suite

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 11dd246f44d0012ce3cda2c421af4b48e7c26113..41e628a97bc3cd8695dd4bed1ffea352591b5be5 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -34,7 +34,7 @@
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 

-\title{Efficient Generation of Pseudo-Random Bumbers based on Chaotic Iterations
+\title{Efficient Generation of Pseudo-Random Numbers based on Chaotic Iterations

 on GPU}
 \begin{document}
 
 on GPU}
 \begin{document}
 


@@ -44,17 +44,63 @@ Guyeux\thanks{Authors in alphabetic order}}
 \maketitle
 
 \begin{abstract}
 \maketitle
 
 \begin{abstract}

-This is the abstract
+

 \end{abstract}
 
 \section{Introduction}
 
 \end{abstract}
 
 \section{Introduction}
 

+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.
+
+
+
+

 Interet des itérations chaotiques pour générer des nombre alea\\
 Interet de générer des nombres alea sur GPU
 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 on GPU based PRNGs}
+
+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.
+
+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 3200000
+random numbers per seconds 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.
+
+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
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}
 This section is devoted to basic definitions and terminologies in the fields of


@@ -389,6 +435,7 @@ 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.
   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
 \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 +456,7 @@ 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}
 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:
 
 Let us consider the discrete dynamical systems in chaotic iterations having 
 the general form:
 


@@ -623,10 +670,27 @@ claimed in the lemma.
 We can now prove the Theorem~\ref{t:chaos des general}...
 
 \begin{proof}[Theorem~\ref{t:chaos des general}]
 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}
 
 
 \end{proof}
 
 


@@ -707,17 +771,16 @@ unsigned int CIprng() {
 
 
 In listing~\ref{algo:seqCIprng}  a sequential version of  our chaotic iterations
 
 
 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$>>$32)} selects the 32
-most  significants bits  of the  variable \texttt{t}.   So to  produce  a random
-number realizes  6 xor operations with  6 32-bits numbers produced  by 3 64-bits
-PRNG.  This version successes the  BigCrush of the TestU01 battery [P.  L’ecuyer
-  and R. Simard. Testu01].
+based PRNG is  presented.  The xor operator is  represented by \textasciicircum.
+This  function uses  three classical  64-bits PRNG:  the  \texttt{xorshift}, the
+\texttt{xor128}  and  the  \texttt{xorwow}.   In  the following,  we  call  them
+xor-like PRNGSs.   These three PRNGs are  presented in~\cite{Marsaglia2003}.  As
+each xor-like PRNG  used works with 64-bits and as our  PRNG works with 32-bits,
+the use of \texttt{(unsigned int)} selects the 32 least significant bits whereas
+\texttt{(unsigned int)(t3$>>$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 prng based on chaotic iterations on GPU}
 
 
 \section{Efficient prng based on chaotic iterations on GPU}
 


@@ -793,7 +856,7 @@ 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,
 
 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
 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 +901,32 @@ version}
 \label{algo:gpu_kernel2}
 \end{algorithm}
 
 \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}
 
 
 \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}
 speed.
 \begin{figure}[t]
 \begin{center}


@@ -864,6 +948,9 @@ 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}
 
 \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.
 \subsection{Impact of the topology's finenesse}
 
 Let us firstly introduce the following notations.