\algnewcommand\Output{\item[\algorithmicoutput]}
\newtheorem{proposition}{Proposition}
+\newtheorem{proof}{Proof}
\begin{document}
%
% use a multiple column layout for up to two different
% affiliations
-\author{\IEEEauthorblockN{Rapha\"el Couturier\IEEEauthorrefmark{1}, Lilia Ziane Khodja \IEEEauthorrefmark{2}, and Christophe Guyeux\IEEEauthorrefmark{1}}
+\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\\
Email: \{raphael.couturier,christophe.guyeux\}@univ-fcomte.fr}
\IEEEauthorblockA{\IEEEauthorrefmark{2} INRIA Bordeaux Sud-Ouest, France\\
\begin{algorithmic}[1]
\Input $A$ (sparse matrix), $b$ (right-hand side)
\Output $x$ (solution vector)\vspace{0.2cm}
- \State Set the initial guess $x^0$
+ \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 $x_k=Solve(A,b,x_{k-1},max\_iter_{kryl})$ \label{algo:solve}
\State retrieve error
- \State $S_{k \mod s}=x^k$ \label{algo:store}
+ \State $S_{k \mod s}=x_k$ \label{algo:store}
\If {$k \mod s=0$ {\bf and} error$>\epsilon_{kryl}$}
\State $R=AS$ \Comment{compute dense matrix} \label{algo:matrix_mul}
- \State $\alpha=Solve\_Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
- \State $x^k=S\alpha$ \Comment{compute new solution}
+ \State $\alpha=Least\_Squares(R,b,max\_iter_{ls})$ \label{algo:}
+ \State $x_k=S\alpha$ \Comment{compute new solution}
\EndIf
\EndFor
\end{algorithmic}
\begin{algorithmic}[1]
\Input $A$ (matrix), $b$ (right-hand side)
\Output $x$ (solution vector)\vspace{0.2cm}
- \State $r=b-Ax$
- \State $p=A'r$
- \State $s=p$
- \State $g=||s||^2_2$
- \For {$k=1,2,3,\ldots$ until convergence (g$<\epsilon_{ls}$)} \label{algo2:conv}
- \State $q=Ap$
- \State $\alpha=g/||q||^2_2$
- \State $x=x+alpha*p$
- \State $r=r-alpha*q$
- \State $s=A'*r$
- \State $g_{old}=g$
- \State $g=||s||^2_2$
- \State $\beta=g/g_{old}$
+ \State Let $x_0$ be an initial approximation
+ \State $r_0=b-Ax_0$
+ \State $p_1=A^Tr_0$
+ \State $s_0=p_1$
+ \State $\gamma=||s_0||^2_2$
+ \For {$k=1,2,3,\ldots$ until convergence ($\gamma<\epsilon_{ls}$)} \label{algo2:conv}
+ \State $q_k=Ap_k$
+ \State $\alpha_k=\gamma/||q_k||^2_2$
+ \State $x_k=x_{k-1}+\alpha_kp_k$
+ \State $r_k=r_{k-1}-\alpha_kq_k$
+ \State $s_k=A^Tr_k$
+ \State $\gamma_{old}=\gamma$
+ \State $\gamma=||s_k||^2_2$
+ \State $\beta_k=\gamma/\gamma_{old}$
+ \State $p_{k+1}=s_k+\beta_kp_k$
\EndFor
\end{algorithmic}
\label{algo:02}
\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
+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, then the TSIRM algorithm is convergent.
+\end{proposition}
+\begin{proof}
+\end{proof}
%%%*********************************************************
%%%*********************************************************
\begin{itemize}
\item ex15 is an example which 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 is equal to -1. This example is
+ 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...
+ 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
- coefficient in embedded circle, it is called $\alpha$.
+ coefficient in embedded circle called $\alpha$.
\end{itemize}
- For more technical details on these applications, interested reader are invited
- to read the codes available in the PETSc sources. Those problem have been
- chosen because they are scalable with many cores. We have tested other problem
- but they are not scalable with many cores.
+ 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...}\\
% that's all folks
\end{document}
-
