new

[book_gpu.git] / BookGPU / Chapters / chapter6 / PartieAsync.tex

diff --git a/BookGPU/Chapters/chapter6/PartieAsync.tex b/BookGPU/Chapters/chapter6/PartieAsync.tex
index 44eadd5609ee4032d59adf459dfb4654dcc60360..3f4a5394d1774fe52f4086dc4955e9590a8aba50 100644 (file)
--- a/BookGPU/Chapters/chapter6/PartieAsync.tex
+++ b/BookGPU/Chapters/chapter6/PartieAsync.tex
@@ -2,50 +2,74 @@
 computation/communication overlapping}
 \label{ch6:part2}
 
 computation/communication overlapping}
 \label{ch6:part2}
 

-In the previous part, we have seen how to efficiently implement overlap of
-computations (CPU and GPU) with communications (GPU transfers and inter-node
+In the previous section, we have seen how to efficiently implement overlap of
+computations (CPU and GPU) with communications (GPU transfers and internode

 communications).  However, we have previously shown that for some parallel
 iterative algorithms, it is sometimes even more efficient to use an asynchronous
 communications).  However, we have previously shown that for some parallel
 iterative algorithms, it is sometimes even more efficient to use an asynchronous

-scheme of iterations\index{iterations!asynchronous} \cite{HPCS2002,ParCo05,Para10}.  In that case, the nodes do
-not wait for each others but they perform their iterations using the last
+scheme of iterations\index{asynchronous iterations} \cite{HPCS2002,ParCo05,Para10}.  In that case, the nodes do
+not wait for each other but they perform their iterations using the last

 external data they have received from the other nodes, even if this
 data was produced \emph{before} the previous iteration on the other nodes.
 
 Formally, if we denote by $f=(f_1,...,f_n)$ the function representing the
 iterative process and by $x^t=(x_1^t,...,x_n^t)$ the values of the $n$ elements of
 the system at iteration $t$, we pass from a synchronous iterative scheme of the
 external data they have received from the other nodes, even if this
 data was produced \emph{before} the previous iteration on the other nodes.
 
 Formally, if we denote by $f=(f_1,...,f_n)$ the function representing the
 iterative process and by $x^t=(x_1^t,...,x_n^t)$ the values of the $n$ elements of
 the system at iteration $t$, we pass from a synchronous iterative scheme of the

-form:
+form given in \Alg{algo:ch6p2sync}
+%% \begin{algorithm}[H]
+%%   \caption{Synchronous iterative scheme}\label{algo:ch6p2sync}
+%%   \begin{Algo}
+%%     $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\\
+%%     \textbf{for} $t=0,1,...$\\
+%%     \>\textbf{for} $i=1,...,n$\\
+%%     \>\>$x_{i}^{t+1}=f_{i}(x_{1}^t,...,x_i^t,...,x_{n}^t)$\\
+%%     \>\textbf{endfor}\\
+%%     \textbf{endfor}
+%%   \end{Algo}
+%% \end{algorithm}

 \begin{algorithm}[H]
 \begin{algorithm}[H]

-  \caption{Synchronous iterative scheme}\label{algo:ch6p2sync}
-  \begin{Algo}
-    $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\\
-    \textbf{for} $t=0,1,...$\\
-    \>\textbf{for} $i=1,...,n$\\
-    \>\>$x_{i}^{t+1}=f_{i}(x_{1}^t,...,x_i^t,...,x_{n}^t)$\\
-    \>\textbf{endfor}\\
-    \textbf{endfor}
-  \end{Algo}
+  \caption{synchronous iterative scheme}\label{algo:ch6p2sync}
+  $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\;
+  \For{ $t=0,1,...$} {
+    \For{ $i=1,...,n$}{
+      $x_{i}^{t+1}=f_{i}(x_{1}^t,...,x_i^t,...,x_{n}^t)$\;
+    }
+  }

 \end{algorithm}
 \end{algorithm}

+

 \noindent
 \noindent

-to an asynchronous iterative scheme of the form:
+to an asynchronous iterative scheme of the form given in \Alg{algo:ch6p2async}.\\
+%% \begin{algorithm}[H]
+%%   \caption{Asynchronous iterative scheme}\label{algo:ch6p2async}
+%%   \begin{Algo}
+%%     $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\\
+%%     \textbf{for} $t=0,1,...$\\
+%%     \>\textbf{for} $i=1,...,n$\\
+%%     \>\>$x_{i}^{t+1}=\left\{
+%%       \begin{array}[h]{ll}
+%%         x_i^t & \text{if } i \text{ is \emph{not} updated at iteration } i\\
+%%         f_i(x_1^{s_1^i(t)},...,x_n^{s_n^i(t)}) & \text{if } i \text{ is updated at iteration } i
+%%       \end{array}
+%%     \right.$\\
+%%     \>\textbf{endfor}\\
+%%     \textbf{endfor}
+%%   \end{Algo}
+%% \end{algorithm}

 \begin{algorithm}[H]
 \begin{algorithm}[H]

-  \caption{Asynchronous iterative scheme}\label{algo:ch6p2async}
-  \begin{Algo}
-    $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\\
-    \textbf{for} $t=0,1,...$\\
-    \>\textbf{for} $i=1,...,n$\\
-    \>\>$x_{i}^{t+1}=\left\{
+  \caption{asynchronous iterative scheme}\label{algo:ch6p2async}
+    $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\;
+    \For {$t=0,1,...$} {
+      \For{ $i=1,...,n$} {
+        $x_{i}^{t+1}=\left\{

       \begin{array}[h]{ll}
         x_i^t & \text{if } i \text{ is \emph{not} updated at iteration } i\\
         f_i(x_1^{s_1^i(t)},...,x_n^{s_n^i(t)}) & \text{if } i \text{ is updated at iteration } i
       \end{array}
       \begin{array}[h]{ll}
         x_i^t & \text{if } i \text{ is \emph{not} updated at iteration } i\\
         f_i(x_1^{s_1^i(t)},...,x_n^{s_n^i(t)}) & \text{if } i \text{ is updated at iteration } i
       \end{array}

-    \right.$\\
-    \>\textbf{endfor}\\
-    \textbf{endfor}
-  \end{Algo}
+    \right.$
+      }
+    }

 \end{algorithm}
 \end{algorithm}

-where $s_j^i(t)$ is the iteration number of the production of the value $x_j$ of
-element $j$  that is  used on element  $i$ at iteration  $t$ (see for example~\cite{BT89,
+In this scheme, $s_j^i(t)$ is the iteration number of the production of the value $x_j$ of
+element $j$  that is  used on element  $i$ at iteration  $t$ (see, for example,~\cite{BT89,

   FS2000} for further details).
 Such schemes  are called AIAC\index{AIAC} for  \emph{Asynchronous Iterations and
   Asynchronous Communications}.  They combine  two aspects that are respectively
   FS2000} for further details).
 Such schemes  are called AIAC\index{AIAC} for  \emph{Asynchronous Iterations and
   Asynchronous Communications}.  They combine  two aspects that are respectively


@@ -62,7 +86,7 @@ which new data arrives on each node. Typically,
 if a node receives newer data only every four or five local iterations, it is
 strongly probable that the evolution of its local iterative process will be
 slower than if it receives data at every iteration. The key point here is that
 if a node receives newer data only every four or five local iterations, it is
 strongly probable that the evolution of its local iterative process will be
 slower than if it receives data at every iteration. The key point here is that

-this frequency does not only depend on the hardware configuration of the
+not only does this frequency  depend on the hardware configuration of the

 parallel system but it also depends on the software that is used to
 implement the algorithmic scheme.
 
 parallel system but it also depends on the software that is used to
 implement the algorithmic scheme.
 


@@ -70,11 +94,11 @@ The impact of the programming environments used to implement asynchronous
 algorithms has already been investigated in~\cite{SuperCo05}.  Although the
 features required to efficiently implement asynchronous schemes have not
 changed, the available programming environments and computing hardware have
 algorithms has already been investigated in~\cite{SuperCo05}.  Although the
 features required to efficiently implement asynchronous schemes have not
 changed, the available programming environments and computing hardware have

-evolved, in particular now GPUs are available.  So, there is a need to reconsider the
+evolved, in particular now that GPUs are available.  So, there is a need to reconsider the

 implementation schemes of AIAC according to the new de facto standards for parallel
 programming (communications and threads) as well as the integration of the GPUs.
 One of the main objective here is to obtain a maximal overlap between the
 implementation schemes of AIAC according to the new de facto standards for parallel
 programming (communications and threads) as well as the integration of the GPUs.
 One of the main objective here is to obtain a maximal overlap between the

-activities of the three types of devices that are the CPU, the GPU and the
+activities of the three types of devices: the CPU, the GPU, and the

 network. Moreover, another objective is to present what we think
 is the best compromise between the simplicity of the implementation and its
 maintainability on one side and its
 network. Moreover, another objective is to present what we think
 is the best compromise between the simplicity of the implementation and its
 maintainability on one side and its


@@ -87,16 +111,16 @@ with full overlap.  Between these two extremes, we propose a synchronization
 mechanism on top of our asynchronous scheme that can be used either statically or
 dynamically during the application execution.
 
 mechanism on top of our asynchronous scheme that can be used either statically or
 dynamically during the application execution.
 

-Although there exist several programming environments for inter-node
-communications, multi-threading and GPU programming, a few of them have
-become \emph{de facto standards}, either due to their good stability, their ease
-of use and/or their wide adoption by the scientific community.
-Therefore, as in the previous section all the schemes presented in the following use MPI~\cite{MPI},
-OpenMP~\cite{openMP} and CUDA~\cite{CUDA}.  However, there is no loss of
-generality as those schemes may easily be implemented with other libraries.
+Although there exist several programming environments for internode
+communications, multithreading, and GPU programming, a few of them have
+become \emph{de facto standards}, due to their good stability, their ease
+of use, and/or their wide adoption by the scientific community.
+Therefore, as in the previous section, all the schemes presented in the following use MPI~\cite{MPI},
+OpenMP~\cite{openMP}, and CUDA~\cite{CUDA}.  However, there is no loss of
+generality as these schemes may easily be implemented with other libraries.

 
 Finally, in order to stay as clear as possible, only the parts of code and
 
 Finally, in order to stay as clear as possible, only the parts of code and

-variables related to the control of parallelism (communications, threads,...)
+variables related to the control of parallelism (communications, threads, etc.)

 are presented in our schemes. The inner organization of data is not detailed as
 it depends on the application.  We only consider that we have two data arrays
 (previous version and current version) and communication buffers.  However, in
 are presented in our schemes. The inner organization of data is not detailed as
 it depends on the application.  We only consider that we have two data arrays
 (previous version and current version) and communication buffers.  However, in


@@ -109,17 +133,17 @@ avoid data copies.
 The first step toward our complete scheme is to implement a basic asynchronous
 scheme that includes an actual overlap of the communications with the
 computations\index{overlap!computation and communication}. In order to ensure that the communications are actually performed
 The first step toward our complete scheme is to implement a basic asynchronous
 scheme that includes an actual overlap of the communications with the
 computations\index{overlap!computation and communication}. In order to ensure that the communications are actually performed

-in parallel of the computations, it is necessary to use different threads.  It
+in parallel with the computations, it is necessary to use different threads.  It

 is important to remember that asynchronous communications provided in
 is important to remember that asynchronous communications provided in

-communication libraries like MPI are not systematically performed in parallel of
+communication libraries such as MPI are not systematically performed in parallel with

 the computations~\cite{ChVCV13,Hoefler08a}.  So, the logical and classical way
 to implement such an overlap is to use three threads: one for
 the computations~\cite{ChVCV13,Hoefler08a}.  So, the logical and classical way
 to implement such an overlap is to use three threads: one for

-computing, one for sending and one for receiving. Moreover, since
-the communication is performed by threads, blocking synchronous communications\index{MPI!communication!blocking}\index{MPI!communication!synchronous}
+computing, one for sending, and one for receiving. Moreover, since
+the communication is performed by threads, blocking synchronous communications\index{MPI!blocking}\index{MPI!synchronous}

 can be used without deteriorating the overall performance.
 
 In this basic version, the termination\index{termination} of the global process is performed
 can be used without deteriorating the overall performance.
 
 In this basic version, the termination\index{termination} of the global process is performed

-individually on each node according to their own termination. This can be guided by either a
+individually on each node according to its own termination. This can be guided by either a

 number of iterations or a local convergence detection\index{convergence detection}. The important step at
 the end of the process is to perform the receptions of all pending
 communications in order to ensure the termination of the two communication
 number of iterations or a local convergence detection\index{convergence detection}. The important step at
 the end of the process is to perform the receptions of all pending
 communications in order to ensure the termination of the two communication


@@ -130,13 +154,17 @@ So, the global organization of this scheme is set up in \Lst{algo:ch6p2BasicAsyn
 % \begin{algorithm}[H]
 %   \caption{Initialization of the basic asynchronous scheme.}
 %   \label{algo:ch6p2BasicAsync}
 % \begin{algorithm}[H]
 %   \caption{Initialization of the basic asynchronous scheme.}
 %   \label{algo:ch6p2BasicAsync}

-\begin{Listing}{algo:ch6p2BasicAsync}{Initialization of the basic asynchronous scheme}
+\begin{Listing}{algo:ch6p2BasicAsync}{initialization of the basic asynchronous scheme}

 // Variables declaration and initialization
 // Variables declaration and initialization

-omp_lock_t lockSend; // Controls the sendings from the computing thread
-omp_lock_t lockRec;  // Ensures the initial reception of external data
-char Finished = 0;   // Boolean indicating the end of the process
-char SendsInProgress = 0; // Boolean indicating if previous data sendings are still in progress
-double Threshold;    // Threshold of the residual for convergence detection
+// Controls the sendings from the computing thread
+omp_lock_t lockSend; 
+// Ensures the initial reception of external data
+omp_lock_t lockRec;  
+char Finished = 0; // Boolean indicating the end of the process
+// Boolean indicating if previous data sendings are still in progress
+char SendsInProgress = 0; 
+// Threshold of the residual for convergence detection
+double Threshold; 

 
 // Parameters reading
 ...
 
 // Parameters reading
 ...


@@ -152,9 +180,10 @@ MPI_Comm_rank(MPI_COMM_WORLD, &numP);
 // OpenMP initialization (mainly declarations and setting up of locks)
 omp_set_num_threads(3);
 omp_init_lock(&lockSend);
 // OpenMP initialization (mainly declarations and setting up of locks)
 omp_set_num_threads(3);
 omp_init_lock(&lockSend);

-omp_set_lock(&lockSend); // Initially locked, unlocked to start sendings
+omp_set_lock(&lockSend);//Initially locked, unlocked to start sendings

 omp_init_lock(&lockRec);
 omp_init_lock(&lockRec);

-omp_set_lock(&lockRec);  // Initially locked, unlocked when initial data are received
+//Initially locked, unlocked when initial data are received
+omp_set_lock(&lockRec);

 
 #pragma omp parallel
 {
 
 #pragma omp parallel
 {


@@ -195,7 +224,7 @@ MPI_Finalize();
 In this scheme, the \texttt{lockRec} mutex\index{OpenMP!mutex} is not mandatory.
 It is only used to ensure that data dependencies are actually exchanged at the
 first iteration of the process.  Data initialization and distribution
 In this scheme, the \texttt{lockRec} mutex\index{OpenMP!mutex} is not mandatory.
 It is only used to ensure that data dependencies are actually exchanged at the
 first iteration of the process.  Data initialization and distribution

-(lines~16-17) are not detailed here because they are directly related to the
+(lines~20--21) are not detailed here because they are directly related to the

 application. The important point is that, in most cases, they should be done
 before the iterative process.  The computing function is given in
 \Lst{algo:ch6p2BasicAsyncComp}.
 application. The important point is that, in most cases, they should be done
 before the iterative process.  The computing function is given in
 \Lst{algo:ch6p2BasicAsyncComp}.


@@ -203,7 +232,7 @@ before the iterative process.  The computing function is given in
 %\begin{algorithm}[H]
 %  \caption{Computing function in the basic asynchronous scheme.}
 %  \label{algo:ch6p2BasicAsyncComp}
 %\begin{algorithm}[H]
 %  \caption{Computing function in the basic asynchronous scheme.}
 %  \label{algo:ch6p2BasicAsyncComp}

-\begin{Listing}{algo:ch6p2BasicAsyncComp}{Computing function in the basic asynchronous scheme}
+\begin{Listing}{algo:ch6p2BasicAsyncComp}{computing function in the basic asynchronous scheme}

 // Variables declaration and initialization
 int iter = 1;      // Number of the current iteration
 double difference; // Variation of one element between two iterations
 // Variables declaration and initialization
 int iter = 1;      // Number of the current iteration
 double difference; // Variation of one element between two iterations


@@ -211,9 +240,10 @@ double residual;   // Residual of the current iteration
 
 // Computation loop
 while(!Finished){
 
 // Computation loop
 while(!Finished){

-  // Sendings of data dependencies if there is no previous sending in progress
+  // Sending of data dependencies if there is no previous sending
+  // in progress

   if(!SendsInProgress){
   if(!SendsInProgress){

-    // Potential copy of data to be sent in additional buffers
+    // Potential copy of data to be sent into additional buffers

     ...
     // Change of sending state
     SendsInProgress = 1;
     ...
     // Change of sending state
     SendsInProgress = 1;


@@ -257,30 +287,29 @@ while(!Finished){
 \end{Listing}
 %\end{algorithm}
 
 \end{Listing}
 %\end{algorithm}
 

-As mentioned above, it can be seen in line~18 of \Lst{algo:ch6p2BasicAsyncComp}
+As mentioned above, it can be seen in lines~19--21 of \Lst{algo:ch6p2BasicAsyncComp}

 that the \texttt{lockRec} mutex is used only at the first iteration to wait for
 the initial data dependencies before the computations. The residual\index{residual}, initialized
 that the \texttt{lockRec} mutex is used only at the first iteration to wait for
 the initial data dependencies before the computations. The residual\index{residual}, initialized

-in line~23 and computed in lines~34-37, is defined by the maximal difference
+in line~24 and computed in lines~35--38, is defined by the maximal difference

 between the elements from two consecutive iterations. It is classically used to
 detect the local convergence of the process on each node.  In the more
 complete schemes presented in the sequel, a global termination detection that
 takes the states of all the nodes into account will be exhibited.
 
 between the elements from two consecutive iterations. It is classically used to
 detect the local convergence of the process on each node.  In the more
 complete schemes presented in the sequel, a global termination detection that
 takes the states of all the nodes into account will be exhibited.
 

-Finally, the local convergence is tested and updated when necessary. In line~44,
+Finally, the local convergence is tested and updated when necessary. In line~45,

 the \texttt{lockSend} mutex is unlocked to allow the sending function to send
 final messages to the dependency nodes.  Those messages are required to keep the
 reception function alive until all the final messages have been received.
 the \texttt{lockSend} mutex is unlocked to allow the sending function to send
 final messages to the dependency nodes.  Those messages are required to keep the
 reception function alive until all the final messages have been received.

-Otherwise, a node could stop its reception function whereas other nodes are
+Otherwise, a node could stop its reception function while other nodes are

 still trying to communicate with it.  Moreover, a local sending of a final
 still trying to communicate with it.  Moreover, a local sending of a final

-message to the node itself is required (line~45) to ensure that the reception
+message to the node itself is required (line~46) to ensure that the reception

 function will not stay blocked in a message probing
 function will not stay blocked in a message probing

-(see~\Lst{algo:ch6p2BasicAsyncReceptions}, line~11). This may happen if the node
-receives the final messages from its dependencies \emph{before} being itself in
+(see~\Lst{algo:ch6p2BasicAsyncReceptions}, line~12). This may happen if the node
+receives the final messages from its dependencies \emph{before} reaching its own

 local convergence.
 
 All the messages but this final local one are performed in the sending function
 described in \Lst{algo:ch6p2BasicAsyncSendings}.
 local convergence.
 
 All the messages but this final local one are performed in the sending function
 described in \Lst{algo:ch6p2BasicAsyncSendings}.

-

 The main loop is only conditioned by the end of the computing process (line~4).
 At each iteration, the thread waits for the permission from the computing thread
 (according to the \texttt{lockSend} mutex).  Then, data are sent with
 The main loop is only conditioned by the end of the computing process (line~4).
 At each iteration, the thread waits for the permission from the computing thread
 (according to the \texttt{lockSend} mutex).  Then, data are sent with


@@ -293,7 +322,7 @@ the main loop, the final messages are sent to the dependencies of the node.
 %\begin{algorithm}[H]
 %  \caption{Sending function in the basic asynchronous scheme.}
 %  \label{algo:ch6p2BasicAsyncSendings}
 %\begin{algorithm}[H]
 %  \caption{Sending function in the basic asynchronous scheme.}
 %  \label{algo:ch6p2BasicAsyncSendings}

-\begin{Listing}{algo:ch6p2BasicAsyncSendings}{Sending function in the basic asynchronous scheme}
+\begin{Listing}{algo:ch6p2BasicAsyncSendings}{sending function in the basic asynchronous scheme}

 // Variables declaration and initialization
 ...
 
 // Variables declaration and initialization
 ...
 


@@ -320,9 +349,10 @@ The last function, detailed in \Lst{algo:ch6p2BasicAsyncReceptions}, does all th
 %  \label{algo:ch6p2BasicAsyncReceptions}
 \begin{Listing}{algo:ch6p2BasicAsyncReceptions}{Reception function in the basic asynchronous scheme}
 // Variables declaration and initialization
 %  \label{algo:ch6p2BasicAsyncReceptions}
 \begin{Listing}{algo:ch6p2BasicAsyncReceptions}{Reception function in the basic asynchronous scheme}
 // Variables declaration and initialization

-char countReceipts = 0; // Boolean indicating whether receptions are counted or not
+char countReceipts = 1; // Boolean indicating whether receptions are 
+                        // counted or not

 int nbEndMsg = 0;       // Number of end messages received
 int nbEndMsg = 0;       // Number of end messages received

-int arrived = 0;        // Boolean indicating if a message is arrived
+int arrived = 0;        // Boolean indicating if a message has arrived

 int srcNd;              // Source node of the message
 int size;               // Message size
 
 int srcNd;              // Source node of the message
 int size;               // Message size
 


@@ -334,10 +364,12 @@ while(!Finished){
     // Management of data messages
     switch(status.MPI_TAG){
       case tagCom: // Management of data messages
     // Management of data messages
     switch(status.MPI_TAG){
       case tagCom: // Management of data messages

-       srcNd = status.MPI_SOURCE; // Get the source node of the message
+       // Get the source node of the message
+       srcNd = status.MPI_SOURCE; 

        // Actual data reception in the corresponding buffer
        MPI_Recv(dataBufferOf(srcNd), nbDataOf(srcNd), dataTypeOf(srcNd), srcNd, tagCom, MPI_COMM_WORLD, &status); 
        // Actual data reception in the corresponding buffer
        MPI_Recv(dataBufferOf(srcNd), nbDataOf(srcNd), dataTypeOf(srcNd), srcNd, tagCom, MPI_COMM_WORLD, &status); 

-       // Unlocking of the computing thread when data are received from all dependencies
+       // Unlocking of the computing thread when data are received 
+       // from all dependencies

        if(countReceipts == 1 && ... @\emph{receptions from ALL dependencies}@ ...){
          omp_unset_lock(&lockRec);
          countReceipts = 0; // No more counting after first iteration
        if(countReceipts == 1 && ... @\emph{receptions from ALL dependencies}@ ...){
          omp_unset_lock(&lockRec);
          countReceipts = 0; // No more counting after first iteration


@@ -352,7 +384,7 @@ while(!Finished){
 }
 
 // Reception of pending messages and counting of end messages
 }
 
 // Reception of pending messages and counting of end messages

-do{ // Loop over the remaining incoming/waited messages
+do{ // Loop over the remaining incoming/end messages

   MPI_Probe(MPI_ANY_SOURCE, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
   MPI_Get_count(&status, MPI_CHAR, &size);
   // Actual reception in dummy buffer
   MPI_Probe(MPI_ANY_SOURCE, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
   MPI_Get_count(&status, MPI_CHAR, &size);
   // Actual reception in dummy buffer


@@ -365,24 +397,26 @@ do{ // Loop over the remaining incoming/waited messages
 \end{Listing}
 %\end{algorithm}
 
 \end{Listing}
 %\end{algorithm}
 

-As in the sending function, the main loop of receptions is done while the
-iterative process is not \texttt{Finished}.  In line~11, the thread waits until
-a message arrives on the node.  Then, it performs the actual reception and the
-corresponding subsequent actions (potential data copies for data messages and
-counting for end messages). Lines 20-23 check that all data dependencies have
-been received before unlocking the \texttt{lockRec} mutex. As mentioned before,
-they are not mandatory and are included only to ensure that all data
-dependencies are received at the first iteration. Lines 25-28 are required to
-manage end messages that arrive on the node \emph{before} it reaches its own
-termination process. As the nodes are \emph{not} synchronized, this may happen.
-Finally, lines 34-43 perform the receptions of all pending communications,
-including the remaining end messages (at least the one from the node itself).
+As  in the  sending function,  the main  loop of  receptions is  done  while the
+iterative process is not \texttt{Finished}.   In line~12, the thread waits until
+a message arrives  on the node.  Then, it performs the  actual reception and the
+corresponding subsequent  actions (potential data  copies for data  messages and
+counting for  end messages). Lines~23--26  check, only at the  first iteration of
+computations, that all data dependencies have been received before unlocking the
+\texttt{lockRec} mutex. %As mentioned previously,  they are not mandatory and are included only  to
+Although this is not mandatory, it ensures that  all data dependencies  are
+received before starting the computations. % at  the first iteration.
+Lines~28--31 are required to  manage end messages that  arrive on the
+node \emph{before}  it reaches  its own termination  process.  As the  nodes are
+\emph{not}  synchronized, this  may happen.   Finally, lines~37--46  perform the
+receptions of  all pending communications, including the  remaining end messages
+(at least the one from the node itself).

 
 \medskip
 
 \medskip

-So, with those algorithms, we obtain a quite simple and efficient asynchronous
+So, with these algorithms, we obtain a quite simple and efficient asynchronous

 iterative scheme. It is interesting to notice that GPU computing can be easily
 included in the computing thread.  This will be fully addressed in
 iterative scheme. It is interesting to notice that GPU computing can be easily
 included in the computing thread.  This will be fully addressed in

-paragraph~\ref{ch6:p2GPUAsync}.  However, before presenting the complete
+Section~\ref{ch6:p2GPUAsync}.  However, before presenting the complete

 asynchronous scheme with GPU computing, we have to detail how our initial scheme
 can be made synchronous.
 
 asynchronous scheme with GPU computing, we have to detail how our initial scheme
 can be made synchronous.
 


@@ -404,11 +438,11 @@ threshold.
 % code, which tends to improve  the implementation and maintenance costs when both
 % versions are required.
 % The  second one  is  that
 % code, which tends to improve  the implementation and maintenance costs when both
 % versions are required.
 % The  second one  is  that

-In our context, the interest of being able to dynamically change the operating
-mode (sync/async) during the process execution, is that this strongly simplifies
+In our context,  being able to dynamically change the operating
+mode (sync/async) during the process execution  strongly simplifies

 the global convergence detection.  In fact, our past experience in the design
 and implementation of global convergence detection in asynchronous
 the global convergence detection.  In fact, our past experience in the design
 and implementation of global convergence detection in asynchronous

-algorithms~\cite{SuperCo05, BCC07, Vecpar08a}, have led us to the conclusion
+algorithms~\cite{SuperCo05, BCC07, Vecpar08a} has led us to the conclusion

 that although a decentralized detection scheme is possible and may be more
 efficient in some situations, its much higher complexity is an
 obstacle to actual use in practice, especially in industrial contexts where
 that although a decentralized detection scheme is possible and may be more
 efficient in some situations, its much higher complexity is an
 obstacle to actual use in practice, especially in industrial contexts where


@@ -420,19 +454,23 @@ consists in dynamically changing the operating mode between asynchronous and syn
 during the execution of the process in order to check the global convergence.
 This is why we need to synchronize our asynchronous scheme.
 
 during the execution of the process in order to check the global convergence.
 This is why we need to synchronize our asynchronous scheme.
 

-In each  algorithm of the  initial scheme, we  only exhibit the  additional code
+In each  algorithm of the  initial scheme, we  only give the  additional code

 required to change the operating mode.
 
 %\begin{algorithm}[H]
 %  \caption{Initialization of the synchronized scheme.}
 %  \label{algo:ch6p2Sync}
 required to change the operating mode.
 
 %\begin{algorithm}[H]
 %  \caption{Initialization of the synchronized scheme.}
 %  \label{algo:ch6p2Sync}

-\begin{Listing}{algo:ch6p2Sync}{Initialization of the synchronized scheme}
+\begin{Listing}{algo:ch6p2Sync}{initialization of the synchronized scheme}

 // Variables declarations and initialization
 ...
 // Variables declarations and initialization
 ...

-omp_lock_t lockStates; // Controls the synchronous exchange of local states 
-omp_lock_t lockIter;   // Controls the synchronization at the end of each iteration
-char localCV = 0;      // Boolean indicating whether the local stabilization is reached or not
-int nbOtherCVs = 0;    // Number of other nodes being in local stabilization
+// Controls the synchronous exchange of local states 
+omp_lock_t lockStates; 
+// Controls the synchronization at the end of each iteration
+omp_lock_t lockIter;   
+//Boolean indicating whether the local stabilization is reached or not
+char localCV = 0;      
+// Number of other nodes being in local stabilization
+int nbOtherCVs = 0;    

 
 // Parameters reading
 ...
 
 // Parameters reading
 ...


@@ -443,9 +481,12 @@ int nbOtherCVs = 0;    // Number of other nodes being in local stabilization
 // OpenMP initialization (mainly declarations and setting up of locks)
 ...
 omp_init_lock(&lockStates);
 // OpenMP initialization (mainly declarations and setting up of locks)
 ...
 omp_init_lock(&lockStates);

-omp_set_lock(&lockStates); // Initially locked, unlocked when all state messages are received
+// Initially locked, unlocked when all state messages are received
+omp_set_lock(&lockStates); 

 omp_init_lock(&lockIter);
 omp_init_lock(&lockIter);

-omp_set_lock(&lockIter);   // Initially locked, unlocked when all "end of iteration" messages are received
+// Initially locked, unlocked when all "end of iteration" messages are
+// received
+omp_set_lock(&lockIter);   

 
 // Threads launching
 #pragma omp parallel
 
 // Threads launching
 #pragma omp parallel


@@ -475,9 +516,9 @@ receptions of all state messages coming from the other nodes.  As shown
 in \Lst{algo:ch6p2SyncComp}, those messages contain only a boolean indicating
 for each node if it is in local convergence\index{convergence!local}. So, once all the states are
 received on a node, it is possible to determine if all the nodes are in local
 in \Lst{algo:ch6p2SyncComp}, those messages contain only a boolean indicating
 for each node if it is in local convergence\index{convergence!local}. So, once all the states are
 received on a node, it is possible to determine if all the nodes are in local

-convergence, and thus to detect the global convergence.  The \texttt{lockIter}
+convergence and, thus, to detect the global convergence.  The \texttt{lockIter}

 mutex is used to synchronize all the nodes at the end of each iteration.  There
 mutex is used to synchronize all the nodes at the end of each iteration.  There

-are also two new variables that respectively represent the local state of the
+are also two new variables that  represent the local state of the

 node (\texttt{localCV}) according to the iterative process (convergence) and the
 number of other nodes that are in local convergence (\texttt{nbOtherCVs}).
 
 node (\texttt{localCV}) according to the iterative process (convergence) and the
 number of other nodes that are in local convergence (\texttt{nbOtherCVs}).
 


@@ -487,18 +528,18 @@ in \Lst{algo:ch6p2SyncComp}.
 %\begin{algorithm}[H]
 %  \caption{Computing function in the synchronized scheme.}
 %  \label{algo:ch6p2SyncComp}
 %\begin{algorithm}[H]
 %  \caption{Computing function in the synchronized scheme.}
 %  \label{algo:ch6p2SyncComp}

-\begin{Listing}{algo:ch6p2SyncComp}{Computing function in the synchronized scheme}
+\begin{Listing}{algo:ch6p2SyncComp}{computing function in the synchronized scheme}

 // Variables declarations and initialization
 ...
 
 // Computation loop
 while(!Finished){
 // Variables declarations and initialization
 ...
 
 // Computation loop
 while(!Finished){

-  // Sendings of data dependencies at @\emph{each}@ iteration
+  // Sending of data dependencies at @\color{white}\emph{\textbf{each}}@ iteration

   // Potential copy of data to be sent in additional buffers
   ...
   omp_unset_lock(&lockSend);
 
   // Potential copy of data to be sent in additional buffers
   ...
   omp_unset_lock(&lockSend);
 

-  // Blocking receptions at @\emph{each}@ iteration
+  // Blocking receptions at @\color{white}\emph{\textbf{each}}@ iteration

   omp_set_lock(&lockRec);
   
   // Local computation 
   omp_set_lock(&lockRec);
   
   // Local computation 


@@ -523,7 +564,7 @@ while(!Finished){
   // Waiting for the state messages receptions from the other nodes
   omp_set_lock(&lockStates);
   
   // Waiting for the state messages receptions from the other nodes
   omp_set_lock(&lockStates);
   

-  // Determination of global convergence (if all nodes are in local CV)
+  //Determination of global convergence (if all nodes are in local CV)

   if(localCV + nbOtherCVs == nbP){
     // Entering global CV state
     Finished = 1;
   if(localCV + nbOtherCVs == nbP){
     // Entering global CV state
     Finished = 1;


@@ -553,39 +594,39 @@ Most of the added code is related to the waiting for specific communications.
 Between lines~6 and~7, the use of the flag \texttt{SendsInProgress} is no longer
 needed since the sends are performed at each iteration.  In line~12, the
 thread waits for the data receptions from its dependencies. In
 Between lines~6 and~7, the use of the flag \texttt{SendsInProgress} is no longer
 needed since the sends are performed at each iteration.  In line~12, the
 thread waits for the data receptions from its dependencies. In

-lines~26-34, the local states are determined and exchanged among all nodes.  A
+lines~27--34, the local states are determined and exchanged among all nodes.  A

 new message tag (\texttt{tagState}) is required for identifying those messages.
 In line~37, the global termination state is determined.  When it is reached,
 new message tag (\texttt{tagState}) is required for identifying those messages.
 In line~37, the global termination state is determined.  When it is reached,

-lines~38-42 change the \texttt{Finished} boolean to stop the iterative process,
+lines~39--42 change the \texttt{Finished} boolean to stop the iterative process

 and send the end messages.  Otherwise each node resets its local state
 and send the end messages.  Otherwise each node resets its local state

-information about the other nodes and a global barrier is done between all the
+information about the other nodes and a global barrier is added between all the

 nodes at the end of each iteration with another new tag (\texttt{tagIter}).
 That barrier is needed to ensure that data messages from successive iterations
 are actually received during the \emph{same} iteration on the destination nodes.
 Nevertheless, it is not useful at the termination of the global process as it is
 replaced by the global exchange of end messages.
 
 nodes at the end of each iteration with another new tag (\texttt{tagIter}).
 That barrier is needed to ensure that data messages from successive iterations
 are actually received during the \emph{same} iteration on the destination nodes.
 Nevertheless, it is not useful at the termination of the global process as it is
 replaced by the global exchange of end messages.
 

-There is no big modification induced by the synchronization in the sending
-function. The only change could be the suppression of line~11 that is not useful
-in this case.  Apart from that, the function stays the same as
-in \Lst{algo:ch6p2BasicAsyncSendings}.
+There  is no  big modification  induced by  the synchronization  in  the sending
+function.     The     function    stays     almost     the     same    as     in
+\Lst{algo:ch6p2BasicAsyncSendings}. The only change  could be the suppression of
+line~11 that is not useful in this case.

 
 In the reception function, given in \Lst{algo:ch6p2SyncReceptions}, there are
 
 In the reception function, given in \Lst{algo:ch6p2SyncReceptions}, there are

-mainly two insertions (in lines~19-30 and 31-40), corresponding to the
+mainly two insertions (in lines~19--31 and 32--42), corresponding to the

 additional types of messages to receive.  There is also the insertion of three
 variables that are used for the receptions of the new message types.  In
 additional types of messages to receive.  There is also the insertion of three
 variables that are used for the receptions of the new message types.  In

-lines~24-29 and 34-39 are located messages counting and mutex unlocking
+lines~24--30 and 35--41 are located messages counting and mutex unlocking

 mechanisms that are used to block the computing thread at the corresponding steps of its
 execution. They are similar to the mechanism used for managing the end messages
 at the end of the entire process.  Line~23 directly updates the
 number of other nodes that are in local convergence by adding the
 received state of the source node. This is possible due to the encoding that is used to
 mechanisms that are used to block the computing thread at the corresponding steps of its
 execution. They are similar to the mechanism used for managing the end messages
 at the end of the entire process.  Line~23 directly updates the
 number of other nodes that are in local convergence by adding the
 received state of the source node. This is possible due to the encoding that is used to

-represent the local convergence (1) and the non-convergence (0).
+represent the local convergence (1) and the non convergence (0).

 
 %\begin{algorithm}[H]
 %  \caption{Reception function in the synchronized scheme.}
 %  \label{algo:ch6p2SyncReceptions}
 
 %\begin{algorithm}[H]
 %  \caption{Reception function in the synchronized scheme.}
 %  \label{algo:ch6p2SyncReceptions}

-\begin{Listing}{algo:ch6p2SyncReceptions}{Reception function in the synchronized scheme}
+\begin{Listing}{algo:ch6p2SyncReceptions}{reception function in the synchronized scheme}

 // Variables declarations and initialization
 ...
 int nbStateMsg = 0; // Number of local state messages received
 // Variables declarations and initialization
 ...
 int nbStateMsg = 0; // Number of local state messages received


@@ -610,7 +651,8 @@ while(!Finished){
        // Updates of numbers of stabilized nodes and received state msgs 
        nbOtherCVs += recvdState;
        nbStateMsg++;
        // Updates of numbers of stabilized nodes and received state msgs 
        nbOtherCVs += recvdState;
        nbStateMsg++;

-       // Unlocking of the computing thread when states of all other nodes are received
+       // Unlocking of the computing thread when states of all other 
+       // nodes are received

        if(nbStateMsg == nbP-1){
          nbStateMsg = 0;
          omp_unset_lock(&lockStates);
        if(nbStateMsg == nbP-1){
          nbStateMsg = 0;
          omp_unset_lock(&lockStates);


@@ -619,8 +661,9 @@ while(!Finished){
       case tagIter: // Management of "end of iteration" messages
        // Actual reception of the message in dummy buffer
        MPI_Recv(dummyBuffer, 1, MPI_CHAR, status.MPI_SOURCE, tagIter, MPI_COMM_WORLD, &status); 
       case tagIter: // Management of "end of iteration" messages
        // Actual reception of the message in dummy buffer
        MPI_Recv(dummyBuffer, 1, MPI_CHAR, status.MPI_SOURCE, tagIter, MPI_COMM_WORLD, &status); 

-       nbIterMsg++; // Update of the nb of iteration messages
-       // Unlocking of the computing thread when iteration messages are received from all other nodes       
+       nbIterMsg++; // Update of the number of iteration messages
+       // Unlocking of the computing thread when iteration messages 
+       // are received from all other nodes       

        if(nbIterMsg == nbP - 1){
          nbIterMsg = 0;
          omp_unset_lock(&lockIter);
        if(nbIterMsg == nbP - 1){
          nbIterMsg = 0;
          omp_unset_lock(&lockIter);


@@ -631,7 +674,7 @@ while(!Finished){
 }
 
 // Reception of pending messages and counting of end messages
 }
 
 // Reception of pending messages and counting of end messages

-do{ // Loop over the remaining incoming/waited messages
+do{ // Loop over the remaining incoming/end messages

   ...
 }while(arrived == 1 || nbEndMsg < nbDeps + 1);
 \end{Listing}
   ...
 }while(arrived == 1 || nbEndMsg < nbDeps + 1);
 \end{Listing}


@@ -640,10 +683,10 @@ do{ // Loop over the remaining incoming/waited messages
 Now that we can synchronize our asynchronous scheme, the final step is to
 dynamically alternate the two operating modes in order to regularly check the
 global convergence of the iterative process. This is detailed in the following
 Now that we can synchronize our asynchronous scheme, the final step is to
 dynamically alternate the two operating modes in order to regularly check the
 global convergence of the iterative process. This is detailed in the following

-paragraph together with the inclusion of GPU computing in the final asynchronous
+section together with the inclusion of GPU computing in the final asynchronous

 scheme.
 
 scheme.
 

-\subsection{Asynchronous scheme using MPI, OpenMP and CUDA}
+\subsection{Asynchronous scheme using MPI, OpenMP, and CUDA}

 \label{ch6:p2GPUAsync}
 
 As mentioned above, the strategy proposed to obtain a good compromise between
 \label{ch6:p2GPUAsync}
 
 As mentioned above, the strategy proposed to obtain a good compromise between


@@ -658,34 +701,34 @@ convergence.
 
 The last problem is to decide \emph{when} to switch from the
 asynchronous to the synchronous mode.  Here again, for the sake of simplicity, any
 
 The last problem is to decide \emph{when} to switch from the
 asynchronous to the synchronous mode.  Here again, for the sake of simplicity, any

-asynchronous mechanism for \emph{detecting} such instant is avoided and we
+asynchronous mechanism for \emph{detecting} such moment is avoided, and we

 prefer to use a mechanism that is local to each node. Obviously, that local system must
 rely neither on the number of local iterations done nor on the local
 convergence.  The former would slow down the fastest nodes according to the
 slowest ones.  The latter would provoke too much synchronization because the
 prefer to use a mechanism that is local to each node. Obviously, that local system must
 rely neither on the number of local iterations done nor on the local
 convergence.  The former would slow down the fastest nodes according to the
 slowest ones.  The latter would provoke too much synchronization because the

-residuals on all nodes commonly do not evolve in the same way and, in most
+residuals on all nodes generally do not evolve in the same way, and in most

 cases, there is a convergence wave phenomenon throughout the elements.  So, a
 good solution is to insert a local timer mechanism on each node with a given
 initial duration. Then, that duration may be modified during the execution
 according to the successive results of the synchronous sections.
 
 Another problem induced by entering synchronous mode from the asynchronous one
 cases, there is a convergence wave phenomenon throughout the elements.  So, a
 good solution is to insert a local timer mechanism on each node with a given
 initial duration. Then, that duration may be modified during the execution
 according to the successive results of the synchronous sections.
 
 Another problem induced by entering synchronous mode from the asynchronous one

-is the possibility to receive some data messages
+is the possibility of receiving some data messages

 from previous asynchronous iterations during synchronous iterations. This could lead to deadlocks. In order
 from previous asynchronous iterations during synchronous iterations. This could lead to deadlocks. In order

-to avoid this, a wait of the end of previous send is added to the
+to avoid this, a wait for the end of previous send is added to the

 transition between the two modes.  This is implemented by replacing the variable
 transition between the two modes.  This is implemented by replacing the variable

-\texttt{SendsInProgress} by a mutex \texttt{lockSendsDone} which is unlocked
+\texttt{SendsInProgress} with a mutex \texttt{lockSendsDone} which is unlocked

 once all the messages have been sent in the sending function.  Moreover, it is
 also necessary to stamp data messages\index{message!stamping} (by the function \texttt{stampData}) with
 once all the messages have been sent in the sending function.  Moreover, it is
 also necessary to stamp data messages\index{message!stamping} (by the function \texttt{stampData}) with

-a Boolean indicating whether they have been sent during a synchronous or
+a boolean indicating whether they have been sent during a synchronous or

 asynchronous iteration.  Then, the \texttt{lockRec} mutex is unlocked only
 after to the complete reception of data messages from synchronous
 iterations.  The message ordering of point-to-point communications in MPI
 asynchronous iteration.  Then, the \texttt{lockRec} mutex is unlocked only
 after to the complete reception of data messages from synchronous
 iterations.  The message ordering of point-to-point communications in MPI

-together with the barrier at the end of each iteration ensure two important
+and the barrier at the end of each iteration ensure two important

 properties of this mechanism. First, data messages from previous
 asynchronous iterations will be received but not taken into account during
 properties of this mechanism. First, data messages from previous
 asynchronous iterations will be received but not taken into account during

-synchronous sections. Then, a data message from a synchronous
-iteration cannot be received in another synchronous iteration.  In the
+synchronous sections. Then, a data message from a given synchronous
+iteration cannot be received during another synchronous iteration.  In the

 asynchronous sections, no additional mechanism is needed as there are no such
 constraints concerning the data receptions.
 
 asynchronous sections, no additional mechanism is needed as there are no such
 constraints concerning the data receptions.
 


@@ -694,30 +737,30 @@ Finally, the required modifications of the previous scheme
 are mainly related to the computing thread.  Small additions or modifications
 are also required in the main process and the other threads.
 
 are mainly related to the computing thread.  Small additions or modifications
 are also required in the main process and the other threads.
 

-In the main process, two new variables are added to store respectively the main
+In the main process, two new variables are added to store the main

 operating mode of the iterative process (\texttt{mainMode}) and the duration of
 asynchronous sections (\texttt{asyncDuration}). Those variables are
 operating mode of the iterative process (\texttt{mainMode}) and the duration of
 asynchronous sections (\texttt{asyncDuration}). Those variables are

-initialized by the programmer. The \texttt{lockSendsDone} mutex is also declared,
-initialized (locked) and destroyed with the other mutex in this process.
+initialized by the programmer. The mutex \texttt{lockSendsDone} is also declared,
+initialized (locked), and destroyed with the other mutex in this process.

 
 In the computing function, shown in \Lst{algo:ch6p2AsyncSyncComp}, the
 
 In the computing function, shown in \Lst{algo:ch6p2AsyncSyncComp}, the

-modifications consist of the insertion of the timer mechanism and of the conditions
+modifications consist of the insertion of the timer mechanism and the tests

 to differentiate the actions to be done in each mode.  Some additional variables
 are also required to store the current operating mode in action during the
 execution (\texttt{curMode}), the starting time of the current asynchronous
 to differentiate the actions to be done in each mode.  Some additional variables
 are also required to store the current operating mode in action during the
 execution (\texttt{curMode}), the starting time of the current asynchronous

-section (\texttt{asyncStart}) and the number of successive synchronous
+section (\texttt{asyncStart}), and the number of successive synchronous

 iterations done (\texttt{nbSyncIter}).
 
 %\begin{algorithm}[H]
 %  \caption{Computing function in the final asynchronous scheme.}% without GPU computing.}
 %  \label{algo:ch6p2AsyncSyncComp}
 %\pagebreak
 iterations done (\texttt{nbSyncIter}).
 
 %\begin{algorithm}[H]
 %  \caption{Computing function in the final asynchronous scheme.}% without GPU computing.}
 %  \label{algo:ch6p2AsyncSyncComp}
 %\pagebreak

-\begin{Listing}{algo:ch6p2AsyncSyncComp}{Computing function in the final asynchronous scheme}% without GPU computing.}
+\begin{Listing}{algo:ch6p2AsyncSyncComp}{computing function in the final asynchronous scheme}% without GPU computing.}

 // Variables declarations and initialization
 ...
 // Variables declarations and initialization
 ...

-OpMode curMode = SYNC; // Current operating mode (always begin in sync)
-double asyncStart;     // Starting time of the current async section
-int nbSyncIter = 0;    // Number of sync iterations done in async mode
+OpMode curMode = SYNC;// Current operating mode (always begin in sync)
+double asyncStart;    // Starting time of the current async section
+int nbSyncIter = 0;   // Number of sync iterations done in async mode

 
 // Computation loop
 while(!Finished){
 
 // Computation loop
 while(!Finished){


@@ -726,14 +769,15 @@ while(!Finished){
     // Entering synchronous mode when asyncDuration is reached
 @%    // (additional conditions can be specified if needed) 
 @    if(MPI_Wtime() - asyncStart >= asyncDuration){
     // Entering synchronous mode when asyncDuration is reached
 @%    // (additional conditions can be specified if needed) 
 @    if(MPI_Wtime() - asyncStart >= asyncDuration){

-      // Waiting for the end of previous sends before starting sync mode
+    // Waiting for the end of previous sends before starting sync mode

       omp_set_lock(&lockSendsDone);
       curMode = SYNC;              // Entering synchronous mode
       stampData(dataToSend, SYNC); // Mark data to send with sync flag
       nbSyncIter = 0;
     }
   }else{
       omp_set_lock(&lockSendsDone);
       curMode = SYNC;              // Entering synchronous mode
       stampData(dataToSend, SYNC); // Mark data to send with sync flag
       nbSyncIter = 0;
     }
   }else{

-    // In main async mode, going back to async mode when the max number of sync iterations are done
+   // In main async mode, going back to async mode when the max number 
+   // of sync iterations are done

     if(mainMode == ASYNC){
       nbSyncIter++; // Update of the number of sync iterations done
       if(nbSyncIter == 2){
     if(mainMode == ASYNC){
       nbSyncIter++; // Update of the number of sync iterations done
       if(nbSyncIter == 2){


@@ -744,7 +788,7 @@ while(!Finished){
     }
   }
 
     }
   }
 

-  // Sendings of data dependencies
+  // Sending of data dependencies

   if(curMode == SYNC || !SendsInProgress){
     ... 
   }
   if(curMode == SYNC || !SendsInProgress){
     ... 
   }


@@ -755,18 +799,18 @@ while(!Finished){
   }  
 
   // Local computation 
   }  
 
   // Local computation 

-  // (init of residual, arrays swapping and iteration computation)
+  // (init of residual, arrays swapping, and iteration computation)

   ...
 
   ...
 

-  // Checking of convergences (local & global) only in sync mode
+  // Checking convergences (local & global) only in sync mode

   if(curMode == SYNC){
     // Local convergence checking (residual under threshold)
     ...
     // Blocking global exchange of local states of the nodes
     ...
     // Determination of global convergence (all nodes in local CV)
   if(curMode == SYNC){
     // Local convergence checking (residual under threshold)
     ...
     // Blocking global exchange of local states of the nodes
     ...
     // Determination of global convergence (all nodes in local CV)

-    //    Stop of the iterative process and sending of end messages
-    // or Re-initialization of state information and iteration barrier
+    //    Stopping the iterative process and sending end messages
+    // or reinitialization of state information and iteration barrier

     ...
     }
   }
     ...
     }
   }


@@ -778,36 +822,38 @@ while(!Finished){
 %\end{algorithm}
 
 In the sending function, the only modification is the replacement in line~11 of
 %\end{algorithm}
 
 In the sending function, the only modification is the replacement in line~11 of

-the assignment of variable \texttt{SendsInProgress} by the unlocking of
+the assignment of variable \texttt{SendsInProgress} with the unlocking of

 \texttt{lockSendsDone}.  Finally, in the reception function, the only
 \texttt{lockSendsDone}.  Finally, in the reception function, the only

-modification is the insertion before line~19
+modification is the insertion before line~21

 of \Lst{algo:ch6p2BasicAsyncReceptions} of the extraction of the stamp from the
 message and its counting among the receipts only if the stamp is \texttt{SYNC}.
 
 The final step to get our complete scheme using GPU is to insert the GPU
 management in the computing thread.  The first possibility, detailed
 in \Lst{algo:ch6p2syncGPU}, is to simply replace the
 of \Lst{algo:ch6p2BasicAsyncReceptions} of the extraction of the stamp from the
 message and its counting among the receipts only if the stamp is \texttt{SYNC}.
 
 The final step to get our complete scheme using GPU is to insert the GPU
 management in the computing thread.  The first possibility, detailed
 in \Lst{algo:ch6p2syncGPU}, is to simply replace the

-CPU kernel (lines~41-43 in \Lst{algo:ch6p2AsyncSyncComp}) by a blocking GPU kernel call.  This includes data
+CPU kernel (lines~42--44 in \Lst{algo:ch6p2AsyncSyncComp}) by a blocking GPU kernel call.  This includes data

 transfers from the node RAM to the GPU RAM, the launching of the GPU kernel, the
 transfers from the node RAM to the GPU RAM, the launching of the GPU kernel, the

-waiting for kernel completion and the results transfers from GPU RAM to
+waiting for kernel completion, and the results transfers from GPU RAM to

 node RAM.
 
 %\begin{algorithm}[H]
 %  \caption{Computing function in the final asynchronous scheme.}
 %  \label{algo:ch6p2syncGPU}
 node RAM.
 
 %\begin{algorithm}[H]
 %  \caption{Computing function in the final asynchronous scheme.}
 %  \label{algo:ch6p2syncGPU}

-\begin{Listing}{algo:ch6p2syncGPU}{Computing function in the final asynchronous scheme}
+\begin{Listing}{algo:ch6p2syncGPU}{computing function in the final asynchronous scheme}

 // Variables declarations and initialization
 ...
 dim3 Dg, Db; // CUDA kernel grids
 
 // Computation loop
 while(!Finished){
 // Variables declarations and initialization
 ...
 dim3 Dg, Db; // CUDA kernel grids
 
 // Computation loop
 while(!Finished){

-  // Determination of the dynamic operating mode, sendings of data dependencies and blocking data receptions in sync mode
+  // Determination of the dynamic operating mode, sendings of data
+  // dependencies, and blocking data receptions in sync mode

   ...
   // Local GPU computation
   // Data transfers from node RAM to GPU
   CHECK_CUDA_SUCCESS(cudaMemcpyToSymbol(dataOnGPU, dataInRAM, inputsSize, 0, cudaMemcpyHostToDevice), "Data transfer");
   ...
   // Local GPU computation
   // Data transfers from node RAM to GPU
   CHECK_CUDA_SUCCESS(cudaMemcpyToSymbol(dataOnGPU, dataInRAM, inputsSize, 0, cudaMemcpyHostToDevice), "Data transfer");

-  ... // There may be several data transfers: typically A and b in linear problems
+  ... // There may be several data transfers: typically A and b in 
+      // linear problems of the form A.x = b

   // GPU grid definition
   Db.x = BLOCK_SIZE_X; // BLOCK_SIZE_# are kernel design dependent
   Db.y = BLOCK_SIZE_Y;
   // GPU grid definition
   Db.x = BLOCK_SIZE_X; // BLOCK_SIZE_# are kernel design dependent
   Db.y = BLOCK_SIZE_Y;


@@ -833,44 +879,44 @@ while(!Finished){
 %\end{algorithm}
 
 This scheme provides asynchronism through a cluster of GPUs as well as a
 %\end{algorithm}
 
 This scheme provides asynchronism through a cluster of GPUs as well as a

-complete overlap of communications with GPU computations (similarly
-to~\Sec{ch6:part1}).  However, the autonomy of GPU devices according to their
+complete overlap of communications with GPU computations (similar
+to the one described in~\Sec{ch6:part1}).  However, the autonomy of GPU devices according to their

 host can be further exploited in order to perform some computations on the CPU
 while the GPU kernel is running. The nature of computations that can be done by
 the CPU may vary depending on the application. For example, when processing data
 host can be further exploited in order to perform some computations on the CPU
 while the GPU kernel is running. The nature of computations that can be done by
 the CPU may vary depending on the application. For example, when processing data

-streams (pipelines), pre-processing of next data item and/or post-processing of
-previous result can be done on the CPU while the GPU is processing the current
+streams (pipelines), pre-processing of the next data item and/or post-processing
+of the previous result can be done on the CPU while the GPU is processing the current

 data item.  In other cases, the CPU can perform \emph{auxiliary}
 data item.  In other cases, the CPU can perform \emph{auxiliary}

-computations\index{computation!auxiliary}
+computations\index{computation auxiliary}

 that are not absolutely required to obtain the result but that may accelerate
 the entire iterative process.  Another possibility would be to distribute the
 main computations between the GPU and CPU. However, this
 that are not absolutely required to obtain the result but that may accelerate
 the entire iterative process.  Another possibility would be to distribute the
 main computations between the GPU and CPU. However, this

-usually leads to poor performance increases. This is mainly due to data
+usually leads to poor performance increases mainly due to data

 dependencies that often require additional transfers between CPU and GPU.
 
 So, if we consider that the application enables such overlap of
 computations, its implementation is straightforward as it consists in inserting
 dependencies that often require additional transfers between CPU and GPU.
 
 So, if we consider that the application enables such overlap of
 computations, its implementation is straightforward as it consists in inserting

-the additional CPU computations between lines~23 and~24
-in \Lst{algo:ch6p2syncGPU}.  Nevertheless, such scheme is fully efficient only
+the additional CPU computations between lines~25 and~26
+in \Lst{algo:ch6p2syncGPU}.  Nevertheless, such a scheme is fully efficient only

 if the computation times on both sides are similar.
 
 In some cases, especially with auxiliary computations, another interesting
 solution is to add a fourth CPU thread to perform them. This suppresses the
 duration constraint over those optional computations as they are performed in
 if the computation times on both sides are similar.
 
 In some cases, especially with auxiliary computations, another interesting
 solution is to add a fourth CPU thread to perform them. This suppresses the
 duration constraint over those optional computations as they are performed in

-parallel of the main iterative process, without blocking it.  Moreover, this
+parallel with the main iterative process, without blocking it.  Moreover, this

 scheme stays coherent with current architectures as most nodes include four CPU
 cores.  The algorithmic scheme of such context of complete overlap of
 CPU/GPU computations and communications is described in
 scheme stays coherent with current architectures as most nodes include four CPU
 cores.  The algorithmic scheme of such context of complete overlap of
 CPU/GPU computations and communications is described in

-Listings~\ref{algo:ch6p2FullOverAsyncMain},~\ref{algo:ch6p2FullOverAsyncComp1}
-and~\ref{algo:ch6p2FullOverAsyncComp2}, where we suppose that  auxiliary
+Listings~\ref{algo:ch6p2FullOverAsyncMain},~\ref{algo:ch6p2FullOverAsyncComp1},
+and~\ref{algo:ch6p2FullOverAsyncComp2}, where we assume that  auxiliary

 computations use intermediate results of the main computation process from any previous iteration. This may be
 different according to the application.
 
 %\begin{algorithm}[H]
 %  \caption{Initialization of the main process of complete overlap with asynchronism.}
 % \label{algo:ch6p2FullOverAsyncMain}
 computations use intermediate results of the main computation process from any previous iteration. This may be
 different according to the application.
 
 %\begin{algorithm}[H]
 %  \caption{Initialization of the main process of complete overlap with asynchronism.}
 % \label{algo:ch6p2FullOverAsyncMain}

-\pagebreak
-\begin{Listing}{algo:ch6p2FullOverAsyncMain}{Initialization of the main process of complete overlap with asynchronism}
+%\pagebreak
+\begin{Listing}{algo:ch6p2FullOverAsyncMain}{initialization of the main process of complete overlap with asynchronism}

 // Variables declarations and initialization
 ...
 omp_lock_t lockAux;   // Informs main thread about new aux results
 // Variables declarations and initialization
 ...
 omp_lock_t lockAux;   // Informs main thread about new aux results


@@ -878,12 +924,13 @@ omp_lock_t lockRes;   // Informs aux thread about new results
 omp_lock_t lockWrite; // Controls exclusion of results access
 ... auxRes ... ;      // Results of auxiliary computations 
 
 omp_lock_t lockWrite; // Controls exclusion of results access
 ... auxRes ... ;      // Results of auxiliary computations 
 

-// Parameters reading, MPI initialization, data initialization and distribution
+// Parameters reading, MPI initialization, and data initialization and
+// distribution

 ...
 // OpenMP initialization
 ...
 omp_init_lock(&lockAux);
 ...
 // OpenMP initialization
 ...
 omp_init_lock(&lockAux);

-omp_set_lock(&lockAux);  // Unlocked when new aux results are available
+omp_set_lock(&lockAux);//Unlocked when new aux results are available

 omp_init_lock(&lockRes);
 omp_set_lock(&lockRes);  // Unlocked when new results are available
 omp_init_lock(&lockWrite);
 omp_init_lock(&lockRes);
 omp_set_lock(&lockRes);  // Unlocked when new results are available
 omp_init_lock(&lockWrite);


@@ -930,30 +977,32 @@ MPI_Finalize();
 %\begin{algorithm}[H]
 %  \caption{Computing function in the final asynchronous scheme with CPU/GPU overlap.}
 %  \label{algo:ch6p2FullOverAsyncComp1}
 %\begin{algorithm}[H]
 %  \caption{Computing function in the final asynchronous scheme with CPU/GPU overlap.}
 %  \label{algo:ch6p2FullOverAsyncComp1}

-\pagebreak
-\begin{Listing}{algo:ch6p2FullOverAsyncComp1}{Computing function in the final asynchronous scheme with CPU/GPU overlap}
+%\pagebreak
+\begin{Listing}{algo:ch6p2FullOverAsyncComp1}{computing function in the final asynchronous scheme with CPU/GPU overlap}

 // Variables declarations and initialization
 ...
 dim3 Dg, Db; // CUDA kernel grids
 
 // Computation loop
 while(!Finished){
 // Variables declarations and initialization
 ...
 dim3 Dg, Db; // CUDA kernel grids
 
 // Computation loop
 while(!Finished){

-  // Determination of the dynamic operating mode, sendings of data dependencies and blocking data receptions in sync mode
+  // Determination of the dynamic operating mode, sending of data 
+  // dependencies, and blocking data receptions in sync mode

   ...
   // Local GPU computation
   ...
   // Local GPU computation

-  // Data transfers from node RAM to GPU, GPU grid definition and init of shared mem
+  // Data transfers from node RAM to GPU, GPU grid definition,  
+  // and init of shared memory

   CHECK_CUDA_SUCCESS(cudaMemcpyToSymbol(dataOnGPU, dataInRAM, inputsSize, 0, cudaMemcpyHostToDevice), "Data transfer");
   ...
   // Kernel call
   gpuKernelName<<<Dg,Db>>>(... @\emph{kernel parameters}@ ...);
   CHECK_CUDA_SUCCESS(cudaMemcpyToSymbol(dataOnGPU, dataInRAM, inputsSize, 0, cudaMemcpyHostToDevice), "Data transfer");
   ...
   // Kernel call
   gpuKernelName<<<Dg,Db>>>(... @\emph{kernel parameters}@ ...);

-  // Potential pre/post-treatments in pipeline like computations
+  // Potential pre-/post-treatments in pipeline-like computations

   ...
   // Waiting for kernel completion
   cudaDeviceSynchronize(); 
   // Results transfer from GPU to node RAM
   omp_set_lock(&lockWrite); // Wait for write access to resultsInRam
   CHECK_CUDA_SUCCESS(cudaMemcpyFromSymbol(resultsInRam, resultsOnGPU, resultsSize, 0, cudaMemcpyDeviceToHost), "Results transfer");
   ...
   // Waiting for kernel completion
   cudaDeviceSynchronize(); 
   // Results transfer from GPU to node RAM
   omp_set_lock(&lockWrite); // Wait for write access to resultsInRam
   CHECK_CUDA_SUCCESS(cudaMemcpyFromSymbol(resultsInRam, resultsOnGPU, resultsSize, 0, cudaMemcpyDeviceToHost), "Results transfer");

-  // Potential post-treatments in non-pipeline computations
+  // Potential post-treatments in non pipeline computations

   ...
   omp_unset_lock(&lockWrite); // Give back read access to aux thread
   omp_test_lock(&lockRes);
   ...
   omp_unset_lock(&lockWrite); // Give back read access to aux thread
   omp_test_lock(&lockRes);


@@ -971,13 +1020,13 @@ while(!Finished){
     ...
     // Determination of global convergence (all nodes in local CV)
     if(cvLocale == 1 && nbCVLocales == nbP-1){
     ...
     // Determination of global convergence (all nodes in local CV)
     if(cvLocale == 1 && nbCVLocales == nbP-1){

-      // Stop of the iterative process and sending of end messages
+      // Stopping the iterative process and sending end messages

       ...
       ...

-      // Unlocking of aux thread for termination
+      // Unlocking aux thread for termination

       omp_test_lock(&lockRes);
       omp_unset_lock(&lockRes);
     }else{
       omp_test_lock(&lockRes);
       omp_unset_lock(&lockRes);
     }else{

-      // Re-initialization of state information and iteration barrier
+      // Reinitialization of state information and iteration barrier

       ...
     }
   }
       ...
     }
   }


@@ -988,8 +1037,8 @@ while(!Finished){
 %\begin{algorithm}[H]
 %  \caption{Auxiliary computing function in the final asynchronous scheme with CPU/GPU overlap.}
 %  \label{algo:ch6p2FullOverAsyncComp2}
 %\begin{algorithm}[H]
 %  \caption{Auxiliary computing function in the final asynchronous scheme with CPU/GPU overlap.}
 %  \label{algo:ch6p2FullOverAsyncComp2}

-\pagebreak
-\begin{Listing}{algo:ch6p2FullOverAsyncComp2}{Auxiliary computing function in the final asynchronous scheme with CPU/GPU overlap}
+%\pagebreak
+\begin{Listing}{algo:ch6p2FullOverAsyncComp2}{auxiliary computing function in the final asynchronous scheme with CPU/GPU overlap}

 // Variables declarations and initialization
 ... auxInput ... // Local array for input data
 
 // Variables declarations and initialization
 ... auxInput ... // Local array for input data
 


@@ -1002,13 +1051,13 @@ while(!Finished){
     for(ind=0; ind<resultsSize; ++ind){
       auxInput[ind] = resultsInRam[ind];
     }
     for(ind=0; ind<resultsSize; ++ind){
       auxInput[ind] = resultsInRam[ind];
     }

-    omp_unset_lock(&lockWrite); // Give back write access to main thread
+    omp_unset_lock(&lockWrite);//Give back write access to main thread

     // Auxiliary computations with possible interruption at the end
     for(ind=0; ind<auxSize && !Finished; ++ind){
       // Computation of auxRes array according to auxInput
       ...
     }
     // Auxiliary computations with possible interruption at the end
     for(ind=0; ind<auxSize && !Finished; ++ind){
       // Computation of auxRes array according to auxInput
       ...
     }

-    // Informs main thread that new aux results are available in auxData
+// Informs main thread that new aux results are available in auxData

     omp_test_lock(&lockAux); // Ensures mutex is locked when unlocking
     omp_unset_lock(&lockAux);
   }
     omp_test_lock(&lockAux); // Ensures mutex is locked when unlocking
     omp_unset_lock(&lockAux);
   }


@@ -1017,23 +1066,23 @@ while(!Finished){
 %\end{algorithm}
 
 As can be seen in \Lst{algo:ch6p2FullOverAsyncMain}, there are three additional
 %\end{algorithm}
 
 As can be seen in \Lst{algo:ch6p2FullOverAsyncMain}, there are three additional

-mutex (\texttt{lockAux}, \texttt{lockRes} and \texttt{lockWrite}) that are used
-respectively to inform the main computation thread that new auxiliary results
-are available (lines~20-21 in \Lst{algo:ch6p2FullOverAsyncComp2} and line~29 in
+mutex (\texttt{lockAux}, \texttt{lockRes}, and \texttt{lockWrite}) that are used
+to inform the main computation thread that new auxiliary results
+are available (lines~20--21 in \Lst{algo:ch6p2FullOverAsyncComp2} and line~31 in

 \Lst{algo:ch6p2FullOverAsyncComp1}), to inform the auxiliary thread that new
 \Lst{algo:ch6p2FullOverAsyncComp1}), to inform the auxiliary thread that new

-results from the main thread are available (lines~25-26
+results from the main thread are available (lines~27--28

 in \Lst{algo:ch6p2FullOverAsyncComp1} and line~7
 in \Lst{algo:ch6p2FullOverAsyncComp2}), and to perform exclusive accesses to the
 in \Lst{algo:ch6p2FullOverAsyncComp1} and line~7
 in \Lst{algo:ch6p2FullOverAsyncComp2}), and to perform exclusive accesses to the

-results from those two threads (lines~20,~24
-in \Lst{algo:ch6p2FullOverAsyncComp1} and 9,~13
+results from those two threads (lines~22 and 26
+in \Lst{algo:ch6p2FullOverAsyncComp1} and 9 and 13

 in \Lst{algo:ch6p2FullOverAsyncComp2}).  Also, an additional array
 (\texttt{auxRes}) is required to store the results of the auxiliary computations
 as well as a local array for the input of the auxiliary function
 (\texttt{auxInput}). That last function has the same general organization as the
 in \Lst{algo:ch6p2FullOverAsyncComp2}).  Also, an additional array
 (\texttt{auxRes}) is required to store the results of the auxiliary computations
 as well as a local array for the input of the auxiliary function
 (\texttt{auxInput}). That last function has the same general organization as the

-send/receive ones, that is a global loop conditioned by the end of the global
+send/receive ones, that is, a global loop conditioned by the end of the global

 process.  At each iteration in this function, the thread waits for the
 availability of new results produced by the main computation thread. This avoids
 process.  At each iteration in this function, the thread waits for the
 availability of new results produced by the main computation thread. This avoids

-to perform the same computations several times with the same input data.
+ performing the same computations several times with the same input data.

 Then, input data of auxiliary computations
 % (as supposed here, they often
 % correspond to the results of the main computations, but may sometimes be
 Then, input data of auxiliary computations
 % (as supposed here, they often
 % correspond to the results of the main computations, but may sometimes be


@@ -1042,24 +1091,26 @@ is copied with a mutual exclusion mechanism. Finally, auxiliary
 computations are performed.  When they are completed, the associated mutex is
 unlocked to signal the availability of those auxiliary results to the main
 computing thread.  The main thread regularly checks this availability at the end
 computations are performed.  When they are completed, the associated mutex is
 unlocked to signal the availability of those auxiliary results to the main
 computing thread.  The main thread regularly checks this availability at the end

-of its iterations and takes them into account whenever this is possible.
+of its iterations and takes them into account whenever possible.

 
 

-Finally, we  obtain an algorithmic  scheme allowing maximal  overlap between
-CPU and  GPU computations as well  as communications. It is  worth noticing that
-such scheme is also usable for systems without GPUs but 4-cores nodes.
+Finally, we  obtain an algorithmic  scheme allowing maximal overlap  between CPU
+and GPU computations  as well as communications. It is  worth noticing that such
+scheme is also efficiently usable for systems without GPUs but with nodes having
+at least four cores. In such contexts, each thread in \Lst{algo:ch6p2FullOverAsyncMain} can
+be executed on distinct cores.

 
 \subsection{Experimental validation}
 \label{sec:ch6p2expes}
 
 As in~\Sec{ch6:part1},  we validate the  feasibility of our  asynchronous scheme
 with  some experiments  performed with  a representative  example  of scientific
 
 \subsection{Experimental validation}
 \label{sec:ch6p2expes}
 
 As in~\Sec{ch6:part1},  we validate the  feasibility of our  asynchronous scheme
 with  some experiments  performed with  a representative  example  of scientific

-application.      It     is    a     three-dimensional     version    of     the
-advection-diffusion-reaction   process\index{PDE example}  that   models  the   evolution   of  the
+application.  This     three-dimensional     version    of     the
+advection-diffusion-reaction   process\index{PDE example}  models  the   evolution   of  the

 concentrations of  two chemical  species in shallow  waters. As this  process is
 dynamic in time,  the simulation is performed for a  given number of consecutive
 time steps.  This  implies two nested loops in the  iterative process, the outer
 one for the time  steps and the inner one for solving  the problem at each time.
 concentrations of  two chemical  species in shallow  waters. As this  process is
 dynamic in time,  the simulation is performed for a  given number of consecutive
 time steps.  This  implies two nested loops in the  iterative process, the outer
 one for the time  steps and the inner one for solving  the problem at each time.

-Full details about this PDE  problem can be found in~\cite{ChapNRJ2011}. That
+Full details about this PDE  problem can be found in~\cite{ChapNRJ2011}. This

 two-stage  iterative process  implies a  few adaptations  of the  general scheme
 presented above in order to include the  outer iterations over  the time steps,  but the
 inner iterative process closely follows the same scheme.
 two-stage  iterative process  implies a  few adaptations  of the  general scheme
 presented above in order to include the  outer iterations over  the time steps,  but the
 inner iterative process closely follows the same scheme.


@@ -1067,7 +1118,7 @@ inner iterative process closely follows the same scheme.
 We show two  series of experiments performed with 16 nodes  of the first cluster
 described  in~\Sec{ch6:p1expes}.  The  first one  deals with  the  comparison of
 synchronous and asynchronous computations.  The second one is related to the use
 We show two  series of experiments performed with 16 nodes  of the first cluster
 described  in~\Sec{ch6:p1expes}.  The  first one  deals with  the  comparison of
 synchronous and asynchronous computations.  The second one is related to the use

-of auxiliary computations. In the context of our PDE application, they consist in
+of auxiliary computations. In the context of our PDE application, they consist of

 the update of the Jacobian of the system.
 
 \subsubsection*{Synchronous and asynchronous computations}
 the update of the Jacobian of the system.
 
 \subsubsection*{Synchronous and asynchronous computations}


@@ -1089,10 +1140,10 @@ corresponds in fact to $30\times30\times30\times2$ values.
   \label{fig:ch6p2syncasync}
 \end{figure}
 
   \label{fig:ch6p2syncasync}
 \end{figure}
 

-The results obtained show that  the asynchronous version is sensibly faster than
+The results obtained show that  the asynchronous version is significantly faster than

 the synchronous one  for smaller problem sizes, then it  becomes similar or even
 a bit slower  for  larger problem  sizes.   A  closer  comparison of computation  and
 the synchronous one  for smaller problem sizes, then it  becomes similar or even
 a bit slower  for  larger problem  sizes.   A  closer  comparison of computation  and

-communication times  in each execution  confirms that this behavior  is consistent.
+communication times  of each execution  confirms that this behavior  is consistent.

 The  asynchronous version is  interesting if communication time
 is similar or  larger than computation time.  In our  example, this is the
 case up  to a problem  size between 50 and 60.   Then, computations  become longer
 The  asynchronous version is  interesting if communication time
 is similar or  larger than computation time.  In our  example, this is the
 case up  to a problem  size between 50 and 60.   Then, computations  become longer


@@ -1112,18 +1163,18 @@ the system.  In such cases, it is  necessary to compute the vector $\Delta x$ in
 $F'\times \Delta  x=-F$ to update $x$ with  $\Delta x$. There are  two levels of
 iterations, the  inner level to get a  stabilized version of $x$,  and the outer
 level to compute $x$ at the successive time steps in the simulation process.  In
 $F'\times \Delta  x=-F$ to update $x$ with  $\Delta x$. There are  two levels of
 iterations, the  inner level to get a  stabilized version of $x$,  and the outer
 level to compute $x$ at the successive time steps in the simulation process.  In

-this context,  classical algorithms either compute  $F'$ only at the  first iteration
+this context,  classic algorithms either compute  $F'$ at only the  first iteration

 of each time step or at some iterations but not all because the  computation of $F'$ is done
 in the main iterative process and it has a relatively high computing cost.
 
 However, with the scheme presented above, it is possible to continuously compute
 of each time step or at some iterations but not all because the  computation of $F'$ is done
 in the main iterative process and it has a relatively high computing cost.
 
 However, with the scheme presented above, it is possible to continuously compute

-new  versions  of  $F'$  in  parallel  to the  main  iterative  process  without
+new  versions  of  $F'$  in  parallel with the  main  iterative  process  without

 penalizing  it. Hence,  $F'$ is  updated  as often  as possible  and taken  into
 account in  the main computations  when it is  relevant. So, the  Newton process
 should be  accelerated a little bit.
 
 We  compare the  performance obtained  with overlapped  Jacobian  updatings and
 penalizing  it. Hence,  $F'$ is  updated  as often  as possible  and taken  into
 account in  the main computations  when it is  relevant. So, the  Newton process
 should be  accelerated a little bit.
 
 We  compare the  performance obtained  with overlapped  Jacobian  updatings and

-non-overlapped ones for several problem sizes, see~\Fig{fig:ch6p2aux}.
+non overlapped ones for several problem sizes (see~\Fig{fig:ch6p2aux}).

 \begin{figure}[h]
   \centering
   \includegraphics[width=.75\columnwidth]{Chapters/chapter6/curves/recouvs.pdf}
 \begin{figure}[h]
   \centering
   \includegraphics[width=.75\columnwidth]{Chapters/chapter6/curves/recouvs.pdf}


@@ -1133,31 +1184,31 @@ non-overlapped ones for several problem sizes, see~\Fig{fig:ch6p2aux}.
 \end{figure}
 
 The  overlap is  clearly efficient  as  the computation  times with  overlapping
 \end{figure}
 
 The  overlap is  clearly efficient  as  the computation  times with  overlapping

-Jacobian updatings  are much better  than without overlap.  Moreover,  the ratio
-between the  two versions tend  to increase with  the problem size, which  is as
+Jacobian updatings  are much better  than the ones without overlap.  Moreover,  the ratio
+between the  two versions tends  to increase with  the problem size, which  is as

 expected. Also, we have tested the application without auxiliary computations at
 all, that is, the  Jacobian is computed only once at the  beginning of each time
 step of the  simulation. The results for this last version  are quite similar to
 the overlapped auxiliary computations, and  even better for small problem sizes.
 expected. Also, we have tested the application without auxiliary computations at
 all, that is, the  Jacobian is computed only once at the  beginning of each time
 step of the  simulation. The results for this last version  are quite similar to
 the overlapped auxiliary computations, and  even better for small problem sizes.

-The fact that no sensible gain can be seen on this range of problem sizes is due
+The fact that no significant gain can be seen on this range of problem sizes is due

 to the limited number of Jacobian updates taken into  account in the main
 computation.  This happens when  the Jacobian update is as long as
 several  iterations of  the main  process. So,  the benefit  is reduced  in this
 particular case.
 
 to the limited number of Jacobian updates taken into  account in the main
 computation.  This happens when  the Jacobian update is as long as
 several  iterations of  the main  process. So,  the benefit  is reduced  in this
 particular case.
 

-Those results show two things; first, auxiliary computations do not induce great
-overhead in  the whole  process.  Second, for  this particular  application the
+Those results show two things. First, auxiliary computations do not induce great
+overhead  in the  whole process.   Second, for  this particular  application the

 choice of updating the Jacobian  matrix as auxiliary computations does not speed
 up the iterative process.  This does  not question the parallel scheme in itself
 choice of updating the Jacobian  matrix as auxiliary computations does not speed
 up the iterative process.  This does  not question the parallel scheme in itself

-but   merely  points  out   the  difficulty   to  identify   relevant  auxiliary
+but  merely  points  out   the  difficulty  of  identifying  relevant  auxiliary

 computations.  Indeed, this identification depends on the considered application
 and requires a profound specialized analysis.
 
 Another  interesting choice  could be  the  computation of  load estimation  for
 dynamic load  balancing, especially in decentralized  diffusion strategies where
 computations.  Indeed, this identification depends on the considered application
 and requires a profound specialized analysis.
 
 Another  interesting choice  could be  the  computation of  load estimation  for
 dynamic load  balancing, especially in decentralized  diffusion strategies where

-loads are  transferred between  neighboring nodes~\cite{BCVG11}.  In  such case,
+loads are  transferred between  neighboring nodes~\cite{BCVG11}.  In  such a case,

 the load evaluation and the comparison  with other nodes can be done in parallel
 the load evaluation and the comparison  with other nodes can be done in parallel

-of the  main computations without perturbing  them.
+with the  main computations without perturbing  them.

 
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