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\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.
+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.
+
+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.
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\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
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. 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.
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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.
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