last pass : the beginning

main.tex

@@ -34,8 +34,8 @@ Comparison of three heuristics for generating points set}
   One of the most important discrepancy notions is the so-called star
   discrepancy. Roughly speaking, a point set of low star discrepancy value
   allows for a small approximation error in quasi-Monte Carlo integration.
-  In this work we present a tool realizing the implantation of three
-  basics heuristics for construction low discrepancy points sets
+  In this work we present a tool realizing the implementation of three
+  basics heuristics for the construction of low discrepancy points sets
   in the generalized Halton model: fully random search, local search with
   simulated annealing and genetic $(5+5)$ search with a ad-hoc
   crossover function.
@@ -57,7 +57,7 @@ presented in figure~\ref{class_flow}.
 Experiments were conducted on two machines:
 \begin{itemize}
   \item 2.4 GHz Intel Dual Core i5 hyper-threaded to 2.8GHz, 8 Go 1600 MHz DDR3.
-  \item 2.7 GHz Intel Quad Core i74800MQ hyper-threaded to 3.7GHz, 16 Go 1600 MHz DDR3.
+  \item 2.7 GHz Intel Quad Core i7 4800MQ hyper-threaded to 3.7GHz, 16 Go 1600 MHz DDR3.
 \end{itemize}
 
 \begin{figure}
@@ -72,14 +72,14 @@ Experiments were conducted on two machines:
   \label{class_flow}
 \end{figure}
 
-On these machines, some basic profiling has make clear that
+On these machines, some basic profiling has made clear that
 the main bottleneck of the computations is hiding in the \emph{computation
 of the discrepancy}. The chosen algorithm and implantation of this
 cost function is the DEM-algorithm~\cite{Dobkin} of
 \emph{Magnus Wahlstr\o m}~\cite{Magnus}.\medskip
 
 All the experiments has been conducted on dimension 2,3, and 4
---- with a fixed Halton basis 7, 13, 29, and 3 ---. Some minor tests have
+--- with a fixed Halton prime basis 7, 13, 29, and 3. Some minor tests have
 been made in order to discuss the dependency of the discrepancy and
 efficiency of the heuristics with regards to the values chosen for the
 prime base. The average results remains roughly identical when taking
@@ -96,10 +96,17 @@ times in order to smooth up the results and obtain more exploitable datas.
 Then an arithmetic mean of the result is performed on the values. In addition
 extremal values are also given in order to construct error bands graphs.
 
+Graph are presented not with the usual box plots to show the
+error bounds, but in a more graphical way with error bands. The graph
+of the mean result is included inside a band of the same color which
+represents the incertitude with regards to the values obtained.
+
 A flowchart of the conduct of one experiment is described in the
 flowchart~\ref{insight_flow}. The number of iteration of the heuristic is
-I and the number of full restart is N. The function Heuristic() correspond to
-a single step of the chosen heuristic. We now present an in-depth view of
+I and the number of full restart is N. The function Heuristic() corresponds to
+a single step of the chosen heuristic.
+
+We now present an in-depth view of
 the implemented heuristics.
 
 \begin{figure}
@@ -110,21 +117,16 @@ the implemented heuristics.
 \end{mdframed}
 \end{figure}
 
-Graph are presented not with the usual box plot to show the
-error bounds, but in a more graphical way with error bands. The graph
-of the mean result is included inside a band of the same color which
-represents the incertitude with regards to the values obtained.
-
 \section{Heuristics developed}
 
 \subsection{Fully random search (Test case)}
  The first heuristic implemented is the random search. We generate
  random permutations, compute the corresponding sets of Halton points
- and select the best set with regard to its discrepancy.
+ and select the best set with regards to its discrepancy.
  The process is wrapped up in the
  flowchart~\ref{random_flow}. In order to generate at each step random
  permutations, we transform them directly from the previous ones.
-  More precisely the permutation is a singleton object which have method
+  More precisely the permutation is a singleton object which have a method
   random, built on the Knuth Fisher Yates shuffle. This algorithm allows
   us to generate an uniformly chosen  permutation at each step. We recall
   this fact and detail the algorithm in the following section.
@@ -170,7 +172,7 @@ the naive implementation.
   an edge $(i,j)$ exists in the graph if and only if the swap of $T[i]$ and
   $T[j]$ had been executed. This graph encodes the permutation represented by
   $T$. To be able to encode any permutation the considered graph must be
-  connected --- in order to allow any pairs of points to be swapped ---.
+  connected --- in order to allow any pairs of points to be swapped.
   Since by construction every points is reached by an edge, and that there
   exists exactly $n-1$ edges, we can conclude directly that any permutation can
   be reached by the algorithm. Since the probability of getting a fixed graph
@@ -183,20 +185,20 @@ the naive implementation.
 \subsubsection{Results and stability}
 We first want to analyze the dependence of the results on the number of
 iterations of the heuristic, in order to discuss its stability.
-The results are compiled in the figures~\ref{rand_iter2},~\ref{rand_iter3},
+The results are compiled in the figures~\ref{rand_iter2} and~\ref{rand_iter3},
 restricted to a number of points between 80 and 180.
-We emphasize on the fact the lots of datas appears on the graphs,
-and the error bands representation make them a bit messy. These graphs
+We emphasize on the fact lot of datas appears on graphs,
+and error bands representation make them a bit messy. These graphs
 were made for extensive internal experiments and parameters researches.
-The final wrap up graphs are much more lighter and only presents the best
+The final wrap up graphs are much more lighter and only present the best
 results obtained.
-As expected from a fully random search, the error bands are very large for
-low number of iterations ($15\%$ of the value for 400 iterations) and tends
+As expected from a fully random search, error bands are very large for
+low number of iterations ($15\%$ of the value for 400 iterations) and tend
 to shrink with a bigger number of iterations (around $5\%$ for 1500 iterations).
 This shrinkage is a direct consequence of well known concentrations bounds
 (Chernoff and Asuma-Hoeffding).
 The average results are quite stable, they decrease progressively with
-the growing number of iterations, but seem to get to a limits after 1000
+the growing number of iterations, but seem to get to a limit after 1000
 iterations. This value acts as a threshold for the interesting number of iterations.
 As such interesting results can be conducted with \emph{only} 1000 iterations,
 without altering too much the quality of the set with regards to its