as inner solver. The current approximation of
-the Krylov method is then stored inside a $n \times s$ matrix $S$, which is
+with the last obtained approximation. The GMRES method~\cite{Saad86}, or any of
+its variants, can potentially be used as inner solver. The current approximation
+of the Krylov method is then stored inside a $n \times s$ matrix $S$, which is
composed by the $s$ last solutions that have been computed during the inner
iterations phase. In the remainder, the $i$-th column vector of $S$ will be
denoted by $S_i$.
At each $s$ iterations, another kind of minimization step is applied in order to
-compute a new solution $x$. For that, the previous residuals of $Ax=b$ are computed by
-the inner iterations with $(b-AS)$. The minimization of the residuals is obtained by
+compute a new solution $x$. For that, the previous residuals of $Ax=b$ are
+computed by the inner iterations with $(b-AS)$. The minimization of the
+residuals is obtained by
\begin{equation}
\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
\label{eq:01}
with $R=AS$. The new solution $x$ is then computed with $x=S\alpha$.
-In practice, $R$ is a dense rectangular matrix belonging in $\mathbb{R}^{n\times s}$,
-with $s\ll n$. In order to minimize~\eqref{eq:01}, a least-squares method such as
-CGLS ~\cite{Hestenes52} or LSQR~\cite{Paige82} is used. Remark that these methods are more
-appropriate than a single direct method in a parallel context.
+In practice, $R$ is a dense rectangular matrix belonging in $\mathbb{R}^{n\times
+ s}$, with $s\ll n$. In order to minimize~\eqref{eq:01}, a least-squares
+method such as CGLS ~\cite{Hestenes52} or LSQR~\cite{Paige82} is used. Remark
+that these methods are more appropriate than a single direct method in a
+parallel context. CGLS has recently been used to improve the performance of multisplitting algorithms \cite{cz15:ij}.
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}
-\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
+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
-columns in practice. As explained previously, at least two methods seem to be
-interesting to solve the least-squares minimization, CGLS and LSQR.
+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, the CGLS and the LSQR
+methods.
-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}
\end{algorithm}
-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.
-
+In each iteration of CGLS, there are 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. 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.
%%%*********************************************************
We can now claim that,
\begin{proposition}
\label{prop:saad}
-If $A$ is either a definite positive or a positive matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent.
+If $A$ is either a definite positive or a positive matrix and GMRES($m$) is used as a solver, then the TSIRM algorithm is convergent.
Furthermore, let $r_k$ be the
$k$-th residue of TSIRM, then
We will now prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$,
$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||$ when $A$ is positive, and $\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|$ when $A$ is positive definite.
-The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ that follows the inductive hypothesis due, to the results recalled above.
+The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ that follows the inductive hypothesis due to the results recalled above.
Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ in the positive case, and $\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_0\|$ in the definite positive one.
We will show that the statement holds too for $r_k$. Two situations can occur:
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{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.
+We have stopped the GMRES every 30
+iterations (\emph{i.e.}, $max\_iter_{kryl}=30$), which is the default
+setting of GMRES restart parameter. The parameter $s$ 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 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.
+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 before stopping GMRES
+and the number of iterations to perform the minimization.
\begin{table}[htbp]
\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 ksp part which is suited for
-scalable linear equations solvers:
+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
-chosen because they are scalable with many cores which is not the case of other
-problems that we have tested.
-
-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.
+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 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 Center in Germany. They belong with other similar architectures
+to 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,
+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
-present results with many preconditioners. In practise, 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}.
+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 further details on the
+preconditioners in PETSc, readers are 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 (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) 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*}
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 at 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
- \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex15_juqueen}
-\caption{Number of iterations per second with ex15 and the same parameters than in Table~\ref{tab:03} (weak scaling)}
+ \includegraphics[width=0.5\textwidth]{nb_iter_sec_ex15_juqueen}
+\caption{Number of iterations per second with ex15 and the same parameters as in Table~\ref{tab:03} (weak scaling)}
\label{fig:01}
\end{figure}
In Figure~\ref{fig:01}, the number of iterations per second corresponding to
Table~\ref{tab:03} is displayed. It can be noticed that the number of
-iterations per second of FMGRES is constant whereas it decreases with TSIRM with
-both preconditioners. This can be explained by the fact that when the number of
-cores increases the time for the least-squares minimization step also increases but, generally,
-when the number of cores increases, the number of iterations to reach the
-threshold also increases, and, in that case, TSIRM is more efficient to reduce
-the number of iterations. So, the overall benefit of using TSIRM is interesting.
+iterations per second of FMGRES is constant whereas it decreases with TSIRM with
+both preconditioners. This can be explained by the fact that when the number of
+cores increases, the time for the least-squares minimization step also increases
+but, generally, when the number of cores increases, the number of iterations to
+reach the threshold also increases, and, in that case, TSIRM is more efficient
+to reduce the number of iterations. So, the overall benefit of using TSIRM is
+interesting.
\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*}
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 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
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 example ex54 on the
+Curie architecture. So, in this case, the number of unknowns is fixed at
$204,919,225$ 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.
-
-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
+of two. The threshold is fixed at $5e-5$ and only the $mg$ preconditioner has
+been tested. Here again we can see that TSIRM is faster than FGMRES. The
+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 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
+important for TSIRM.
+
+In Figure~\ref{fig:02} we report the number of iterations per second for the
+experiments reported in Table~\ref{tab:05}. This figure highlights that the
+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-sqaure steps are not
+have previously explained, 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*}
\begin{figure}[htbp]
\centering
- \includegraphics[width=0.45\textwidth]{nb_iter_sec_ex54_curie}
-\caption{Number of iterations per second with ex54 and the same parameters than in Table~\ref{tab:05} (strong scaling)}
+ \includegraphics[width=0.5\textwidth]{nb_iter_sec_ex54_curie}
+\caption{Number of iterations per second with ex54 and the same parameters as in Table~\ref{tab:05} (strong scaling)}
\label{fig:02}
\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 have 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 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 decreasing 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 which is stopped and
+ restarted every 16 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 new 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.
%%%*********************************************************
\section*{Acknowledgment}
This paper is partially funded by the Labex ACTION program (contract
-ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resources
+ANR-11-LABX-01-01). We acknowledge PRACE for awarding us access to resources
Curie and Juqueen respectively based in France and Germany.
