X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/c1ba143536007ddfded6ec4043d1a77536e27d75..3a4d92d48d8e34ab9e636f7eb092235bcfa0215d:/prng_gpu.tex?ds=inline

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 6409faf..41e628a 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}
 
@@ -50,24 +50,28 @@ Guyeux\thanks{Authors in alphabetic order}}
 \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
+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.  Devaney~\cite{Devaney} proposed a  common mathematical
-formulation of chaotic dynamical systems.
+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).   In this  paper we  propose a  faster
-version which is also proven to be chaotic with the Devaney formulation.
+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 image, they are  nowadays commonly used  in many scientific
+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 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.
+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.
 
 
 
@@ -76,10 +80,26 @@ Interet des itÃ©rations chaotiques pour gÃ©nÃ©rer des nombre alea\\
 Interet de gÃ©nÃ©rer des nombres alea sur GPU
 
 
-\section{Related works}
+\section{Related works on GPU based PRNGs}
 
-In this section we review some GPU based PRNGs.
-\alert{RC, un petit state-of-the-art sur les PRNGs sur GPU ?}
+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}
@@ -751,17 +771,16 @@ 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$>>$32)} selects the 32
-most  significants bits  of the  variable \texttt{t}.   So to  produce  a random
-number realizes  6 xor operations with  6 32-bits numbers produced  by 3 64-bits
-PRNG.  This version successes the  BigCrush of the TestU01 battery [P.  Lâecuyer
-  and R. Simard. Testu01].
+based PRNG is  presented.  The xor operator is  represented by \textasciicircum.
+This  function uses  three classical  64-bits PRNG:  the  \texttt{xorshift}, the
+\texttt{xor128}  and  the  \texttt{xorwow}.   In  the following,  we  call  them
+xor-like PRNGSs.   These three PRNGs are  presented in~\cite{Marsaglia2003}.  As
+each xor-like PRNG  used works with 64-bits and as our  PRNG works with 32-bits,
+the use of \texttt{(unsigned int)} selects the 32 least significant bits whereas
+\texttt{(unsigned int)(t3$>>$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}
 
