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).
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
+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 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, 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 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}.
+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
-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.
+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.
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
\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.
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}. 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.
+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 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
\end{algorithm}
-In each iteration of CGLS, there is two matrix-vector multiplications and some
+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
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:
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
+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
$\epsilon_{tsirm}=1e-10$. These experiments have been performed on an Intel(R)
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
+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
-of the PRACE initiative (Partnership for Advanced Computing in Europe), which
+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,
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, reader is referred to~\cite{petsc-web-page}.
+preconditioners in PETSc, readers are referred to~\cite{petsc-web-page}.
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
+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
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 $max\_iter_{kryl}*s$
+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)}
+\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.
significant. Both can be good but it seems not possible to know in advance which
one will be the best.
-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
+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 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 than one whereas the efficiency of TSIRM is lower than
-one. Nevertheless, the ratio of TSIRM with any version of the least-squares
+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
-experiments reported in Table~\ref{tab:05}. This Figure highlights that the
+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-squares steps are not
+have previously explained, the iterations of the least-squares steps are not
taken into account with TSIRM.
\begin{table*}[htbp]
\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)}
+\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.
\begin{itemize}
-\item We have tested other examples of PETSc (ex29, ex45, ex49). For all these
- 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 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 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 far more efficient. Some
- restarted CG or CG variant versions exist and may be interesting to study in
- future works.
+ 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}
%%%*********************************************************
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%%%*********************************************************
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-A novel two-stage iterative algorithm TSIRM has been proposed in this article,
+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.
