maj de la prop

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index a2a3e9df873435c7a5acf9ff42c8779a6b3eafbe..436909ae599c7cc81316f5d2bfd8bdce47fed693 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -364,7 +364,6 @@
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \newtheorem{proposition}{Proposition}
 \algnewcommand\Output{\item[\algorithmicoutput]}
 
 \newtheorem{proposition}{Proposition}

-\newtheorem{proof}{Proof}

 
 \begin{document}
 %
 
 \begin{document}
 %


@@ -649,15 +648,15 @@ appropriate than a single direct method in a parallel context.
 \begin{algorithmic}[1]
   \Input $A$ (sparse matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
 \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}
   \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 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}
     \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}
     \EndIf
   \EndFor
 \end{algorithmic}


@@ -670,8 +669,8 @@ called for a  maximum of $max\_iter_{kryl}$ iterations.  In practice, we  sugges
 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.}
 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  $S$. After  the
+$\epsilon_{tsirm}$).  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
 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


@@ -687,13 +686,13 @@ Let us summarize the most important parameters of TSIRM:
 \end{itemize}
 
 
 \end{itemize}
 
 

-The  parallelisation  of  TSIRM  relies   on  the  parallelization  of  all  its
+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
 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

-colums in  practice. As explained  previously, at least  two methods seem  to be
+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
 interesting to solve the least-squares minimization, CGLS and LSQR.
 
 In the following  we remind the CGLS algorithm. The LSQR  method follows more or


@@ -704,19 +703,21 @@ less the same principle but it takes more place, so we briefly explain the paral
 \begin{algorithmic}[1]
   \Input $A$ (matrix), $b$ (right-hand side)
   \Output $x$ (solution vector)\vspace{0.2cm}
 \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}
   \EndFor
 \end{algorithmic}
 \label{algo:02}


@@ -736,22 +737,47 @@ these operations are easy to implement in PETSc or similar environment.
 \label{sec:04}
 Let us recall the following result, see~\cite{Saad86}.
 \begin{proposition}
 \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}
 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 
+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}
 
 the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
 \end{proposition}
 

+

 We can now claim that,
 \begin{proposition}
 We can now claim that,
 \begin{proposition}

-If $A$ is a positive real matrix, then the TSIRM algorithm is convergent.
+If $A$ is a positive real matrix and GMRES($m$) is used as solver, then the TSIRM algorithm is convergent. Furthermore, we still have 
+\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}.

 \end{proposition}
 
 \begin{proof}
 \end{proposition}
 
 \begin{proof}

+Let $r_k = b-Ax_k$, where $x_k$ is the approximation of the solution after the
+$k$-th iterate of TSIRM.
+We will prove that $r_k \rightarrow 0$ when $k \rightarrow +\infty$.
+
+Each step of the TSIRM algorithm \\
+
+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 vectors $S$. So,\\
+$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
+
+$\begin{array}{ll}
+& = \min_{x \in span\left(S_{k-s}, S_{k-s+1}, \hdots, S_{k-1} \right)} ||b-AS\alpha ||_2\\
+& = \min_{x \in span\left(x_{k-s}, x_{k-s}+1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\
+& \leqslant \min_{x \in span\left( x_{k-1} \right)} ||b-Ax ||_2\\
+& \leqslant \min_{\lambda \in \mathbb{R}} ||b-\lambda Ax_{k-1} ||_2\\
+& \leqslant ||b-Ax_{k-1}||_2 .
+\end{array}$

 \end{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.
+

 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using PETSc}
 %%%*********************************************************
 %%%*********************************************************
 \section{Experiments using PETSc}


@@ -822,7 +848,7 @@ torso3             & fgmres / sor  & 37.70 & 565 & 34.97 & 510 \\
 \hline
 
 \end{tabular}
 \hline
 
 \end{tabular}

-\caption{Comparison of (F)GMRES and 2 stage (F)GMRES algorithms in sequential with some matrices, time is expressed in seconds.}
+\caption{Comparison of (F)GMRES and TSIRM with (F)GMRES in sequential with some matrices, time is expressed in seconds.}

 \label{tab:02}
 \end{center}
 \end{table}
 \label{tab:02}
 \end{center}
 \end{table}


@@ -837,22 +863,21 @@ scalable linear equations solvers:
 \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
 \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
   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
 \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}
 \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...}\\
 
 
 
 In the following larger experiments are described on two large scale architectures: Curie and Juqeen... {\bf description...}\\
 
 

-{\bf Description of preconditioners}
+{\bf Description of preconditioners}\\

 
 \begin{table*}[htbp]
 \begin{center}
 
 \begin{table*}[htbp]
 \begin{center}


@@ -873,27 +898,27 @@ In the following larger experiments are described on two large scale architectur
 \hline
 
 \end{tabular}
 \hline
 
 \end{tabular}

-\caption{Comparison of FGMRES and TSIRM with FGMRES for example ex15 of PETSc with two preconditioner (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) with 25,000 components per core on Juqueen (threshold 1e-3, restart=30, s=12),  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
 \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. Differents  number of cores
-are  studied rangin  from  2,048  upto 16,383.   Two  preconditioners have  been
-tested.   For those experiments,  the number  of components  (or unknown  of the
-problems)  per processor  is fixed  to 25,000,  also called  weak  scaling. This
+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.
 
 
 
 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 this Table, we  can notice that TSIRM is always faster  than FGMRES. The last
+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
 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  alsways  faster than
-FGMRES. For this example, the  multigrid preconditionner is faster than SOR. The
+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
 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


@@ -911,10 +936,10 @@ corresponds to 30*12, there are $max\_iter_{ls}$ which corresponds to 15.
 
 
 In  Figure~\ref{fig:01}, the number  of iterations  per second  corresponding to
 
 
 In  Figure~\ref{fig:01}, the number  of iterations  per second  corresponding to

-Table~\ref{tab:01}  is  displayed.   It  can  be  noticed  that  the  number  of
-iterations per second of FMGRES is  constant whereas it decrease with TSIRM with
-both preconditioner. This  can be explained by the fact that  when the number of
-core increases the time for the minimization step also increases but, generally,
+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.
 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.


@@ -929,7 +954,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting.
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|} 
 \hline
 

-  nb. cores & threshold   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} &  \multicolumn{2}{c|}{TSIRM LSQR} & best gain \\ 
+  nb. cores & threshold   & \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 \\
 \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 \\


@@ -942,7 +967,7 @@ the number of iterations. So, the overall benefit of using TSIRM is interesting.
 \hline
 
 \end{tabular}
 \hline
 
 \end{tabular}

-\caption{Comparison of FGMRES  and 2 stage FGMRES algorithms for ex54 of Petsc (both with the MG preconditioner) with 25000 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 (restart=30, s=12),  time is expressed in seconds.}

 \label{tab:04}
 \end{center}
 \end{table*}
 \label{tab:04}
 \end{center}
 \end{table*}


@@ -956,9 +981,9 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} 
 \hline
 
 \begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|} 
 \hline
 

-  nb. cores   & \multicolumn{2}{c|}{GMRES} & \multicolumn{2}{c|}{TSIRM CGLS} &  \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ 
+  nb. cores   & \multicolumn{2}{c|}{FGMRES} & \multicolumn{2}{c|}{TSIRM CGLS} &  \multicolumn{2}{c|}{TSIRM LSQR} & best gain & \multicolumn{3}{c|}{efficiency} \\ 

 \cline{2-7} \cline{9-11}
 \cline{2-7} \cline{9-11}

-                    & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. &   & GMRES & TS CGLS & TS LSQR\\\hline \hline
+                    & Time  & \# Iter.  & Time  & \# Iter. & Time  & \# Iter. &   & FGMRES & TS CGLS & TS LSQR\\\hline \hline

    512              & 3,969.69 & 33,120 & 709.57 & 5,790  & 622.76 & 5,070  & 6.37  &   1    &    1    &     1     \\
    1024             & 1,530.06  & 25,860 & 290.95 & 4,830  & 307.71 & 5,070 & 5.25  &  1.30  &    1.21  &   1.01     \\
    2048             & 919.62    & 31,470 & 237.52 & 8,040  & 194.22 & 6,510 & 4.73  & 1.08   &    .75   &   .80\\
    512              & 3,969.69 & 33,120 & 709.57 & 5,790  & 622.76 & 5,070  & 6.37  &   1    &    1    &     1     \\
    1024             & 1,530.06  & 25,860 & 290.95 & 4,830  & 307.71 & 5,070 & 5.25  &  1.30  &    1.21  &   1.01     \\
    2048             & 919.62    & 31,470 & 237.52 & 8,040  & 194.22 & 6,510 & 4.73  & 1.08   &    .75   &   .80\\


@@ -968,7 +993,7 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect
 \hline
 
 \end{tabular}
 \hline
 
 \end{tabular}

-\caption{Comparison of FGMRES  and 2 stage FGMRES algorithms 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, threshol 5e-5),  time is expressed in seconds.}
+\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.}

 \label{tab:05}
 \end{center}
 \end{table*}
 \label{tab:05}
 \end{center}
 \end{table*}


@@ -996,7 +1021,7 @@ In Table~\ref{tab:04}, some experiments with example ex54 on the Curie architect
 
 future plan : \\
 - study other kinds of matrices, problems, inner solvers\\
 
 future plan : \\
 - study other kinds of matrices, problems, inner solvers\\

-- test the influence of all the parameters\\
+- test the influence of all parameters\\

 - adaptative number of outer iterations to minimize\\
 - other methods to minimize the residuals?\\
 - implement our solver inside PETSc
 - adaptative number of outer iterations to minimize\\
 - other methods to minimize the residuals?\\
 - implement our solver inside PETSc


@@ -1011,7 +1036,7 @@ future plan : \\
 %%%*********************************************************
 \section*{Acknowledgment}
 This  paper  is   partially  funded  by  the  Labex   ACTION  program  (contract
 %%%*********************************************************
 \section*{Acknowledgment}
 This  paper  is   partially  funded  by  the  Labex   ACTION  program  (contract

-ANR-11-LABX-01-01).   We acknowledge PRACE  for awarding  us access  to resource
+ANR-11-LABX-01-01).   We acknowledge PRACE  for awarding  us access  to resources

 Curie and Juqueen respectively based in France and Germany.
 
 
 Curie and Juqueen respectively based in France and Germany.
 
 


@@ -1054,4 +1079,3 @@ Curie and Juqueen respectively based in France and Germany.
 
 % that's all folks
 \end{document}
 
 % that's all folks
 \end{document}

-