Avancées dans la preuve

[GMRES2stage.git] / paper.tex

diff --git a/paper.tex b/paper.tex
index e3d19ec0d604e10229be70a5643a4b4c08aeb92b..ceffa3d7903def8c6c51d0395e382a9487e67e2a 100644 (file)
--- a/paper.tex
+++ b/paper.tex
@@ -425,16 +425,17 @@ Email: lilia.ziane@inria.fr}
 
 
 \begin{abstract}
 
 
 \begin{abstract}

-In  this article,  a  two-stage  iterative algorithm is proposed to improve  the
-convergence of Krylov based iterative methods,  typically those of GMRES variants. The
-principle of  the proposed approach  is to  build an external  iteration over  the Krylov
-method, and to  frequently store its current  residual   (at  each
-GMRES restart for instance). After a given number of outer iterations, a minimization
-step  is applied  on the  matrix composed by the  saved residuals,  in  order to
-compute a better solution and to make  new iterations if required.  It is proven that
-the proposal has  the same convergence properties than the  inner embedded method itself. 
-Experiments using up  to 16,394 cores also show that the proposed algorithm
-runs around 5 or 7 times faster than GMRES.
+In  this article, a  two-stage iterative  algorithm is  proposed to  improve the
+convergence  of  Krylov  based  iterative  methods,  typically  those  of  GMRES
+variants.  The  principle of  the  proposed approach  is  to  build an  external
+iteration over the  Krylov method, and to frequently  store its current residual
+(at each GMRES restart for instance).  After a given number of outer iterations,
+a least-squares minimization step is applied on the matrix composed by the saved
+residuals, in order  to compute a better solution and to  make new iterations if
+required.  It  is proven that the  proposal has the  same convergence properties
+than the  inner embedded  method itself.  Experiments  using up to  16,394 cores
+also  show that the  proposed algorithm  runs around  5 or  7 times  faster than
+GMRES.

 \end{abstract}
 
 \begin{IEEEkeywords}
 \end{abstract}
 
 \begin{IEEEkeywords}


@@ -647,15 +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}


@@ -702,19 +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}


@@ -743,6 +746,26 @@ the convergence of GMRES($m$) for all $m$ under that assumption regarding $A$.
 \end{proposition}
 
 
 \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.
+\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 \\
+$\min_{\alpha \in \mathbb{R}^s} ||b-R\alpha ||_2 = \min_{\alpha \in \mathbb{R}^s} ||b-AS\alpha ||_2$
+
+$\begin{array}{ll}
+& = \min_{x \in Vect\left(x_0, x_1, \hdots, x_{k-1} \right)} ||b-AS\alpha ||_2\\
+& \leqslant \min_{x \in Vect\left( S_{k-1} \right)} ||b-Ax ||_2\\
+& \leqslant ||b-Ax_{k-1}||
+\end{array}$
+\end{proof}
+

 
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@@ -829,22 +852,21 @@ 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}


@@ -865,15 +887,15 @@ 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
+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 processor  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
 problems)  per processor  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


@@ -881,11 +903,11 @@ 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


@@ -903,10 +925,10 @@ 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.


@@ -1047,4 +1069,3 @@ Curie and Juqueen respectively based in France and Germany.
 % that's all folks
 \end{document}
 
 % that's all folks
 \end{document}
 

-