\section{The asynchronous iteration model and the motivations of our work}
\label{sec:asynchro}
-Asynchronous iterative methods have been studied for many years theoritecally and
+Asynchronous iterative methods have been studied for many years theoretically and
practically. Many methods have been considered and convergence results have been
proved. These methods can be used to solve, in parallel, fixed point problems
(i.e. problems for which the solution is $x^\star =f(x^\star)$. In practice,
-asynchronous iterations methods can be used to solve, for example, linear and
+asynchronous iteration methods can be used to solve, for example, linear and
non-linear systems of equations or optimization problems, interested readers are
invited to read~\cite{BT89,bahi07}.
requires very few modifications to be able to be executed in both variants. In
practice, only the communications and convergence detection are different. In
the synchronous mode, iterations are synchronized whereas in the asynchronous
-one, they are not. It should be noticed that non blocking communications can be
+one, they are not. It should be noticed that non-blocking communications can be
used in both modes. Concerning the convergence detection, synchronous variants
can use a global convergence procedure which acts as a global synchronization
point. In the asynchronous model, the convergence detection is more tricky as
consult~\cite{myBCCV05c,bahi07,ccl09:ij}.
The number of iterations required to reach the convergence is generally greater
-for the asynchronous scheme (this number depends depends on the delay of the
+for the asynchronous scheme (this number depends on the delay of the
messages). Note that, it is not the case in the synchronous mode where the
number of iterations is the same than in the sequential mode. In this way, the
set of the parameters of the platform (number of nodes, power of nodes,
-inter and intra clusters bandwidth and latency, \ldots) and of the
+inter and intra clusters bandwidth and latency,~\ldots) and of the
application can drastically change the number of iterations required to get the
convergence. It follows that asynchronous iterative algorithms are difficult to
optimize since the financial and deployment costs on large scale multi-core
-architecture are often very important. So, prior to delpoyment and tests it
+architectures are often very important. So, prior to deployment and tests it
seems very promising to be able to simulate the behavior of asynchronous
-iterative algorithms. The problematic is then to show that the results produce
+iterative algorithms. The problematic is then to show that the results produced
by simulation are in accordance with reality i.e. of the same order of
magnitude. To our knowledge, there is no study on this problematic.
