X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/2e8e9b41154df43b7a17bb19733abea069a7f20e..7165f1d084760cf67c1cf3a0881e1478eba90b55:/prng_gpu.tex?ds=sidebyside

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 5431409..bc5b3e5 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -34,7 +34,7 @@
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 
-\title{Efficient generation of pseudo random numbers based on chaotic iterations
+\title{Efficient Generation of Pseudo-Random Numbers based on Chaotic Iterations
 on GPU}
 \begin{document}
 
@@ -44,26 +44,97 @@ Guyeux\thanks{Authors in alphabetic order}}
 \maketitle
 
 \begin{abstract}
-This is the abstract
+
 \end{abstract}
 
 \section{Introduction}
 
-Interet des itÃ©rations chaotiques pour gÃ©nÃ©rer des nombre alea\\
-Interet de gÃ©nÃ©rer des nombres alea sur GPU
-\alert{RC, un petit state-of-the-art sur les PRNGs sur GPU ?}
-...
-
+Random  numbers are  used in  many scientific  applications and  simulations. On
+finite  state machines,  as computers,  it is  not possible  to  generate random
+numbers but only pseudo-random numbers. In practice, a good pseudo-random number
+generator (PRNG) needs  to verify some features to be used  by scientists. It is
+important  to  be  able  to  generate  pseudo-random  numbers  efficiently,  the
+generation  needs to  be reproducible  and a  PRNG needs  to satisfy  many usual
+statistical properties. Finally, from our point a view, it is essential to prove
+that  a PRNG  is  chaotic.  Concerning  the  statistical tests,  TestU01 is  the
+best-known public-domain statistical testing package.   So we use it for all our
+PRNGs, especially the {\it BigCrush}  which provides the largest serie of tests.
+Concerning  the  chaotic properties,  Devaney~\cite{Devaney}  proposed a  common
+mathematical formulation of chaotic dynamical systems.
+
+In a  previous work~\cite{bgw09:ip}  we have proposed  a new familly  of chaotic
+PRNG  based on  chaotic iterations  (IC). We  have proven  that these  PRNGs are
+chaotic in the Devaney's sense.  In this paper we propose a faster version which
+is also proven to be chaotic.
+
+Although graphics  processing units (GPU)  was initially designed  to accelerate
+the manipulation of  images, they are nowadays commonly  used in many scientific
+applications. Therefore,  it is important  to be able to  generate pseudo-random
+numbers inside a GPU when a scientific application runs in a GPU. That is why we
+also provide  an efficient  PRNG for  GPU respecting based  on IC.  Such devices
+allows us to generated almost 20 billions of random numbers per second.
+
+In order  to establish  that our  PRNGs are chaotic  according to  the Devaney's
+formulation, we extend what we have proposed in~\cite{guyeux10}. Moreover,  we define a new distance to measure the disorder in the chaos and we prove some interesting properties with this distance.
+
+The rest of this paper  is organised as follows. In Section~\ref{section:related
+  works}  we review  some GPU  implementions of  PRNG.  Section~\ref{section:BASIC RECALLS}  gives some  basic recalls  on  Devanay's formation  of chaos  and
+chaotic iterations. In Section~\ref{sec:pseudo-random} the proof of chaos of our
+PRNGs  is  studied.   Section~\ref{sec:efficient  prng}  presents  an  efficient
+implementation of  our chaotic PRNG  on a CPU.   Section~\ref{sec:efficient prng
+  gpu}   describes   the  GPU   implementation   of   our   chaotic  PRNG.    In
+Section~\ref{sec:experiments}     some    experimentations     are    presented.
+Section~\ref{sec:de  la  relativitÃ© du  dÃ©sordre}  describes  the relativity  of
+disorder.  In Section~\ref{sec:  chaos order  topology} the  proof  that chaotic
+iterations can be described by iterations on a real interval is established. Finally, we give a conclusion and some perspectives.
+
+
+
+
+\section{Related works on GPU based PRNGs}
+\label{section:related works}
+In the litterature many authors have work on defining GPU based PRNGs. We do not
+want to be exhaustive and we just give the most significant works from our point
+of view. When authors mention the  number of random numbers generated per second
+we mention  it. We  consider that  a million numbers  per second  corresponds to
+1MSample/s and than a billion numbers per second corresponds to 1GSample/s.
+
+In \cite{Pang:2008:cec},  the authors define  a PRNG based on  cellular automata
+which  does   not  require  high  precision  integer   arithmetics  nor  bitwise
+operations. There is no mention of statistical tests nor proof that this PRNG is
+chaotic.  Concerning   the  speed  of   generation,  they  can   generate  about
+3.2MSample/s on a GeForce 7800 GTX GPU (which is quite old now).
+
+In \cite{ZRKB10}, the authors propose  different versions of efficient GPU PRNGs
+based on  Lagged Fibonacci, Hybrid  Taus or Hybrid  Taus.  They have  used these
+PRNGs   for  Langevin   simulations   of  biomolecules   fully  implemented   on
+GPU. Performance of  the GPU versions are far better than  those obtained with a
+CPU and these PRNGs succeed to pass the {\it BigCrush} test of TestU01. There is
+no mention that their PRNGs have chaos mathematical properties.
+
+
+Authors of~\cite{conf/fpga/ThomasHL09}  have studied the  implementation of some
+PRNGs on  diferrent computing architectures: CPU,  field-programmable gate array
+(FPGA), GPU and massively parallel  processor. This study is interesting because
+it  shows the  performance  of the  same  PRNGs on  different architeture.   For
+example,  the FPGA  is globally  the  fastest architecture  and it  is also  the
+efficient one because it provides the fastest number of generated random numbers
+per joule. Concerning the GPU,  authors can generate betweend 11 and 16GSample/s
+with a GTX 280  GPU. The drawback of this work is  that those PRNGs only succeed
+the {\it Crush} test which is easier than the {\it Big Crush} test.
+\newline
+\newline
+To the best of our knowledge no GPU implementation have been proven to have chaotic properties.
 
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}
 This section is devoted to basic definitions and terminologies in the fields of
 topological chaos and chaotic iterations.
-\subsection{Devaney's chaotic dynamical systems}
+\subsection{Devaney's Chaotic Dynamical Systems}
 
 In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
 denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$
-denotes the $k^{th}$ composition of a function $f$. Finally, the following
+is for the $k^{th}$ composition of a function $f$. Finally, the following
 notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.
 
 
@@ -89,7 +160,7 @@ necessarily the same period).
 \end{definition}
 
 
-\begin{definition}
+\begin{definition}[Devaney's formulation of chaos~\cite{Devaney}]
 $f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and
 topologically transitive.
 \end{definition}
@@ -119,7 +190,7 @@ possible and occur in an unpredictable way.
 
 
 
-\subsection{Chaotic iterations}
+\subsection{Chaotic Iterations}
 \label{sec:chaotic iterations}
 
 
@@ -135,7 +206,7 @@ denoted by $\llbracket 1, \mathsf{N} \rrbracket^\mathds{N}.$
 \label{Def:chaotic iterations}
 The      set       $\mathds{B}$      denoting      $\{0,1\}$,      let
 $f:\mathds{B}^{\mathsf{N}}\longrightarrow  \mathds{B}^{\mathsf{N}}$ be
-a  function  and  $S\in  \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$  be  a  strategy.  The  so-called
+a  function  and  $S\in  \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$  be  a  ``strategy''.  The  so-called
 \emph{chaotic      iterations}     are     defined      by     $x^0\in
 \mathds{B}^{\mathsf{N}}$ and
 \begin{equation}
@@ -155,7 +226,7 @@ $\left(f(x^{n-1})\right)_{S^{n}}$      can     be      replaced     by
 $\left(f(x^{k})\right)_{S^{n}}$, where  $k<n$, describing for example,
 delays  transmission~\cite{Robert1986,guyeux10}.  Finally,  let us  remark that
 the term  ``chaotic'', in  the name of  these iterations,  has \emph{a
-priori} no link with the mathematical theory of chaos, recalled above.
+priori} no link with the mathematical theory of chaos, presented above.
 
 
 Let us now recall how to define a suitable metric space where chaotic iterations
@@ -185,8 +256,8 @@ G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), \label{Gf}
 (S^{n})_{n\in \mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow (S^{n+1})_{n\in
 \mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$ and $i$ is the \emph{initial function} 
 $i:(S^{n})_{n\in \mathds{N}} \in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow S^{0}\in \llbracket
-1;\mathsf{N}\rrbracket$. Then the chaotic iterations defined in
-(\ref{sec:chaotic iterations}) can be described by the following iterations:
+1;\mathsf{N}\rrbracket$. Then the chaotic iterations proposed in
+Definition \ref{Def:chaotic iterations} can be described by the following iterations:
 \begin{equation}
 \left\{
 \begin{array}{l}
@@ -200,7 +271,6 @@ With this formulation, a shift function appears as a component of chaotic
 iterations. The shift function is a famous example of a chaotic
 map~\cite{Devaney} but its presence is not sufficient enough to claim $G_f$ as
 chaotic. 
-
 To study this claim, a new distance between two points $X = (S,E), Y =
 (\check{S},\check{E})\in
 \mathcal{X}$ has been introduced in \cite{guyeux10} as follows:
@@ -238,22 +308,24 @@ measure of the differences between strategies $S$ and $\check{S}$. More
 precisely, this floating part is less than $10^{-k}$ if and only if the first
 $k$ terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is
 nonzero, then the $k^{th}$ terms of the two strategies are different.
+The impact of this choice for a distance will be investigate at the end of the document.
 
 Finally, it has been established in \cite{guyeux10} that,
 
 \begin{proposition}
-Let $f$ be a map from $\mathds{B}^n$ to itself. Then $G_{f}$ is continuous in
+Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. Then $G_{f}$ is continuous in
 the metric space $(\mathcal{X},d)$.
 \end{proposition}
 
 The chaotic property of $G_f$ has been firstly established for the vectorial
-Boolean negation \cite{guyeux10}. To obtain a characterization, we have secondly
+Boolean negation $f(x_1,\hdots, x_\mathsf{N}) =  (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly
 introduced the notion of asynchronous iteration graph recalled bellow.
 
-Let $f$ be a map from $\mathds{B}^n$ to itself. The
+Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The
 {\emph{asynchronous iteration graph}} associated with $f$ is the
 directed graph $\Gamma(f)$ defined by: the set of vertices is
-$\mathds{B}^n$; for all $x\in\mathds{B}^n$ and $i\in \llbracket1;n\rrbracket$,
+$\mathds{B}^\mathsf{N}$; for all $x\in\mathds{B}^\mathsf{N}$ and 
+$i\in \llbracket1;\mathsf{N}\rrbracket$,
 the graph $\Gamma(f)$ contains an arc from $x$ to $F_f(i,x)$. 
 The relation between $\Gamma(f)$ and $G_f$ is clear: there exists a
 path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
@@ -265,21 +337,21 @@ We have finally proven in \cite{bcgr11:ip} that,
 
 \begin{theorem}
 \label{Th:CaractÃ©risation   des   IC   chaotiques}  
-Let $f:\mathds{B}^n\to\mathds{B}^n$. $G_f$ is chaotic  (according to  Devaney) 
+Let $f:\mathds{B}^\mathsf{N}\to\mathds{B}^\mathsf{N}$. $G_f$ is chaotic  (according to  Devaney) 
 if and only if $\Gamma(f)$ is strongly connected.
 \end{theorem}
 
 This result of chaos has lead us to study the possibility to build a
 pseudo-random number generator (PRNG) based on the chaotic iterations. 
-As $G_f$, defined on the domain   $\llbracket 1 ;  n \rrbracket^{\mathds{N}} 
-\times \mathds{B}^n$, is build from Boolean networks $f : \mathds{B}^n
-\rightarrow \mathds{B}^n$, we can preserve the theoretical properties on $G_f$
+As $G_f$, defined on the domain   $\llbracket 1 ;  \mathsf{N} \rrbracket^{\mathds{N}} 
+\times \mathds{B}^\mathsf{N}$, is build from Boolean networks $f : \mathds{B}^\mathsf{N}
+\rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$
 during implementations (due to the discrete nature of $f$). It is as if
-$\mathds{B}^n$ represents the memory of the computer whereas $\llbracket 1 ;  n
+$\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ;  \mathsf{N}
 \rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).
 
 \section{Application to Pseudo-Randomness}
-
+\label{sec:pseudo-random}
 \subsection{A First Pseudo-Random Number Generator}
 
 We have proposed in~\cite{bgw09:ip} a new family of generators that receives 
@@ -350,9 +422,9 @@ We have proven in \cite{bcgr11:ip} that,
   if and only if $M$ is a double stochastic matrix.
 \end{theorem} 
 
+This former generator as successively passed various batteries of statistical tests, as the NIST tests~\cite{bcgr11:ip}.
 
-
-\subsection{Improving the speed of the former generator}
+\subsection{Improving the Speed of the Former Generator}
 
 Instead of updating only one cell at each iteration, we can try to choose a
 subset of components and to update them together. Such an attempt leads
@@ -388,6 +460,7 @@ to the following discrete dynamical system in chaotic iterations:
   x_i^{n-1} &  \text{ if  } i \notin \mathcal{S}^n \\
   \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
 \end{array}\right.
+\label{eq:generalIC}
 \end{equation}
 where $f$ is the vectorial negation and $\forall n \in \mathds{N}$, 
 $\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ is such that
@@ -404,11 +477,11 @@ the vectorial negation, leads to a speed improvement. However, proofs
 of chaos obtained in~\cite{bg10:ij} have been established
 only for chaotic iterations of the form presented in Definition 
 \ref{Def:chaotic iterations}. The question is now to determine whether the
-use of more general chaotic iterations to generate pseudo-random numbers more 
-fastly, does not deflate their topological chaos properties.
-
-\subsection{Proofs of chaos of the general formulation of the chaotic iterations}
+use of more general chaotic iterations to generate pseudo-random numbers 
+faster, does not deflate their topological chaos properties.
 
+\subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
+\label{deuxiÃ¨me def}
 Let us consider the discrete dynamical systems in chaotic iterations having 
 the general form:
 
@@ -477,7 +550,7 @@ Another time, a shift function appears as a component of these general chaotic
 iterations. 
 
 To study the Devaney's chaos property, a distance between two points 
-$X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be introduced.
+$X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined.
 Let us introduce:
 \begin{equation}
 d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
@@ -622,15 +695,33 @@ claimed in the lemma.
 We can now prove the Theorem~\ref{t:chaos des general}...
 
 \begin{proof}[Theorem~\ref{t:chaos des general}]
- On the one hand, strong transitivity implies transitivity. On the other hand, 
-the regularity is exactly Lemma~\ref{strongTrans} with $Y=X$. As the sensitivity
-to the initial condition is implied by these two properties, we thus have
-the theorem.
+Firstly, strong transitivity implies transitivity.
+
+Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
+prove that $G_f$ is regular, it is sufficient to prove that
+there exists a strategy $\tilde S$ such that the distance between
+$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
+$(\tilde S,E)$ is a periodic point.
+
+Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
+configuration that we obtain from $(S,E)$ after $t_1$ iterations of
+$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$ 
+and $t_2\in\mathds{N}$ such
+that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
+
+Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
+of $S$ and the first $t_2$ terms of $S'$: $$\tilde
+S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
+is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after
+$t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic
+point. Since $\tilde S_t=S_t$ for $t<t_1$, by the choice of $t_1$, we
+have $d((S,E),(\tilde S,E))<\epsilon$.
 \end{proof}
 
 
 
 \section{Efficient PRNG based on Chaotic Iterations}
+\label{sec:efficient prng}
 
 In  order to  implement efficiently  a PRNG  based on  chaotic iterations  it is
 possible to improve  previous works [ref]. One solution  consists in considering
@@ -706,19 +797,19 @@ unsigned int CIprng() {
 
 
 In listing~\ref{algo:seqCIprng}  a sequential version of  our chaotic iterations
-based   PRNG    is   presented.   The    xor   operator   is    represented   by
-\textasciicircum.  This   function  uses  three  classical   64-bits  PRNG:  the
-\texttt{xorshift},  the   \texttt{xor128}  and  the   \texttt{xorwow}.   In  the
-following,  we call  them  xor-like  PRNGSs.  These  three  PRNGs are  presented
-in~\cite{Marsaglia2003}.  As each  xor-like PRNG used works with  64-bits and as
-our PRNG works  with 32-bits, the use of \texttt{(unsigned  int)} selects the 32
-least significant bits whereas  \texttt{(unsigned int)(t3$>>$32)} selects the 32
-most  significants bits  of the  variable \texttt{t}.   So to  produce  a random
-number realizes  6 xor operations with  6 32-bits numbers produced  by 3 64-bits
-PRNG.  This version successes the  BigCrush of the TestU01 battery [P.  Lâecuyer
-  and R. Simard. Testu01].
-
-\section{Efficient prng based on chaotic iterations on GPU}
+based PRNG is  presented.  The xor operator is  represented by \textasciicircum.
+This  function uses  three classical  64-bits PRNG:  the  \texttt{xorshift}, the
+\texttt{xor128}  and  the  \texttt{xorwow}.   In  the following,  we  call  them
+xor-like PRNGSs.   These three PRNGs are  presented in~\cite{Marsaglia2003}.  As
+each xor-like PRNG  used works with 64-bits and as our  PRNG works with 32-bits,
+the use of \texttt{(unsigned int)} selects the 32 least significant bits whereas
+\texttt{(unsigned int)(t3$>>$32)}  selects the 32 most significants  bits of the
+variable \texttt{t}.   So to produce a  random number realizes  6 xor operations
+with 6 32-bits  numbers produced by 3 64-bits PRNG.   This version successes the
+BigCrush of the TestU01 battery~\cite{LEcuyerS07}.
+
+\section{Efficient PRNGs based on chaotic iterations on GPU}
+\label{sec:efficient prng gpu}
 
 In  order to benefit  from computing  power of  GPU, a  program needs  to define
 independent blocks of threads which  can be computed simultaneously. In general,
@@ -726,8 +817,8 @@ the larger the number of threads is,  the more local memory is used and the less
 branching  instructions are  used (if,  while, ...),  the better  performance is
 obtained  on  GPU.  So  with  algorithm  \ref{algo:seqCIprng}  presented in  the
 previous section, it is possible to  build a similar program which computes PRNG
-on  GPU. In  the CUDA  [ref] environment,  threads have  a  local identificator,
-called \texttt{ThreadIdx} relative to the block containing them.
+on   GPU.  In  the   CUDA~\cite{Nvid10}  environment,   threads  have   a  local
+identificator, called \texttt{ThreadIdx} relative to the block containing them.
 
 
 \subsection{Naive version for GPU}
@@ -737,14 +828,14 @@ The principe consists in assigning the computation of a PRNG as in sequential to
 each thread  of the  GPU.  Of course,  it is  essential that the  three xor-like
 PRNGs  used for  our computation  have different  parameters. So  we  chose them
 randomly with  another PRNG. As the  initialisation is performed by  the CPU, we
-have chosen to use the ISAAC PRNG  [ref] to initalize all the parameters for the
-GPU version  of our  PRNG.  The  implementation of the  three xor-like  PRNGs is
-straightforward  as soon  as their  parameters have  been allocated  in  the GPU
-memory. Each xor-like  PRNGs used works with an internal  number $x$ which keeps
-the last generated random numbers. Other internal variables are also used by the
-xor-like PRNGs. More  precisely, the implementation of the  xor128, the xorshift
-and  the xorwow  respectively  require 4,  5  and 6  unsigned  long as  internal
-variables.
+have  chosen  to  use  the  ISAAC  PRNG~\ref{Jenkins96}  to  initalize  all  the
+parameters for  the GPU version  of our PRNG.   The implementation of  the three
+xor-like  PRNGs  is  straightforward  as  soon as  their  parameters  have  been
+allocated in  the GPU memory.  Each xor-like PRNGs  used works with  an internal
+number  $x$  which keeps  the  last  generated  random numbers.  Other  internal
+variables  are   also  used   by  the  xor-like   PRNGs.  More   precisely,  the
+implementation of the  xor128, the xorshift and the  xorwow respectively require
+4, 5 and 6 unsigned long as internal variables.
 
 \begin{algorithm}
 
@@ -792,7 +883,7 @@ for all the differents nodes involves in the computation.
 
 As GPU cards using CUDA have shared memory between threads of the same block, it
 is possible  to use this  feature in order  to simplify the  previous algorithm,
-i.e. using less  than 3 xor-like PRNGs. The solution  consists in computing only
+i.e., using less  than 3 xor-like PRNGs. The solution  consists in computing only
 one xor-like PRNG by thread, saving  it into shared memory and using the results
 of some  other threads in the  same block of  threads. In order to  define which
 thread uses the result of which other  one, we can use a permutation array which
@@ -837,18 +928,53 @@ version}
 \label{algo:gpu_kernel2}
 \end{algorithm}
 
-
+\subsection{Theoretical Evaluation of the Improved Version}
+
+A run of Algorithm~\ref{algo:gpu_kernel2} consists in four operations having 
+the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
+system of Eq.~\ref{eq:generalIC}. That is, four iterations of the general chaotic
+iterations are realized between two stored values of the PRNG.
+To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
+we must guarantee that this dynamical system iterates on the space 
+$\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.
+The left term $x$ obviously belongs into $\mathds{B}^ \mathsf{N}$.
+To prevent from any flaws of chaotic properties, we must check that each right 
+term, corresponding to terms of the strategies,  can possibly be equal to any
+integer of $\llbracket 1, \mathsf{N} \rrbracket$. 
+
+Such a result is obvious for the two first lines, as for the xor-like(), all the
+integers belonging into its interval of definition can occur at each iteration.
+It can be easily stated for the two last lines by an immediate mathematical
+induction.
+
+Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
+chaotic iterations presented previously, and for this reason, it satisfies the 
+Devaney's formulation of a chaotic behavior.
 
 \section{Experiments}
-
-Differents experiments have been performed in order to measure the generation
-speed.
-\begin{figure}[t]
+\label{sec:experiments}
+
+Different experiments  have been  performed in order  to measure  the generation
+speed. We have used  a computer equiped with Tesla C1060 NVidia  GPU card and an
+Intel Xeon E5530 cadenced at 2.40 GHz for our experiments.
+
+In Figure~\ref{fig:time_gpu}  we compare the number of  random numbers generated
+per second.   In order  to obtain the  optimal number  we remove the  storage of
+random numbers  in the GPU memory. This  step is time consumming  and slows down
+the random number  generation.  Moreover, if you are  interested by applications
+that consome  random number directly when  they are generated,  their storage is
+completely useless. In this figure we can see that when the number of threads is
+greater than approximately  30,000 upto 5 millions the  number of random numbers
+generated per second is almost constant.   With the naive version, it is between
+2.5  and  3GSample/s.   With the  optimized  version,  it  is almost  equals  to
+20GSample/s.
+
+\begin{figure}[htbp]
 \begin{center}
   \includegraphics[scale=.7]{curve_time_gpu.pdf}
 \end{center}
 \caption{Number of random numbers generated per second}
-\label{fig:time_naive_gpu}
+\label{fig:time_gpu}
 \end{figure}
 
 
@@ -863,6 +989,9 @@ Faire une courbe du nombre de random en fonction du nombre de threads,
 \section{The relativity of disorder}
 \label{sec:de la relativitÃ© du dÃ©sordre}
 
+In the next two sections, we investigate the impact of the choices that have
+lead to the definitions of measures in Sections \ref{sec:chaotic iterations} and \ref{deuxiÃ¨me def}.
+
 \subsection{Impact of the topology's finenesse}
 
 Let us firstly introduce the following notations.
@@ -968,7 +1097,7 @@ sets $A$ and $B$, an integer $n$ must satisfy $f^{(n)}(A) \cap B \neq
 
 
 \section{Chaos on the order topology}
-
+\label{sec: chaos order topology}
 \subsection{The phase space is an interval of the real line}
 
 \subsubsection{Toward a topological semiconjugacy}
