%%%*********************************************************
\section{Related works}
\label{sec:02}
-%Wherever Times is specified, Times Roman or Times New Roman may be used. If neither is available on your system, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc.
+Krylov subspace iteration methods have increasingly become useful and successful techniques for solving linear and nonlinear systems and eigenvalue problems, especially since the increase development of the preconditioners~\cite{Saad2003,Meijerink77}. One reason of the popularity of these methods is their generality, simplicity and efficiency to solve systems of equations arising from very large and complex problems. %A Krylov method is based on a projection process onto a Krylov subspace spanned by vectors and it forms a sequence of approximations by minimizing the residual over the subspace formed~\cite{}.
+
+GMRES is one of the most widely used Krylov iterative method for solving sparse and large linear systems. It is developed by Saad and al.~\cite{Saad86} as a generalized method to deal with unsymmetric and non-Hermitian problems, and indefinite symmetric problems too. In its original version called full GMRES, it minimizes the residual over the current Krylov subspace until convergence in at most $n$ iterations, where $n$ is the size of the sparse matrix. It should be noted that full GMRES is too expensive in the case of large matrices since the required orthogonalization process per iteration grows quadratically with the number of iterations. For that reason, in practice GMRES is restarted after each $m\ll n$ iterations to avoid the storage of a large orthonormal basis. However, the convergence behavior of the restarted GMRES, called GMRES($m$), in many cases depends quite critically on the value of $m$~\cite{Huang89}. Therefore in most cases, a preconditioning technique is applied to the restarted GMRES method in order to improve its convergence.
+
+In order to enhance the robustness of Krylov iterative solvers, some techniques have been proposed allowing the use of different preconditioners, if necessary, within the iteration instead of restarting. Those techniques may lead to considerable savings in CPU time and memory requirements. Van der Vorst in~\cite{Vorst94} has proposed variants of the GMRES algorithm in which a different preconditioner is applied in each iteration, so-called GMRESR family of nested methods. In fact, the GMRES method is effectively preconditioned with other iterative schemes (or GMRES itself), where the iterations of the GMRES method are called outer iterations while the iterations of the preconditioning process referred to as inner iterations. Saad in~\cite{Saad:1993} has proposed FGMRES which is another variant of the GMRES algorithm using a variable preconditioner. In FGMRES the search directions are preconditioned whereas in GMRESR the residuals are preconditioned. However in practice the good preconditioners are those based on direct methods, as ILU preconditioners, which are not easy to parallelize and suffer from the scalability problems on large clusters of thousands of cores.
+
+Recently, communication-avoiding methods have been developed to reduce the communication overheads in Krylov subspace iterative solvers. On modern computer architectures, communications between processors are much slower than floating-point arithmetic operations on a given processor. Communication-avoiding techniques reduce either communications between processors or data movements between levels of the memory hierarchy, by reformulating the communication-bound kernels (more frequently SpMV kernels) and the orthogonalization operations within the Krylov iterative solver. Different works have studied the communication-avoiding methods for multicore processors and multi-GPU machines~\cite{}.
+
+
%%%*********************************************************
%%%*********************************************************
\Input $A$ (sparse matrix), $b$ (right-hand side)
\Output $x$ (solution vector)\vspace{0.2cm}
\State Set the initial guess $x_0$
- \For {$k=1,2,3,\ldots$ until convergence (error$<\epsilon_{tsirm}$)} \label{algo:conv}
+ \For {$k=1,2,3,\ldots$ until convergence ($error<\epsilon_{tsirm}$)} \label{algo:conv}
\State $[x_k,error]=Solve(A,b,x_{k-1},max\_iter_{kryl})$ \label{algo:solve}
- \State $S_{k \mod s}=x_k$ \label{algo:store} \Comment{update column (k mod s) of S}
- \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
+ \State $S_{k \mod s}=x_k$ \label{algo:store} \Comment{update column ($k \mod s$) of $S$}
+ \If {$k \mod s=0$ {\bf and} $error>\epsilon_{kryl}$}
\State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
\State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
\State $x_k=S\alpha$ \Comment{compute new solution}
practice, this threshold must be much smaller than the convergence threshold of
the TSIRM algorithm (\emph{i.e.}, $\epsilon_{tsirm}$). We also consider that
after the call of the $Solve$ function, we obtain the vector $x_k$ and the error
-which is defined by $||Ax^k-b||_2$.
+which is defined by $||Ax_k-b||_2$.
Line~\ref{algo:store},
-$S_{k \mod s}=x^k$ consists in copying the solution $x_k$ into the column $k
+$S_{k \mod s}=x_k$ consists in copying the solution $x_k$ into the column $k
\mod s$ of $S$. After the minimization, the matrix $S$ is reused with the new
values of the residuals. To solve the minimization problem, an iterative method
is used. Two parameters are required for that: the maximum number of iterations
\section{Convergence results}
\label{sec:04}
-Let us recall the following result, see~\cite{Saad86} for further readings.
-\begin{proposition}
-\label{prop:saad}
-Suppose that $A$ is a positive real matrix with symmetric part $M$. Then the residual norm provided at the $m$-th step of GMRES satisfies:
-\begin{equation}
-||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
-\end{equation}
-where $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$, which proves
-the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
-\end{proposition}
We can now claim that,
\begin{proposition}
-If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore,
-let $r_k$ be the
+\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.
+
+Furthermore, let $r_k$ be the
$k$-th residue of TSIRM, then
-we still have:
+we have the following boundaries:
+\begin{itemize}
+\item when $A$ is positive:
\begin{equation}
||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0|| ,
\end{equation}
-where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}.
+where $M$ is the symmetric part of $A$, $\alpha = \lambda_{min}(M)^2$ and $\beta = \lambda_{max}(A^T A)$;
+\item when $A$ is positive definite:
+\begin{equation}
+\|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\|.
+\end{equation}
+\end{itemize}
+%In the general case, where A is not positive definite, we have
+%$\|r_n\| \le \inf_{p \in P_n} \|p(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)| \|r_0\|, .$
\end{proposition}
\begin{proof}
-We will prove by a mathematical induction that, for each $k \in \mathbb{N}^\ast$,
-$||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{mk}{2}} ||r_0||.$
+Let us first recall that the residue is under control when considering the GMRES algorithm on a positive definite matrix, and it is bounded as follows:
+\begin{equation*}
+\|r_k\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{k/2} \|r_0\| .
+\end{equation*}
+Additionally, when $A$ is a positive real matrix with symmetric part $M$, then the residual norm provided at the $m$-th step of GMRES satisfies:
+\begin{equation*}
+||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0|| ,
+\end{equation*}
+where $\alpha$ and $\beta$ are defined as in Proposition~\ref{prop:saad}, which proves
+the convergence of GMRES($m$) for all $m$ under such assumptions regarding $A$.
+These well-known results can be found, \emph{e.g.}, in~\cite{Saad86}.
-The base case is obvious, as for $k=1$, the TSIRM algorithm simply consists in applying GMRES($m$) once, leading to a new residual $r_1$ which follows the inductive hypothesis due to Proposition~\ref{prop:saad}.
+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.
-Suppose now that the claim holds for all $m=1, 2, \hdots, k-1$, that is, $\forall m \in \{1,2,\hdots, k-1\}$, $||r_m|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$.
+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:
\begin{itemize}
-\item If $k \mod m \neq 0$, then the TSIRM algorithm consists in executing GMRES once. In that case, we obtain $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ by the inductive hypothesis.
-\item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$, and a least squares resolution.
+\item If $k \not\equiv 0 ~(\textrm{mod}\ m)$, then the TSIRM algorithm consists in executing GMRES once. In that case and by using the inductive hypothesis, we obtain either $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ if $A$ is positive, or $\|r_k\| \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{m/2} \|r_{k-1}\|$ $\leqslant$ $\left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|$ in the positive definite case.
+\item Else, the TSIRM algorithm consists in two stages: a first GMRES($m$) execution leads to a temporary $x_k$ whose residue satisfies:
+\begin{itemize}
+\item $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||$ in the positive case,
+\item $\|r_k\| \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{m/2} \|r_{k-1}\|$ $\leqslant$ $\left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|$ in the positive definite one,
+\end{itemize}
+and a least squares resolution.
Let $\operatorname{span}(S) = \left \{ {\sum_{i=1}^k \lambda_i v_i \Big| k \in \mathbb{N}, v_i \in S, \lambda _i \in \mathbb{R}} \right \}$ be the linear span of a set of real vectors $S$. So,\\
$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k} ||_2\\
& \leqslant ||b-Ax_{k}||_2\\
& = ||r_k||_2\\
-& \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||,
+& \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{km}{2}} ||r_0||, \textrm{ if $A$ is positive,}\\
+& \leqslant \left( 1-\frac{\lambda_{\mathrm{min}}^2(1/2(A^T + A))}{ \lambda_{\mathrm{max}}(A^T A)} \right)^{km/2} \|r_{0}\|, \textrm{ if $A$ is}\\
+& \textrm{positive definite,}
\end{array}$
\end{itemize}
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.
+%We can remark that, at each iterate, the residue of the TSIRM algorithm is lower
+%than the one of the GMRES method.
%%%*********************************************************
%%%*********************************************************
\label{sec:05}
-In order to see the influence of our algorithm with only one processor, we first
-show a comparison with GMRES or FGMRES and our algorithm. In Table~\ref{tab:01},
-we show the matrices we have used and some of them characteristics. Those
-matrices are chosen from the Davis collection of the University of
-Florida~\cite{Dav97}. They are matrices arising in real-world applications. For
-all the matrices, the name, the field, the number of rows and the number of
-nonzero elements are given.
+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
+Florida~\cite{Dav97}.
\begin{table}[htbp]
\begin{center}
\label{tab:01}
\end{center}
\end{table}
-
-The following parameters have been chosen for our experiments. As by default
+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:
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
+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}.
+
-{\bf Description of preconditioners}\\
\begin{table*}[htbp]
\begin{center}
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. Two preconditioners have been
-tested: {\it mg} and {\it sor}. For those experiments, the number of components (or unknowns of the
+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
\end{table*}
-In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architecture are reported.
-
+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
+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
+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
+core. It can also be observed that the difference between CGLS and LSQR is not
+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
+$204,919,225$ 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
+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
+taken into account with TSIRM.
\begin{table*}[htbp]
\begin{center}
\label{fig:02}
\end{figure}
+
+Concerning the experiments some other remarks are interesting.
+\begin{itemize}
+\item We can tested other examples of PETSc (ex29, ex45, ex49). For all these
+ examples, we also obtained similar gain between GMRES and TSIRM but those
+ examples are not scalable with many cores. In general, we had some problems
+ with more than $4,096$ cores.
+\item We have tested many iterative solvers available in PETSc. In fast, it is
+ possible to use most of them with TSIRM. From our point of view, the condition
+ to use a solver inside TSIRM is that the solver must have a restart
+ feature. More precisely, the solver must support to be stoped and restarted
+ without decrease its converge. That is why with GMRES we stop it when it is
+ naturraly restarted (i.e. with $m$ the restart parameter). The Conjugate
+ Gradient (CG) and all its variants do not have ``restarted'' version in PETSc,
+ so they are not efficient. They will converge with TSIRM but not quickly
+ because if we compare a normal CG with a CG for which we stop it each 16
+ iterations for example, the normal CG will be for more efficient. Some
+ restarted CG or CG variant versions exist and may be interested to study in
+ future works.
+\end{itemize}
%%%*********************************************************
%%%*********************************************************
5 or 7 times faster than GMRES.
-For future work, the authors' intention is to investigate
-other kinds of matrices, problems, and inner solvers. 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 overall performances of the proposal.
-Finally, this solver will be implemented inside PETSc.
+For future work, the authors' intention is to investigate other kinds of
+matrices, problems, and inner solvers. 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
+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
+all the non-linear examples and compare our algorithm with the other algorithm
+implemented in PETSc.
% conference papers do not normally have an appendix
