% affiliations
\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 Comte, France\\
+\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}
\end{abstract}
\begin{IEEEkeywords}
-Iterative Krylov methods; sparse linear systems; residual minimization; PETSc; %à voir...
+Iterative Krylov methods; sparse linear systems; two stage iteration; least-squares residual minimization; PETSc
\end{IEEEkeywords}
% You must have at least 2 lines in the paragraph with the drop letter
% (should never be an issue)
-Iterative methods have recently become more attractive than direct ones to solve very large
-sparse linear systems. They are more efficient in a parallel
-context, supporting thousands of cores, and they require less memory and arithmetic
-operations than direct methods. This is why new iterative methods are frequently
-proposed or adapted by researchers, and the increasing need to solve very large sparse
-linear systems has triggered the development of such efficient iterative techniques
-suitable for parallel processing.
-
-Most of the successful iterative methods currently available are based on 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 methods are conjugate
-gradient and GMRES ones (Generalized Minimal RESidual).
-
-
-However, iterative methods suffer from scalability problems on parallel
-computing platforms with many processors, due to their need of reduction
-operations, and to collective communications to achive matrix-vector
+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
+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
+the increasing need to solve very large sparse linear systems has triggered the
+development of such efficient iterative techniques suitable for parallel
+processing.
+
+Most of the successful iterative methods currently available are based on
+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
+methods are conjugate gradient and GMRES ones (Generalized Minimal RESidual).
+
+
+However, iterative methods suffer from scalability problems on parallel
+computing platforms with many processors, due to their need of reduction
+operations, and to collective communications to achieve matrix-vector
multiplications. The communications on large clusters with thousands of cores
-and large sizes of messages can significantly affect the performances of these
-iterative methods. As a consequence, Krylov subspace iteration methods are often used
-with preconditioners in practice, to increase their convergence and accelerate their
-performances. However, most of the good preconditioners are not scalable on
-large clusters.
-
-In this research work, a two-stage algorithm based on two nested iterations
-called inner-outer iterations is proposed. This algorithm consists in solving the sparse
-linear system iteratively with a small number of inner iterations, and restarting
-the outer step with a new solution minimizing some error functions over some
-previous residuals. This algorithm is iterative and easy to parallelize on large
-clusters. Furthermore, the minimization technique improves its convergence and
-performances.
+and large sizes of messages can significantly affect the performances of these
+iterative methods. As a consequence, Krylov subspace iteration methods are often
+used with preconditioners in practice, to increase their convergence and
+accelerate their performances. However, most of the good preconditioners are
+not scalable on large clusters.
+
+In this research work, a two-stage algorithm based on two nested iterations
+called inner-outer iterations is proposed. This algorithm consists in solving
+the sparse linear system iteratively with a small number of inner iterations,
+and restarting the outer step with a new solution minimizing some error
+functions over some previous residuals. For further information on two-stage
+iteration methods, interested readers are invited to
+consult~\cite{Nichols:1973:CTS}. Two-stage algorithms are easy to parallelize on
+large clusters. Furthermore, the least-squares minimization technique improves
+its convergence and performances.
The present article is organized as follows. Related works are presented in
Section~\ref{sec:02}. Section~\ref{sec:03} details the two-stage algorithm using
%%%*********************************************************
\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 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
+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}.
+
+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.
+
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
-\section{Two-stage iteration with least-squares residuals minimization algorithm}
+\section{TSIRM: Two-stage iteration with least-squares residuals minimization algorithm}
\label{sec:03}
A two-stage algorithm is proposed to solve large sparse linear systems of the
form $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is a sparse and square
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.
-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
-best known approximation previously obtained.
-GMRES method~\cite{Saad86}, or any of its variants, can be used for instance as an
-inner solver. The current approximation of the Krylov method is then stored inside a matrix
-$S$ composed by the successive solutions that are computed during inner iterations.
+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
+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, the minimization step is applied in order to
+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
\begin{equation}
\underset{\alpha\in\mathbb{R}^{s}}{min}\|b-R\alpha\|_2
\label{eq:01}
\end{equation}
-with $R=AS$. Then the new solution $x$ is computed with $x=S\alpha$.
+with $R=AS$. The new solution $x$ is then computed with $x=S\alpha$.
In practice, $R$ is a dense rectangular matrix belonging in $\mathbb{R}^{n\times s}$,
\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}
- \State $x_k=Solve(A,b,x_{k-1},max\_iter_{kryl})$ \label{algo:solve}
- \State retrieve error
- \State $S_{k \mod s}=x_k$ \label{algo:store}
- \If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
+ \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 $R=AS$ \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}
\label{algo:01}
\end{algorithm}
-Algorithm~\ref{algo:01} summarizes the principle of our method. The outer
-iteration is inside the for loop. Line~\ref{algo:solve}, the Krylov method is
-called for a maximum of $max\_iter_{kryl}$ iterations. In practice, we suggest to set this parameter
-equals to the restart number of 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.}
-$\epsilon_{tsirm}$). Line~\ref{algo:store}, $S_{k \mod s}=x^k$ consists in copying the
-solution $x_k$ into the column $k \mod s$ of the matrix $S$, where $S$ is a matrix of size $n\times s$ whose column vector $i$ is denoted by $S_i$. 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.
+Algorithm~\ref{algo:01} summarizes the principle of the proposed method. The
+outer iteration is inside the \emph{for} loop. Line~\ref{algo:solve}, the Krylov
+method is called for a maximum of $max\_iter_{kryl}$ iterations. In practice,
+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 $||Ax_k-b||_2$.
+
+ 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
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 the following we remind the CGLS algorithm. The LSQR method follows more or
-less the same principle but it takes more place, so we briefly explain the parallelization of CGLS which is similar to LSQR.
+In Algorithm~\ref{algo:02} we remind the CGLS algorithm. The LSQR method follows
+more or less the same principle but it takes more place, so we briefly explain
+the parallelization of CGLS which is similar to LSQR.
\begin{algorithm}[t]
\caption{CGLS}
In each iteration of CGLS, there is two matrix-vector multiplications and some
-classical operations: dot product, norm, multiplication and addition on vectors. All
-these operations are easy to implement in PETSc or similar environment.
-
+classical operations: dot product, norm, multiplication, and addition on
+vectors. All these operations are easy to implement in PETSc or similar
+environment. It should be noticed that LSQR follows the same principle, it is a
+little bit longer but it performs more or less the same operations.
%%%*********************************************************
\section{Convergence results}
\label{sec:04}
-Let us recall the following result, see~\cite{Saad86}.
-\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 $||r_k|| \leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_{k-1}||\leqslant \left(1-\dfrac{\alpha}{\beta}\right)^{\frac{m}{2}} ||r_0||$.
-
-\item Else, 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,\\
+\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,\\
$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
$\begin{array}{ll}
-& = \min_{x \in span\left(S_{k-s}, S_{k-s+1}, \hdots, S_{k-1} \right)} ||b-AS\alpha ||_2\\
-& = \min_{x \in span\left(x_{k-s}, x_{k-s}+1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\
-& \leqslant \min_{x \in span\left( x_{k-1} \right)} ||b-Ax ||_2\\
-& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k-1} ||_2\\
-& \leqslant ||b-Ax_{k-1}||_2 .
+& = \min_{x \in span\left(S_{k-s+1}, S_{k-s+2}, \hdots, S_{k} \right)} ||b-AS\alpha ||_2\\
+& = \min_{x \in span\left(x_{k-s+1}, x_{k-s}+2, \hdots, x_{k} \right)} ||b-AS\alpha ||_2\\
+& \leqslant \min_{x \in span\left( x_{k} \right)} ||b-Ax ||_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||, \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.
+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 influence of our algorithm with only one processor, we first
-show a comparison with the standard version of GMRES and our algorithm. In
-Table~\ref{tab:01}, we show the matrices we have used and some of them
-characteristics. 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 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]
\begin{center}
\label{tab:01}
\end{center}
\end{table}
-
-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. $s$, 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 out 2
-stage algorithm are given. In the second column, it can be noticed that either
-gmres or fgmres 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:
+In order to perform larger experiments, we have tested some example applications
+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... {\bf description...}\\
+For more technical details on these applications, interested readers are invited
+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
+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,
+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.
+However, for parallel applications, all the preconditioners based on matrix factorization
+are not available. In our experiments, we have tested different kinds of
+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}.
-{\bf Description of preconditioners}\\
\begin{table*}[htbp]
\begin{center}
\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. Two preconditioners have been
-tested: {\it mg} and {\it sor}. For those experiments, the number of components (or unknowns of the
-problems) per core is fixed to 25,000, also called weak scaling. This
-number can seem relatively small. In fact, for some applications that need a lot
-of memory, the number of components per processor requires sometimes to be
-small.
-
-
-
-In Table~\ref{tab:03}, we can notice that TSIRM is always faster than FGMRES. The last
-column shows the ratio between FGMRES and the best version of TSIRM according to
-the minimization procedure: CGLS or LSQR. Even if we have computed the worst
-case between CGLS and LSQR, it is clear that TSIRM is always faster than
-FGMRES. For this example, the multigrid preconditioner is faster than SOR. The
-gain between TSIRM and FGMRES is more or less similar for the two
+are studied ranging from 2,048 up-to 16,383 with the two preconditioners {\it
+ mg} and {\it sor}. For those experiments, the number of components (or
+unknowns of the problems) per core is fixed to 25,000, also called weak
+scaling. This number can seem relatively small. In fact, for some applications
+that need a lot of memory, the number of components per processor requires
+sometimes to be small. Other parameters for this application are described in
+the legend of this table.
+
+
+
+In Table~\ref{tab:03}, we can notice that TSIRM is always faster than
+FGMRES. The last column shows the ratio between FGMRES and the best version of
+TSIRM according to the minimization procedure: CGLS or LSQR. Even if we have
+computed the worst case between CGLS and LSQR, it is clear that TSIRM is always
+faster than FGMRES. For this example, the multigrid preconditioner is faster
+than SOR. The gain between TSIRM and FGMRES is more or less similar for the two
preconditioners. Looking at the number of iterations to reach the convergence,
it is obvious that TSIRM allows the reduction of the number of iterations. It
should be noticed that for TSIRM, in those experiments, only the iterations of
the Krylov solver are taken into account. Iterations of CGLS or LSQR were not
-recorded but they are time-consuming. In general each $max\_iter_{kryl}*s$ which
-corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.
+recorded but they are time-consuming. In general each $max\_iter_{kryl}*s$
+iterations which corresponds to 30*12, there are $max\_iter_{ls}$ iterations for
+the least-squares method which corresponds to 15.
\begin{figure}[htbp]
\centering
\begin{tabular}{|r|r|r|r|r|r|r|r|r|}
\hline
- nb. cores & threshold & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\
+ nb. cores & $\epsilon_{tsirm}$ & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} & \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\
\cline{3-8}
& & Time & \# Iter. & Time & \# Iter. & Time & \# Iter. & \\\hline \hline
2,048 & 8e-5 & 108.88 & 16,560 & 23.06 & 3,630 & 22.79 & 3,630 & 4.77 \\
\hline
\end{tabular}
-\caption{Comparison of FGMRES and TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 components per core on Curie (restart=30, s=12), time is expressed in seconds.}
+\caption{Comparison of FGMRES and TSIRM with FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25,000 components per core on Curie ($max\_iter_{kryl}=30$, $s=12$, $max\_iter_{ls}=15$, $\epsilon_{ls}=1e-40$), time is expressed in seconds.}
\label{tab:04}
\end{center}
\end{table*}
-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 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
+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} 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 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
+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 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.
\begin{table*}[htbp]
\begin{center}
\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*}
\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 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.
+\end{itemize}
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
%%%*********************************************************
-
-future plan : \\
-- study other kinds of matrices, problems, inner solvers\\
-- test the influence of all parameters\\
-- adaptative number of outer iterations to minimize\\
-- other methods to minimize the residuals?\\
-- implement our solver inside PETSc
+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
+experiments up to 16,394 cores have been led to verify that TSIRM runs
+5 or 7 times faster than GMRES.
+
+
+For future work, the authors' intention is to investigate other kinds of
+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 adaptive to improve the
+overall performances of the proposal. Finally, this solver will be implemented
+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.
% conference papers do not normally have an appendix
