X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/9b6557073a4a37c514adbdccffe506f12cede15e..7165f1d084760cf67c1cf3a0881e1478eba90b55:/prng_gpu.tex?ds=inline

diff --git a/prng_gpu.tex b/prng_gpu.tex
index ccb0c95..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 Bumbers based on Chaotic Iterations
+\title{Efficient Generation of Pseudo-Random Numbers based on Chaotic Iterations
 on GPU}
 \begin{document}
 
@@ -44,16 +44,87 @@ 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}
@@ -280,7 +351,7 @@ $\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracke
 \rrbracket^{\mathds{N}}$ 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 
@@ -650,6 +721,7 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 
 
 \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
@@ -725,19 +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,
@@ -745,8 +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}
@@ -756,14 +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}
 
@@ -880,15 +952,29 @@ chaotic iterations presented previously, and for this reason, it satisfies the
 Devaney's formulation of a chaotic behavior.
 
 \section{Experiments}
-
-Different 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}
 
 
@@ -1011,7 +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}
