a given matrix size of 170$^3$ elements, a non-variation in the number of
iterations for the classical GMRES algorithm, which is not the case of the
Krylov two-stage algorithm. In fact, with multisplitting algorithms, the number
-of splitting (in our case, it is equal to the number of clusters) influences on the
-convergence speed. The higher the number of splitting is, the slower the
+of splittings (in our case, it is equal to the number of clusters) influences on the
+convergence speed. The higher the number of splittings is, the slower the
convergence of the algorithm is (see the output results obtained from
configurations 2$\times$16 vs. 4$\times$8 and configurations 4$\times$16 vs.
8$\times$8).
-The execution times between both algorithms is significant with different grid
+The execution times between both algorithms are significant with different grid
architectures. The synchronous Krylov two-stage algorithm presents better
-performances than the GMRES algorithm, even for a high number of clusters (about
-$32\%$ more efficient on a grid of 8$\times$8 than GMRES). In addition, we can
+performances than the GMRES algorithm, even for a high number of clusters (it is about
+$32\%$ more efficient on a grid of 8$\times$8 than the GMRES). In addition, we can
observe a better sensitivity of the Krylov two-stage algorithm (compared to the
GMRES one) when scaling up the number of the processors in the computational
grid: the Krylov two-stage algorithm is about $48\%$ and the GMRES algorithm is
Figure~\ref{fig:04} reports the results obtained for the simulation of a grid of
$2\times16$ processors interconnected by a network of latency $lat=50\mu$s to
-solve a 3D Poisson problem of size $150^3$. The results of increasing the
-network bandwidth from $1$Gbit/s to $10$Gbit/s show the performances improvement for
-both algorithms by reducing the execution times. However, the Krylov two-stage
+solve a 3D Poisson problem of size $150^3$. Increasing the
+network bandwidth from $1$Gbit/s to $10$Gbit/s results in improving the performances of both algorithms by reducing the execution times. However, the Krylov two-stage
algorithm presents a better performance gain in the considered bandwidth
interval with a gain of $40\%$ compared to only about $24\%$ for the classical
GMRES algorithm.
\subsubsection{Matrix size impacts on performances\\}
-In these experiments, the matrix size of the 3D Poisson problem is varied from
+In these experiments, the matrix size of the 3D Poisson problem varies from
$50^3$ to $190^3$ elements. The simulated computational grid is composed of $4$
clusters of $8$ processors each interconnected by the network $N2$ (see
Table~\ref{tab:01}). As shown in Figure~\ref{fig:05}, the execution
observe that for some problem sizes, the convergence (and thus the execution
time) of the Krylov two-stage algorithm varies quite a lot.
%This is due to the 3D partitioning of the 3D matrix of the Poisson problem.
-These findings may help a lot end users to setup the best and the optimal targeted environment for the application deployment when focusing on the problem size scale up.
+These findings may greatly help a lot end users to setup the best and the optimal targeted environment for the application deployment when focusing on the problem size scale up.
\begin{figure}[ht]
\centering
To conclude these series of experiments, with SimGrid we have been able to make
many simulations with many parameters variations. Doing all these experiments
-with a real platform is most of the time not possible or very costly. Moreover
+with a real platform is most of the time impossible or very costly. Moreover
the behavior of both GMRES and Krylov two-stage algorithms is in accordance
with larger real executions on large scale supercomputers~\cite{couturier15}.
\subsection{Comparison between synchronous GMRES and asynchronous two-stage multisplitting algorithms}
-The previous paragraphs put in evidence the interests to simulate the behavior
+The previous paragraphs put in evidence the interest to simulate the behavior
of the application before any deployment in a real environment. In this
section, following the same previous methodology, our goal is to compare the
efficiency of the multisplitting method in \textit{ asynchronous mode} compared with the
The interest of using an asynchronous algorithm is that there is no more
synchronization. With geographically distant clusters, this may be essential.
In this case, each processor can compute its iterations freely without any
-synchronization with the other processors. Thus, the asynchronous may
-theoretically reduce the overall execution time and can improve the algorithm
+synchronization with the other processors. Thus, an asynchronous algorithm may
+theoretically reduce the overall execution time and can also improve the algorithm
performance.
In this section, the SimGrid simulator is used to compare the behavior of the
Table~\ref{tab:03} reports the relative gains between both algorithms. It is
defined by the ratio between the execution time of GMRES and the execution time
of the multisplitting. The ratio is greater than one because the asynchronous
-multisplitting version is faster than GMRES. In average, the two-stage
-multisplitting algorithm to be more than $2.5$ times faster than the classical
+multisplitting version is faster than GMRES. On average, the two-stage
+multisplitting algorithm is more than $2.5$ times faster than the classical
GMRES. These experiments also show the relative tolerance of the multisplitting
algorithm when using a low speed network as usually observed with geographically
distant clusters through the internet.
These results are important since it is very time consuming to find optimal
configuration and deployment requirements for a given application on a given
-multi-core architecture. Finding good resource allocations policies under
+multi-core architecture. Finding good resource allocation policies under
varying CPU power, network speeds and loads is very challenging and labor
intensive. This problematic is even more difficult for the asynchronous
scheme where a small parameter variation of the execution platform and of the
application data can lead to very different numbers of iterations to reach the
-converge and so to very different execution times.
+convergeence and consequently to very different execution times.
In future works, we plan to investigate how to simulate the behavior of really
-large scale applications. For example, if we are interested to simulate the
+large scale applications. For example, if we are interested in simulating the
execution of the solvers of this paper with thousand or even dozens of thousands
of cores, it is not possible to do that with SimGrid. In fact, this tool will
-make the real computation. So we plan to focus our research on that problematic.
+make the real computation. That is why, we plan to focus our research on that problematic.
