X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/bce6fe373543bc9037b3de8504d9599882257bf5..c1ba143536007ddfded6ec4043d1a77536e27d75:/prng_gpu.tex?ds=inline

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 11dd246..6409faf 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -44,17 +44,43 @@ Guyeux\thanks{Authors in alphabetic order}}
 \maketitle
 
 \begin{abstract}
-This is the abstract
+
 \end{abstract}
 
 \section{Introduction}
 
+Random  numbers are  used in  many scientific  applications and  simulations. On
+finite state  machines, like  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.  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).   In this  paper we  propose a  faster
+version which is also proven to be chaotic with the Devaney formulation.
+
+Although graphics  processing units (GPU)  was initially designed  to accelerate
+the manipulation  of image, they are  nowadays commonly used  in many scientific
+applications. Therefore,  it is important  to be able to  generate pseudo-random
+numbers in  a GPU when a  scientific application runs in  a GPU. That  is why we
+also provie an efficient PRNG for GPU respecting based on IC.
+
+
+
+
 Interet des itérations chaotiques pour générer des nombre alea\\
 Interet de générer des nombres alea sur GPU
-\alert{RC, un petit state-of-the-art sur les PRNGs sur GPU ?}
-...
 
 
+\section{Related works}
+
+In this section we review some GPU based PRNGs.
+\alert{RC, un petit state-of-the-art sur les PRNGs sur GPU ?}
+
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}
 This section is devoted to basic definitions and terminologies in the fields of
@@ -389,6 +415,7 @@ 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
@@ -409,7 +436,7 @@ 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{deuxième def}
 Let us consider the discrete dynamical systems in chaotic iterations having 
 the general form:
 
@@ -623,10 +650,27 @@ 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}
 
 
@@ -793,7 +837,7 @@ 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
@@ -838,11 +882,32 @@ 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
+Different experiments have been performed in order to measure the generation
 speed.
 \begin{figure}[t]
 \begin{center}
@@ -864,6 +929,9 @@ Faire une courbe du nombre de random en fonction du nombre de threads,
 \section{The relativity of disorder}
 \label{sec:de la relativité du désordre}
 
+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{deuxième def}.
+
 \subsection{Impact of the topology's finenesse}
 
 Let us firstly introduce the following notations.