correct english

[Krylov_multi.git] / krylov_multi_reviewed.tex

diff --git a/krylov_multi_reviewed.tex b/krylov_multi_reviewed.tex
index f12f3ae7547bafd66a607802d9cc8622434a2207..02d1acaaa59d40a61b4227935bf1475445990db9 100644 (file)
--- a/krylov_multi_reviewed.tex
+++ b/krylov_multi_reviewed.tex
@@ -98,8 +98,15 @@ process can drastically improve the convergence.\\
 
 %%% AJOUTE************************
 %%%*******************************
 
 %%% AJOUTE************************
 %%%*******************************

-\noindent {\bf Contributions:}\\ 
-In this work we develop a new parallel two-stage algorithm for large-scale clusters. Our objective is to mix between Krylov based iterative methods and the multisplitting method to improve the scalability. In fact Krylov subspace methods are well-known for their good convergence compared to other iterative methods. So our main contribution is to use the multisplitting method which splits the problem to solve into different blocks in order to reduce the large amount of communications and, to implement both inner and outer iterations as Krylov subspace iterations improving the convergence of the multisplitting algorithm.\\
+\noindent  {\bf  Contributions:}\\  In  this  work we  develop  a  new  parallel
+two-stage algorithm for large-scale clusters.   Our objective is to create a mix
+between Krylov based iterative methods  and the multisplitting method to improve
+scalability.   In fact  Krylov subspace  methods are  well-known for  their good
+convergence compared to  other iterative methods.  So, our  main contribution is
+to  use  the  multisplitting method  which  splits  the  problem to  solve  into
+different blocks in  order to reduce the large amount  of communications and, to
+implement both inner and outer iterations as Krylov subspace iterations in order
+to improve the convergence of the multisplitting algorithm.\\

 %%%*******************************
 %%%*******************************
 
 %%%*******************************
 %%%*******************************
 


@@ -183,7 +190,7 @@ In the case where the diagonal weighting matrices $E_\ell$ have only zero and on
 
 %%% MODIFIE ************************
 %%%*********************************
 
 %%% MODIFIE ************************
 %%%*********************************

-Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce the communications. Let the computing platform be composed of $L$ blocks of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting to the linear system as follows
+Let $Ax=b$ be a given large and sparse linear system of $n$ equations where $A\in\mathbb{R}^{n\times n}$ is a sparse square and non-singular matrix, $x\in\mathbb{R}^{n}$ is the solution vector and $b\in\mathbb{R}^{n}$ is the right-hand side vector. We use a multisplitting method to solve the linear system on a large computing platform in order to reduce communications. Let the computing platform be composed of $L$ blocks of processors physically adjacent or geographically distant. In this work we apply the block Jacobi multisplitting method to the linear system as follows

 %%%*********************************
 %%%*********************************
 
 %%%*********************************
 %%%*********************************
 


@@ -201,7 +208,7 @@ b & = & [B_{1}, \ldots, B_{L}]
 where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$. 
 %%% MODIFIE ***********************
 %%%********************************
 where for $\ell\in\{1,\ldots,L\}$, $A_\ell$ is a rectangular block of size $n_\ell\times n$ and $X_\ell$ and $B_\ell$ are sub-vectors of size $n_\ell$ each, such that $\sum_\ell n_\ell=n$. 
 %%% MODIFIE ***********************
 %%%********************************

-The splitting is performed row-by-row without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to one block of processors. 
+The splitting is performed row-by-row without overlapping in such a way that successive rows of sparse matrix $A$ and both vectors $x$ and $b$ are assigned to a block of processors. 

 %%%********************************
 %%%********************************
 So, the multisplitting format of the linear system is defined as follows
 %%%********************************
 %%%********************************
 So, the multisplitting format of the linear system is defined as follows


@@ -253,9 +260,9 @@ S=\{x^1,x^2,\ldots,x^s\},~s\leq n,
 where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system. 
 %%% MODIFIE ***********************
 %%%********************************
 where for $j\in\{1,\ldots,s\}$, $x^j=[X_1^j,\ldots,X_L^j]$ is a solution of the global linear system. 
 %%% MODIFIE ***********************
 %%%********************************

-The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between the different blocks is necessary to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.
+The advantage of such a Krylov subspace is that we neither need an orthonormal basis nor any synchronization between the different blocks to orthogonalize the generated basis. This avoids to perform other synchronizations between the blocks of processors.

 
 

-The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes an error function, in our case it minimizes the residuals $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system
+The multisplitting method is periodically restarted every $s$ iterations with a new initial guess $\tilde{x}=S\alpha$ which minimizes an error function, in our case it minimizes the residuals $\|b-Ax\|_2$ over the Krylov subspace spanned by vectors of $S$. So $\alpha$ is defined as the solution of the large overdetermined linear system.

 %%%********************************
 %%%********************************
 
 %%%********************************
 %%%********************************
 


@@ -349,7 +356,19 @@ preconditioners are not scalable when using many cores.
 
 %%% AJOUTE ************************
 %%%********************************
 
 %%% AJOUTE ************************
 %%%********************************

-We have performed some experiments on an infiniband cluster of three Intel Xeon quad-core E5620 CPUs of 2.40 GHz and 12 GB of memory. For the GMRES code (alone and in both multisplitting versions) the restart parameter is fixed to 16. The precision of the GMRES version is fixed to 1e-6. For the multisplitting versions, there are two precisions, one for the external solver which is fixed to 1e-6 and another one for the inner solver (GMRES) which is fixed to 1e-10. It should be noted that a high precision is used but we also fixed a maximum number of iterations for each internal step. In practice, we limit the number of iterations in the internal step to 10. So an internal iteration is finished when the precision is reached or when the maximum internal number of iterations is reached. The precision and the maximum number of iterations of CGNR method used by our Krylov multisplitting algorithm are fixed to 1e-25 and 20 respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.
+We have performed some experiments on  an infiniband cluster of three Intel Xeon
+quad-core E5620 CPUs of 2.40 GHz and  12 GB of memory. For the GMRES code (alone
+and in both  multisplitting versions) the restart parameter is  fixed to 16. The
+precision  of  the  GMRES version  is  fixed  to  1e-6. For  the  multisplitting
+versions, there are  two precisions, one for the external  solver which is fixed
+to 1e-6 and another one for the inner solver (GMRES) which is fixed to 1e-10. It
+should be noted that a high precision is used but we also fixed a maximum number
+of  iterations for  each internal  step.  In practice,  we limit  the number  of
+iterations in the internal step to 10. So an internal iteration is finished when
+the precision  is reached or when  the maximum internal number  of iterations is
+reached. The precision and the maximum  number of iterations of CGNR method used
+by   our  Krylov   multisplitting  algorithm   are   fixed  to   1e-25  and   20
+respectively. The size of the Krylov subspace basis $S$ is fixed to 10 vectors.

 
 \begin{figure}[htbp]
 \centering
 
 \begin{figure}[htbp]
 \centering


@@ -368,7 +387,19 @@ We have performed some experiments on an infiniband cluster of three Intel Xeon
 \end{figure}
 
 %%The experiments are performed on 3 different clusters of cores interconnected by an infiniband network (each cluster is a quad-core CPU). 
 \end{figure}
 
 %%The experiments are performed on 3 different clusters of cores interconnected by an infiniband network (each cluster is a quad-core CPU). 

-Figures~\ref{fig:001} and~\ref{fig:002} show the scalability performances of GMRES, classical multisplitting and Krylov multisplitting methods: strong and weak scaling are presented respectively. We can remark from these figures that the performances of our Krylov multisplitting method are better than those of GMRES and classical multisplitting methods. In the experiments conducted in this work, our method is about twice faster than the GMRES method and about 9 times faster than the classical multisplitting method. Our multisplitting method uses a minimization step over a Krylov subspace which reduces the number of iterations and accelerates the convergence. We can also remark that the performances of the classical block Jacobi multisplitting method are the worst compared with those of the other two methods. This is why in the following experiments we compare the performances of our Krylov multisplitting method with only those of the GMRES method.
+Figures~\ref{fig:001}  and~\ref{fig:002} show  the  scalability performances  of
+GMRES, classical  multisplitting and  Krylov multisplitting methods:  strong and
+weak scaling are  presented respectively. We can remark  from these figures that
+the performances  of our Krylov multisplitting  method are better  than those of
+GMRES and classical multisplitting methods. In the experiments conducted in this
+work, our method is approximately twice faster than the GMRES method and about 9
+times faster than the classical multisplitting method. Our multisplitting method
+uses a  minimization step  over a  Krylov subspace which  reduces the  number of
+iterations  and  accelerates the  convergence.   We  can  also remark  that  the
+performances of the  classical block Jacobi multisplitting method  are the worst
+compared with  those of  the other two  methods.  This  is why in  the following
+experiments we compare the performances of our Krylov multisplitting method with
+only those of the GMRES method.

 %%%********************************
 %%%********************************
 
 %%%********************************
 %%%********************************
 


@@ -377,19 +408,19 @@ Figures~\ref{fig:001} and~\ref{fig:002} show the scalability performances of GMR
 %%%*********************************
 %Doing many experiments  with many cores is  not easy and requires to  access to a supercomputer  with several  hours for  developing  a code  and then  improving it. 
 In the following we present some experiments we could achieve out on the Hector
 %%%*********************************
 %Doing many experiments  with many cores is  not easy and requires to  access to a supercomputer  with several  hours for  developing  a code  and then  improving it. 
 In the following we present some experiments we could achieve out on the Hector

-architecture,  a UK's  high-end computing  resource, funded  by the  UK Research
+architecture,  a UK  high-end computing  resource, funded  by the  UK Research

 Councils~\cite{hector}.  This is  a Cray  XE6 supercomputer,  equipped  with two
 16-core AMD  Opteron 2.3 GHz  and 32 GB  of memory. Machines  are interconnected
 with a 3D torus. The different parameters used by the GMRES and the Krylov multisplitting codes are as those previously mentioned. 
 
 Table~\ref{tab1} shows  the result of  the experiments.  The first  column shows
 Councils~\cite{hector}.  This is  a Cray  XE6 supercomputer,  equipped  with two
 16-core AMD  Opteron 2.3 GHz  and 32 GB  of memory. Machines  are interconnected
 with a 3D torus. The different parameters used by the GMRES and the Krylov multisplitting codes are as those previously mentioned. 
 
 Table~\ref{tab1} shows  the result of  the experiments.  The first  column shows

-the  size of  the  3D Poisson  problem.  The size  is chosen  in  order to  have
+the  size of  the 3D  Poisson problem.   The  size is  chosen in  order to  have

 approximately  50,000 components  per core.   The second  column  represents the
 approximately  50,000 components  per core.   The second  column  represents the

-number of  cores used. In brackets,  one can find the decomposition  used for the
-Krylov multisplitting. The  third column and the sixth  column respectively show
-the execution time for the GMRES  and the Krylov multisplitting codes. The fourth
-and  the   seventh  column  describe   the  number  of  iterations.    For  the
-multisplitting  code, the  total number  of inner  iterations is  represented in
+number of cores used. Between brackets,  one can find the decomposition used for
+the Krylov  multisplitting. The third  column and the sixth  column respectively
+show the execution  time for the GMRES and the  Krylov multisplitting codes. The
+fourth  and the  seventh  column describe  the  number of  iterations.  For  the
+multisplitting code, the total number of inner iterations is represented between

 brackets.
 %%%********************************
 %%%******************************** 
 brackets.
 %%%********************************
 %%%******************************** 


@@ -435,12 +466,12 @@ under the specified threshold.
 %%%*******************************
 In Figure~\ref{fig:01}, the number of iterations per second is reported for both
 GMRES and the  multisplitting methods. It should be noted that  we took only the
 %%%*******************************
 In Figure~\ref{fig:01}, the number of iterations per second is reported for both
 GMRES and the  multisplitting methods. It should be noted that  we took only the

-inner number  of iterations (i.e.  the GMRES iterations) for  the multisplitting
+inner number of  iterations (i.e.  the GMRES iterations)  for the multisplitting

 method. Iterations of CGNR are not  taken into account. From this figure, it can
 method. Iterations of CGNR are not  taken into account. From this figure, it can

-be seen that the  number of iterations per second is higher  with GMRES but it is
-not  so different  with the  multisplitting method.  For the  case  with $8,192$
-cores,  the number of  iterations per  second with  4 blocks  is approximately
-equals to 115. So it is not different from GMRES.
+be seen that the number of iterations  per second is higher with GMRES but it is
+not  so different with  the multisplitting  method.  For  the case  with $8,192$
+cores, the number of iterations per second with 4 blocks is approximately equal
+to 115. So it is not different from GMRES.

 
 \begin{figure}[htbp]
 \centering
 
 \begin{figure}[htbp]
 \centering


@@ -449,16 +480,17 @@ equals to 115. So it is not different from GMRES.
 \label{fig:01}
 \end{figure}
 
 \label{fig:01}
 \end{figure}
 

-\noindent {\bf Final remarks:}\\
-It should  be noted, on  the one  hand, that the  development of a  complete new
-method usable with any  kind of problem is a really long  and fastidious task if
-one is working from  scratch. On the other hand, using an  existing tool for the
-inner solver is also not easy because it requires to make link between the inner
-solver  and the outer  one.  We  plan to  do that  later with  engineers working
-specifically on  that point.  Moreover,  we think that  it is very  important to
-analyze the convergence  of this method compared to other  methods. In this work,
-we have  focused on the  description of this  method and its performance  with a
-typical application. Many other investigations are required for this method as explained in the next section.
+\noindent {\bf Final  remarks:}\\ It should be noted, on the  one hand, that the
+development of a complete  new code usable with any kind of  problem is a really
+long and  fastidious task  if one is  working from  scratch. On the  other hand,
+using an existing  tool for the inner solver is also  quite difficult because it
+requires to  establish a link  between the inner  solver and the outer  one.  We
+plan  to  do that  later  with engineers  working  specifically  on that  point.
+Moreover, we think that it is  very important to analyze the convergence of this
+method  compared  to  other methods.  In  this  work,  we  have focused  on  the
+description of this method and its performances with a typical application.  Many
+other  investigations are  required for  this method  as explained  in  the next
+section.

 %%%*******************************
 %%%*******************************
 
 %%%*******************************
 %%%*******************************
 


@@ -496,7 +528,9 @@ preconditioning  techniques  for  Krylov  iterative methods  and  multisplitting
 methods with overlapping blocks.
 
 \section{Acknowledgement}
 methods with overlapping blocks.
 
 \section{Acknowledgement}

-The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR.
+The authors would like to thank Mark Bull of the EPCC his fruitful remarks and the facilities of HECToR. This work has been partially supported by the Labex 
+ACTION project (contract “ANR-11-LABX-01-01”). 
+

 
 %Other applications (=> other matrices)\\
 %Larger experiments\\
 
 %Other applications (=> other matrices)\\
 %Larger experiments\\