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

\section{General scheme of asynchronous parallel code with
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
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
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:
%% \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]
  \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}


\noindent
to an asynchronous iterative scheme of the form:\\
%% \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]
  \caption{Asynchronous iterative scheme}\label{algo:ch6p2async}
    $x^{0}=(x_{1}^{0},...,x_{n}^{0})$\;
    \For {$t=0,1,...$} {
      \For{ $i=1,...,n$} {
        $x_{i}^{t+1}=\left\{
      \begin{array}[h]{ll}
        x_i^t & \text{if } i \text{ is \emph{not} updated at iteration } i\\
        f_i(x_1^{s_1^i(t)},...,x_n^{s_n^i(t)}) & \text{if } i \text{ is updated at iteration } i
      \end{array}
    \right.$
      }
    }
\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,
  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
different computation speeds of  the computing elements and communication delays
between them.

The key feature of such algorithmic schemes
is  that they  may be  faster  than their  synchronous counterparts  due to  the
implied total overlap of computations  with communications: in fact, this scheme
suppresses all the  idle times induced by nodes  synchronizations between each iteration.

However, the efficiency of such a scheme is directly linked to the frequency at
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
this frequency does not only depend on the hardware configuration of the
parallel system but it also depends on the software that is used to
implement the algorithmic scheme.

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
evolved, in particular now 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
activities of the three types of devices that are 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
performance on the other side.  This is especially important for industries where
implementation and maintenance costs are strong constraints.

For the sake of clarity, we present the different algorithmic schemes in a progressive
order of complexity, from the basic asynchronous scheme to the complete scheme
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.

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.

Finally, in order to stay as clear as possible, only the parts of code and
variables related to the control of parallelism (communications, threads,...)
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
most of the cases, those buffers can correspond to the data arrays themselves to
avoid data copies.

\subsection{A basic asynchronous scheme}
\label{ch6:p2BasicAsync}

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
is important to remember that asynchronous communications provided in
communication libraries like MPI are not systematically performed in parallel of
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}
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
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
threads.

So, the global organization of this scheme is set up in \Lst{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}
// 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

// Parameters reading
...

// MPI initialization
MPI_Init_thread(argc, argv, MPI_THREAD_MULTIPLE, &provided);
MPI_Comm_size(MPI_COMM_WORLD, &nbP);
MPI_Comm_rank(MPI_COMM_WORLD, &numP);

// Data initialization and distribution among the nodes
...

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

#pragma omp parallel
{
  switch(omp_get_thread_num()){
    case COMPUTATION :
    computations(... @\emph{relevant parameters}@ ...);
    break;
    
    case SENDINGS :
    sendings();
    break;
    
    case RECEPTIONS :
    receptions();
    break;
  }
}

// Cleaning of OpenMP locks
omp_test_lock(&lockSend);
omp_unset_lock(&lockSend);
omp_destroy_lock(&lockSend);
omp_test_lock(&lockRec);
omp_unset_lock(&lockRec);
omp_destroy_lock(&lockRec);

// MPI termination
MPI_Finalize();
\end{Listing}
%\end{algorithm}

% \ANNOT{Est-ce qu'on laisse le 1er échange de données ou pas dans
%   l'algo~\ref{algo:ch6p2BasicAsyncComp} ? (lignes 16-19)\\
%   (ça n'est pas nécessaire si les données de
%   départ sont distribuées avec les dépendances qui correspondent, sinon il faut
%   le faire)} 

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
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}.

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

// Computation loop
while(!Finished){
  // Sendings of data dependencies if there is no previous sending in progress
  if(!SendsInProgress){
    // Potential copy of data to be sent in additional buffers
    ...
    // Change of sending state
    SendsInProgress = 1;
    omp_unset_lock(&lockSend);
  }
  
  // Blocking receptions at the first iteration
  if(iter == 1){
    omp_set_lock(&lockRec);
  }
  
  // Initialization of the residual
  residual = 0.0;
  // Swapping of data arrays (current and previous)
  tmp = current;      // Pointers swapping to avoid
  current = previous; // actual data copies between
  previous = tmp;     // the two data versions
  // Computation of current iteration over local data
  for(ind=0; ind<localSize; ++ind){
    // Updating of current array using previous array 
    ...
    // Updating of the residual
    // (max difference between two successive iterations)
    difference = fabs(current[ind] - previous[ind]);
    if(difference > residual){
      residual = difference;
    }
  }
    
  // Checking of the end of the process (residual under threshold)
  // Other conditions can be added to the termination detection
  if(residual <= Threshold){
    Finished = 1;
    omp_unset_lock(&lockSend); // Activation of end messages sendings
    MPI_Ssend(&Finished, 1, MPI_CHAR, numP, tagEnd, MPI_COMM_WORLD);
  }
  
  // Updating of the iteration number
  iter++;
}
\end{Listing}
%\end{algorithm}

As mentioned above, it can be seen in line~18 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
in line~23 and computed in lines~34-37, 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.

Finally, the local convergence is tested and updated when necessary. In line~44,
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
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
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
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
blocking synchronous communications.  The \texttt{SendsInProgress} boolean
allows the computing thread to skip data sendings as long as a previous sending
is in progress. This skip is possible due to the nature of asynchronous
algorithms that allows such \emph{message loss}\index{message!loss/miss} or \emph{message miss}. After
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{Listing}{algo:ch6p2BasicAsyncSendings}{Sending function in the basic asynchronous scheme}
// Variables declaration and initialization
...

while(!Finished){
  omp_set_lock(&lockSend); // Waiting for signal from the comp. thread
  if(!Finished){
    // Blocking synchronous sends to all dependencies
    for(i=0; i<nbDeps; ++i){
      MPI_Ssend(&dataToSend[deps[i]], nb_data, type_of_data, deps[i], tagCom, MPI_COMM_WORLD);
    }
    SendsInProgress = 0; // Indicates that the sendings are done
  }
}
// At the end of the process, sendings of final messages
for(i=0; i<nbDeps; ++i){
  MPI_Ssend(&Finished, 1, MPI_CHAR, deps[i], tagEnd, MPI_COMM_WORLD);
}
\end{Listing}
%\end{algorithm}

The last function, detailed in \Lst{algo:ch6p2BasicAsyncReceptions}, does all the messages receptions.
%\begin{algorithm}[H]
%  \caption{Reception function in the basic asynchronous scheme.}
%  \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
int nbEndMsg = 0;       // Number of end messages received
int arrived = 0;        // Boolean indicating if a message is arrived
int srcNd;              // Source node of the message
int size;               // Message size

// Main loop of receptions
while(!Finished){
  // Waiting for an incoming message
  MPI_Probe(MPI_ANY_SOURCE, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
  if(!Finished){
    // 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
       // 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
       if(countReceipts == 1 && ... @\emph{receptions from ALL dependencies}@ ...){
         omp_unset_lock(&lockRec);
         countReceipts = 0; // No more counting after first iteration
       }
       break;
      case tagEnd: // Management of end messages
       // Actual end message reception in dummy buffer
       MPI_Recv(dummyBuffer, 1, MPI_CHAR, status.MPI_SOURCE, tagEnd, MPI_COMM_WORLD, &status); 
       nbEndMsg++;
    }
  }
}

// Reception of pending messages and counting of end messages
do{ // Loop over the remaining incoming/waited 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_Recv(dummyBuffer, size, MPI_CHAR, status.MPI_SOURCE, status.MPI_TAG, MPI_COMM_WORLD, &status); 
  if(status.MPI_TAG == tagEnd){ // Counting of end messages
    nbEndMsg++;
  }
  MPI_Iprobe(MPI_ANY_SOURCE, MPI_ANY_TAG, MPI_COMM_WORLD, &arrived, &status);
}while(arrived == 1 || nbEndMsg < nbDeps + 1);
\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).

\medskip
So, with those 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
paragraph~\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.

\subsection{Synchronization of the  asynchronous scheme}
\label{ch6:p2SsyncOverAsync}

The presence of synchronization in the previous scheme may seem contradictory to our goal,
and obviously, it is neither the simplest way to obtain a synchronous scheme nor
the most efficient (as presented in~\Sec{ch6:part1}). However, it is necessary
for our global convergence detection strategy\index{convergence detection!global}. Recall that the
global convergence\index{convergence!global} is the extension of the local convergence\index{convergence!local} concept to all the
nodes. This implies that all the nodes have to be in local convergence at the
same time to achieve global convergence. Typically, if we use the residual and a
threshold to stop the iterative process, all the nodes have to continue their
local iterative process until \emph{all} of them obtain a residual under the
threshold.

% Moreover, it  allows the gathering  of both operating  modes in a  single source
% 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
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
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
implementation/maintenance costs are strong constraints. Moreover, although the
decentralized scheme does not slow down the computations, it requires more iterations than a synchronous version and
thus may induce longer detection times in some cases.  So, the solution we
present below is a good compromise between simplicity and efficiency.  It
consists in dynamically changing the operating mode between asynchronous and synchronous
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
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}
// 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

// Parameters reading
...
// MPI initialization
...
// Data initialization and distribution among the nodes
...
// 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
omp_init_lock(&lockIter);
omp_set_lock(&lockIter);   // Initially locked, unlocked when all "end of iteration" messages are received

// Threads launching
#pragma omp parallel
{
  switch(omp_get_thread_num()){
    ...
  }
}

// Cleaning of OpenMP locks
...
omp_test_lock(&lockStates);
omp_unset_lock(&lockStates);
omp_destroy_lock(&lockStates);
omp_test_lock(&lockIter);
omp_unset_lock(&lockIter);
omp_destroy_lock(&lockIter);

// MPI termination
MPI_Finalize();
\end{Listing}
%\end{algorithm}

As can be seen in \Lst{algo:ch6p2Sync}, the synchronization implies two
additional mutex.  The \texttt{lockStates} mutex is used to wait for the
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
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
are also two new variables that respectively 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}).

The computation thread is where most of the modifications take place, as shown
in \Lst{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}
// Variables declarations and initialization
...

// Computation loop
while(!Finished){
  // Sendings of data dependencies at @\emph{each}@ iteration
  // Potential copy of data to be sent in additional buffers
  ...
  omp_unset_lock(&lockSend);

  // Blocking receptions at @\emph{each}@ iteration
  omp_set_lock(&lockRec);
  
  // Local computation 
  // (init of residual, arrays swapping and iteration computation)
  ...
  
  // Checking of the stabilization of the local process
  // Other conditions than the residual can be added
  if(residual <= Threshold){
    localCV = 1;
  }else{
    localCV = 0;
  }
      
  // Global exchange of local states of the nodes
  for(ind=0; ind<nbP; ++ind){
    if(ind != numP){
      MPI_Ssend(&localCV, 1, MPI_CHAR, ind, tagState, MPI_COMM_WORLD);
    }
  }
  
  // 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)
  if(localCV + nbOtherCVs == nbP){
    // Entering global CV state
    Finished = 1;
    // Unlocking of sending thread to start sendings of end messages
    omp_unset_lock(&lockSend);
    MPI_Ssend(&Finished, 1, MPI_CHAR, numP, tagEnd, MPI_COMM_WORLD);
  }else{
    // Resetting of information about the states of the other nodes
    ...
    // Global barrier at the end of each iteration during the process
    for(ind=0; ind<nbP; ++ind){
      if(ind != numP){
        MPI_Ssend(&Finished, 1, MPI_CHAR, ind, tagIter, MPI_COMM_WORLD);
      }
    }
    omp_set_lock(&lockIter);
  }
  

  // Updating of the iteration number
  iter++;
}
\end{Listing}
%\end{algorithm}

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
lines~26-34, 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,
lines~38-42 change the \texttt{Finished} boolean to stop the iterative process,
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
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}.

In the reception function, given in \Lst{algo:ch6p2SyncReceptions}, there are
mainly two insertions (in lines~19-30 and 31-40), 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
lines~24-29 and 34-39 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
represent the local convergence (1) and the non-convergence (0).

%\begin{algorithm}[H]
%  \caption{Reception function in the synchronized scheme.}
%  \label{algo:ch6p2SyncReceptions}
\begin{Listing}{algo:ch6p2SyncReceptions}{Reception function in the synchronized scheme}
// Variables declarations and initialization
...
int nbStateMsg = 0; // Number of local state messages received
int nbIterMsg = 0;  // Number of "end of iteration" messages received
char recvdState;    // Received state from another node (0 or 1)

// Main loop of receptions
while(!Finished){
  // Waiting for an incoming message
  MPI_Probe(MPI_ANY_SOURCE, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
  if(!Finished){
    switch(status.MPI_TAG){ // Actions related to message type
      case tagCom: // Management of data messages
       ...
       break;
      case tagEnd: // Management of termination messages
       ...
       break;
      case tagState: // Management of local state messages
       // Actual reception of the message
       MPI_Recv(&recvdState, 1, MPI_CHAR, status.MPI_SOURCE, tagState, MPI_COMM_WORLD, &status); 
       // 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
       if(nbStateMsg == nbP-1){
         nbStateMsg = 0;
         omp_unset_lock(&lockStates);
       }
       break;
      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       
       if(nbIterMsg == nbP - 1){
         nbIterMsg = 0;
         omp_unset_lock(&lockIter);
       }
       break;
    }
  }
}

// Reception of pending messages and counting of end messages
do{ // Loop over the remaining incoming/waited messages
  ...
}while(arrived == 1 || nbEndMsg < nbDeps + 1);
\end{Listing}
%\end{algorithm}

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
scheme.

\subsection{Asynchronous scheme using MPI, OpenMP and CUDA}
\label{ch6:p2GPUAsync}

As mentioned above, the strategy proposed to obtain a good compromise between
simplicity and efficiency in the asynchronous scheme is to dynamically change
the operating mode of the process.  A good way to obtain a maximal
simplification of the final scheme while preserving good performance is to
perform local and global convergence detections only in synchronous
mode. Moreover, as two successive iterations are sufficient in synchronous mode
to detect local and global convergences, the key is to alternate some
asynchronous iterations with two synchronous iterations until
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
asynchronous mechanism for \emph{detecting} such instant 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
residuals on all nodes commonly 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
is the possibility to receive some data messages
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
transition between the two modes.  This is implemented by replacing the variable
\texttt{SendsInProgress} by 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
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
together with 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
synchronous sections. Then, a data message from a synchronous
iteration cannot be received in another synchronous iteration.  In the
asynchronous sections, no additional mechanism is needed as there are no such
constraints concerning the data receptions.

Finally, the required modifications of the previous scheme
% to make it dynamically change its operating mode (asynchronous or synchronous)
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
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.

In the computing function, shown in \Lst{algo:ch6p2AsyncSyncComp}, the
modifications consist of the insertion of the timer mechanism and of the conditions
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
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.}
// 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

// Computation loop
while(!Finished){
  // Determination of the dynamic operating mode
  if(curMode == ASYNC){
    // 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
      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
    if(mainMode == ASYNC){
      nbSyncIter++; // Update of the number of sync iterations done
      if(nbSyncIter == 2){
        curMode = ASYNC;              // Going back to async mode
        stampData(dataToSend, ASYNC); // Mark data to send
        asyncStart = MPI_Wtime();     // Get the async starting time
      }
    }
  }

  // Sendings of data dependencies
  if(curMode == SYNC || !SendsInProgress){
    ... 
  }

  // Blocking data receptions in sync mode
  if(curMode == SYNC){
    omp_set_lock(&lockRec);
  }  

  // Local computation 
  // (init of residual, arrays swapping and iteration computation)
  ...

  // Checking of 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)
    //    Stop of the iterative process and sending of end messages
    // or Re-initialization of state information and iteration barrier
    ...
    }
  }
    
  // Updating of the iteration number
  iter++;
}
\end{Listing}
%\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
\texttt{lockSendsDone}.  Finally, in the reception function, the only
modification is the insertion before line~19
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
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
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}
// 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
  ...
  // 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
  // GPU grid definition
  Db.x = BLOCK_SIZE_X; // BLOCK_SIZE_# are kernel design dependent
  Db.y = BLOCK_SIZE_Y;
  Db.z = BLOCK_SIZE_Z;
  Dg.x = localSize/BLOCK_SIZE_X + (localSize%BLOCK_SIZE_X ? 1 : 0);
  Dg.y = localSize/BLOCK_SIZE_Y + (localSize%BLOCK_SIZE_Y ? 1 : 0);
  Dg.z = localSize/BLOCK_SIZE_Z + (localSize%BLOCK_SIZE_Z ? 1 : 0);
  // Use of shared memory (when possible)
  cudaFuncSetCacheConfig(gpuKernelName, cudaFuncCachePreferShared);
  // Kernel call
  gpuKernelName<<<Dg,Db>>>(... @\emph{kernel parameters}@ ...);
  // Waiting for kernel completion
  cudaDeviceSynchronize(); 
  // Results transfer from GPU to node RAM
  CHECK_CUDA_SUCCESS(cudaMemcpyFromSymbol(resultsInRam, resultsOnGPU, resultsSize, 0, cudaMemcpyDeviceToHost), "Results transfer");
  // Potential post-treatment of results on the CPU
  ...

  // Convergences checking
  ...
}
\end{Listing}
%\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
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
data item.  In other cases, the CPU can perform \emph{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
usually leads to poor performance increases. This is 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
the additional CPU computations between lines~23 and~24
in \Lst{algo:ch6p2syncGPU}.  Nevertheless, such 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
parallel of 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
Listings~\ref{algo:ch6p2FullOverAsyncMain},~\ref{algo:ch6p2FullOverAsyncComp1}
and~\ref{algo:ch6p2FullOverAsyncComp2}, where we suppose 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}
\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
omp_lock_t lockRes;   // Informs aux thread about new results 
omp_lock_t lockWrite; // Controls exclusion of results access
... auxRes ... ;      // Results of auxiliary computations 

// Parameters reading, MPI initialization, data initialization and distribution
...
// OpenMP initialization
...
omp_init_lock(&lockAux);
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_unset_lock(&lockWrite); // Controls access to results from threads

#pragma omp parallel
{
  switch(omp_get_thread_num()){
    case COMPUTATION :
    computations(... @\emph{relevant parameters}@ ...);
    break;

    case AUX_COMPS :
    auxComps(... @\emph{relevant parameters}@ ...);
    break;
    
    case SENDINGS :
    sendings();
    break;
    
    case RECEPTIONS :
    receptions();
    break;
  }
}

// Cleaning of OpenMP locks
...
omp_test_lock(&lockAux);
omp_unset_lock(&lockAux);
omp_destroy_lock(&lockAux);
omp_test_lock(&lockRes);
omp_unset_lock(&lockRes);
omp_destroy_lock(&lockRes);
omp_test_lock(&lockWrite);
omp_unset_lock(&lockWrite);
omp_destroy_lock(&lockWrite);

// MPI termination
MPI_Finalize();
\end{Listing}
%\end{algorithm}

%\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}
// 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
  ...
  // Local GPU computation
  // Data transfers from node RAM to GPU, GPU grid definition and init of shared mem
  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
  ...
  // 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
  ...
  omp_unset_lock(&lockWrite); // Give back read access to aux thread
  omp_test_lock(&lockRes);
  omp_unset_lock(&lockRes);   // Informs aux thread of new results

  // Auxiliary computations availability checking
  if(omp_test_lock(&lockAux)){
    // Use auxRes to update the iterative process
    ... // May induce additional GPU transfers
  }

  // Convergences checking
  if(curMode == SYNC){
    // Local convergence checking and global exchange of local states
    ...
    // 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
      ...
      // Unlocking of aux thread for termination
      omp_test_lock(&lockRes);
      omp_unset_lock(&lockRes);
    }else{
      // Re-initialization of state information and iteration barrier
      ...
    }
  }
}
\end{Listing}
%\end{algorithm}

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

// Computation loop
while(!Finished){
  // Data copy from resultsInRam into auxInput
  omp_set_lock(&lockRes);   // Waiting for new results from main comps
  if(!Finished){
    omp_set_lock(&lockWrite); // Waiting for access to results
    for(ind=0; ind<resultsSize; ++ind){
      auxInput[ind] = resultsInRam[ind];
    }
    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
      ...
    }
    // 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);
  }
}
\end{Listing}
%\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
\Lst{algo:ch6p2FullOverAsyncComp1}), to inform the auxiliary thread that new
results from the main thread are available (lines~25-26
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
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
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.
Then, input data of auxiliary computations
% (as supposed here, they often
% correspond to the results of the main computations, but may sometimes be
% something else)
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
of its iterations and takes them into account whenever this is 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.

\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
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
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.

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
the update of the Jacobian of the system.

\subsubsection*{Synchronous and asynchronous computations}

The first experiment allows us  to check that the asynchronous behavior obtained
with our  scheme corresponds  to the expected  one according to  its synchronous
counterpart. So,  we show  in~\Fig{fig:ch6p2syncasync} the computation  times of
our test application  in both modes for different problem  sizes. The size shown
is the number of discrete spatial elements on each side of the cube representing
the  3D   volume.  Moreover,  for  each   of  these  elements,   there  are  the
concentrations of the two chemical species considered.  So, for example, size 30
corresponds in fact to $30\times30\times30\times2$ values.

\begin{figure}[h]
  \centering
  \includegraphics[width=.75\columnwidth]{Chapters/chapter6/curves/syncasync.pdf}
  \caption{Computation times of the test application in synchronous and
    asynchronous modes.}
  \label{fig:ch6p2syncasync}
\end{figure}

The results obtained show that  the asynchronous version is sensibly 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
communication times  in 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
than communications. Since asynchronous computations  often require more
iterations to converge, the gain  obtained on the communication side becomes smaller
than the  overhead generated on the computation side, and the  asynchronous version
takes longer.
% So, the observed behavior is fully coherent with the expected behavior.

\subsubsection*{Overlap of auxiliary computations}

In this experiment, we use only the asynchronous version of the application.  In
the context of  our test application, we have an iterative  PDE solver based on
Netwon    resolution.    Such    solvers    are   written    under   the    form
$x=T(x),~x\in\Reals^n$ where $T(x)=x-F'(x)^{-1}F(x)$ and $F'$ is the Jacobian of
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
this context,  classical algorithms either compute  $F'$ only at 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
new  versions  of  $F'$  in  parallel  to 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
non-overlapped ones for several problem sizes, see~\Fig{fig:ch6p2aux}.
\begin{figure}[h]
  \centering
  \includegraphics[width=.75\columnwidth]{Chapters/chapter6/curves/recouvs.pdf}
  \caption{Computation times  with or without overlap  of Jacobian updatings
    in asynchronous mode.}
  \label{fig:ch6p2aux}
\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
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
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
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
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,
the load evaluation and the comparison  with other nodes can be done in parallel
of the  main computations without perturbing  them.

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