• Aucun résultat trouvé

A Matrix-Free Preconditioner Applied to CFD

N/A
N/A
Protected

Academic year: 2021

Partager "A Matrix-Free Preconditioner Applied to CFD"

Copied!
35
0
0

Texte intégral

(1)

HAL Id: inria-00074080

https://hal.inria.fr/inria-00074080

Submitted on 24 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

A Matrix-Free Preconditioner Applied to CFD

Rémi Choquet

To cite this version:

Rémi Choquet. A Matrix-Free Preconditioner Applied to CFD. [Research Report] RR-2605, INRIA.

1995. �inria-00074080�

(2)

a p p o r t

d e r e c h e r c h e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A matrix-free preconditioner applied to CFD

Re´mi Choquet

N˚ 2605

Juin 1995

PROGRAMME 6

(3)
(4)

-,.

/103254#687:9<;=/?>:@

A8BDCFEFBG'HIHJKMLON5G'PRQSTPVUWQXYJZ?[\X^]`_SaJbaHCac dJePfXYUG'[\XYCZJ[5PYCFEFXRQXYJPVZ?SaH,dJB\XR_SaJ

A8BDCgJ[5hiVhjk\l

mnG'oaopCFBD[c?JB\JQDqaJBQDqaJrZMs1tFKuFvwLyxFSaXYZ{z||FvwLO}aznoG'EJU

~€e‚„ƒ'…†'‚‡‰ˆ{JQŠCFZaUDX‹cŒJB[DqaJ XfHIoaPYXRQXY[#[\XYHJXYZ?[\JeEFBWG[\XYCFZŽC'#[\qTJQŠCFHoTB\JUDU\XY‘aPYJ$lG„’?XYJB”“\•?[\C–FJU

J_SG[\XYCFZTU— ˜GFQDq™XYHIoaPYX‹QŠXf[5UD[\JoŽXYZ1’CFPY’FJeU![DqaJU\CPYSa[\XYCFZC'G$QCU\[DPfšZaCZ{PfXYZaJ„G'BU\š?U\[DJH›—Ik”Z{[\qTXfU

oGoJeBbœJc?JŠ]ZaJGoaBDJQCZc?XY[\XYCFZTJB[\CUDCFPY’FJ JGFQDqPYXYZaJGB=UDš?U\[\JeHžHICFBDJQDqaJG'oTPfšŸœ5XY[\qaXYZ#[DqaJ  BG'HJŠ“

œ CFBD–›C¡H G'[\BDX^¢1“£ BDJJ¤GPYEFCFBDXY[\qaH—#¥5qaJH G'XYZ$XRc?JGIXYU#[\C BDJSaU\J [DqaJ¦nB\š?PYC'’$U\Sa‘TU\oGQJ$GFQBDCFU\UM[\qaJ

[DXfHIJ#U\[DJoaU„—#ˆ{Jc?JHCZaU\[DBWG[\J#UDCFHJMoaB\CoJBD[\XYJU#C[\qTJ§oaBDJQCZc?XY[\XYCFZTJB§G'Zpc U\qaC'œ¨[\qaJŸoaBGFQ[DX‹QeG'P

’FGPfSTJC'[\qaXYUG'oaoTB\C1GQDq™£CFBŸU\[\X^©,oaBDCF‘aPYJHIU§œ5qaXRQDq™ZTJJc?UIJ¢QJU\UDXY’FJ$UDHG'PYP[DXYHJU\[DJoaUIœ5Xf[DqªGZ

J¢aoaPYXRQXY[5HJe[\qaCacV—

«I¬?­V®\¯#°pƒF±8€'‡QCFHIoaB\JUDU\XY‘aPYJlG„’?XYJB”“\•?[DCF–FJeUb'm “\¦™GZc²j³UQDqaJHJeUbFl´Jœ5[\CZ:b'µ#¶$m˜·•VbHG[\B\X^¢?“

 BDJJFbpoaBDJQCFZpc?XY[\XYCFZaJeB—

¸R¹º

»½¼”¾¿

º

»À=ÁY¼””ÃÄ

Å\Æ£ÇFÈÉeÊFË\̔Í<ÎÐÏ ÎÒÑ£Ó'ÔÕÐÏ

(5)

¬Œ€¬Ÿ‡ l´CFSaU8QCFZaUDXRcdJB\CZaU:PXYZ?[adJEBWG'[DXYCFZXYHIoaPYX‹QŠXf[DJ:JZM[\JHIoaU:c?JUŸdJ_SG[\XYCFZaU8c?JlnG„’1XYJB:•Œ[\CF–JU

QCHoaBDJU\UDXY‘aPYJU—FN qpGF_SaJoGUc?J[DJHoTU:Z5dJQŠJU\UDXf[DJPRG U\CPYSa[\XYCFZŸQCFSa[DJSaUDJcSaZ#U\š?UD[ JHIJZaCZ?“”PfXYZ5dJ„G'XYB\JF—

iGoaoaBDCŒQDqaJQŠCFZaUDX‹c dJBadJJIJU\[ŸQJPYPYJ§CS›PYJŸoaB\C‘aPJHJ#JeU\[ŸXf[adJBWG[\XY’FJeHJZ?[PYXYZ5dJGB\XYUadJ—`jGZaU#QJoG'oTXfJeBb

ZaCSaU$c dJ]ZTXfUDU\CZaU$SaZ,oTBadJQCZc?XY[\XYCZaZaJSaBoCSaB BadJUDCFSc?BDJ™QDqGF_SaJU\š?U\[ JHJPYXYZ¡dJG'XYBDJ›c?JHG'ZTXJeB\J

JQeGFQJ$caGZaU#PYJQGFcŒB\JcGPYEFCFBDXY[\qaHIJ#UWGZaU§H G'[DB\XRQJF—:iX‹c dJJJU\[c?JBadJSa[\XYPYXYU\JBŸPYJU\CSaU”“”JU\opGFQJ$c?J

¦nB\š?PYC'’ Gn[\BWG„’JB\U8PYJU8oGU=c?J [\JHIoaU—'l´CFSaU HCFZ?[DB\CFZTU=_SaJPR_SaJU8oaBDCFoaBDXdJe[adJUcŒJ¡QJ oaBadJQCFZpc?XY[\XYCFZaZTJSaB

J[MPfJeU!JŠ¢ŒoTPfCXY[\CFZTUJZHCZ1[DBWGZ1[5PJQGQXY[adJ§cŒJQJ§oTBadJQCZc?XY[\XYCZaZaJSaBŸG'oaoaPYXR_S5dJJ#JeZ1[DB\JGSa[\BDJGIPRG

UDXfHSTP‹G[\XYCFZcSaZ oaBDCF‘aPJHIJBWGXRc?JF—

! °p‚ €e®W†#"$¬¨‡ lnG„’?XfJeB”“\•?[\C–FJU QCFHIoaB\JUDU\XY‘aPYJFb·UWQDq5dJH G'Um¡“ ¦ J[² j³ b¡l´Jœ5[\CZ:b5µ#¶$mn˜•Vb8UWGZaU

H G'[\BDXRQJFb?oaBadJQCFZpc?XY[\XYCFZaZTJSaB—

(6)

" # §I5

¥5qaJHG'XYZ{HICF[\XY’FG[\XYCFZ{[DC{U\CFPY’J[DqaJ›QCHoaBDJU\UDXY‘aPYJlnG„’?XfJeB”“\•?[\C–FJUJ_SG[\XYCFZTU‘?š™HIJG'ZaUŸC'§GZ

XYHoTPfXRQXY[#HIJ[\qTCŒc XYU[\qaJ J¢QJUDU\XY’FJU\H G'PYP[DXfHIJUD[\JoaUIPYXfHIXY[\Jc‘?š™UD[WG‘aXYPfXY[”š{QCFZTU\[\BG'XYZ?[\U CFBJ¢?“

oaPYXRQXY[HIJ[\qTCŒcŒU—¥5q?SaU„b  CBU\[DB\SQŠ[\SaBDJc HJUDqaJU„bœ CFBD–,qGU‘pJJZ c?CZaJ$‘pCF[\q CFZ [\qaJ{c?J’JPYCFo?“

HIJZ?[C'#ZaJœ"XYHoaPYXRQXY[#UWQDqaJeHJU%$i •'&(&!)\b$Е?JZ|u!)wGZcŽCFZ3[\qaJSaU\J$C#ZaJœ [\C?CPfU[DC{U\CPf’J P‹GB\EJ

UDoG'BDU\JŸZaCFZ?“”U\š?HIHJ[DB\XRQ5PYXYZaJGB5U\š?U\[DJHU*$ˆ,+-+.&v!)\b/$10´¥•a|F}!)\—?hPYUDC CB5SaZaU\[DB\SQŠ[\SaBDJcHJUDqaJU„baZaJœ

UQ\qTJHJU*$10³8¶2&FK)\b/$Еa³8|3*)=G'ZcI[\J„Q\qTZaXR_FSTJU4$x50n³8|az)\b$ÐjnSa[W|Tz)\b$Ði:³8|3!)\b/$Ði G'Z|3):qG„’J‘JJeZ›c?J’JPYCFo?“

opJcV—605C'œ J’JBbTXfZ HICFU\[5QGU\JU„bJ„GFQDq$XYHoaPYXRQXY[UD[\Jo$PYJGc?U[DCCFZaJ#QCU\[DPfš ZaCZ PfXYZaJ„G'B5U\š?UD[\JH [DC‘J

UDCFPY’FJc<—

ˆ,qTJZ™[DqaB\JJŠ“c?XYHIJZaU\XYCZG'PUDoGFQŠJoaBDCF‘aPYJHIUwGZc'7CFB§QDqaJHIXRQG'PQŠCFZ?[\J¢a[G'BDJQCFZaUDXRc?JB\J„cVb8[\qaJIU\X8J

C¡[DqaJU\š?U\[DJHUXYUSaU\SpG'PYPYš$PRG'B\EJF—p¥5q?SaUbV[DqaJHJeHCFBDš“½UDX/8eJ#C'¡QŠSaB\BDJZ?[¤QCHoaST[\JBDU!oTB\CFqTXf‘TXf[DU[\qaJ

UD[\CFBG'EJC'5[\qTJHG[\BDXÒ¢›QCHXYZaEI BDCFH‰[DqaJPYXYZaJGB\X8JcoTB\CF‘TPfJeH›—pˆ{JG'BDJXfZ?[DJB\JUD[\JcŽXYZ]ZcŒXfZTEG

oaBDJQCZc?XY[\XYCFZTJBn CFBnU\CFPY’?XYZaE[\qTJU\JPYXYZaJGBU\š?UD[\JHIUœ5qaJZHG[\BDXÒ¢?“£ B\JeJwGPYEFCFBDXY[\qaHIU¡GB\J#SaUDJcV—V¥5qTXfU

oaBDJQCZc?XY[\XYCFZTJBU\qaCSaPRc ‘JMJG'UDš[DCQCHoaST[\J‘C[\q$[G'–?XYZaEXYZ?[\CGFQeQCFSaZ?[n[\qaJŸHXYU\U C'[DqaJ§H G'[DB\X^¢

GZc [\qTJ¤QCHoaST[\JBnU\š?U\[DJH›—pk”Z$oGB\[DX‹QŠSaPRG'Bbpœ5XY[\qGQCZ1’JZaXYJZ?[!cTG'[WGU\[DB\SQ[DSaB\Jba[DqaJ§oTB\Cac?SQ[C

[DqaXYU§oaBDJQCZc?XY[\XYCFZTJB§‘?š™G ’JQ[DCFBŸU\qaCSaPRc{‘pJc?CZaJJGU\XYPYš›XYZ{oGBWGPfPYJP”—V¥5qTJoaBDCF‘aPYJH QXY[\J„c{oaBDJ“

’?XYCFSaUDPYš™CaQQSaBDUGPfUDC{XYZ™PYJGFcŒXfZTE™G'B\J„G'UC'#Z?SaHIJB\XRQGPUDXfHSTP‹G[\XYCFZ{UDSQDq G'U:9QJ„G'Z GZc3N PfXYH G'[\J

HICŒcŒJPYXfZTE—

i<J[SaU c?JUQB\XY‘pJ›HICFB\JIoaB\J„QXYU\JPYš3[\qaJ BWGHJœ CFBD–{C'#[\qaJU\[\Spc?š™œ5qaJeB\J[\qaJ £CFPYPYC'œ5XfZTEQŠCFZaUDX‹cŒJ“

BG'[\XYCZaU5[WG–FJoaPRGFQJ—aˆ{JqpG„’FJ[\CUDCFPY’FJ

;6<=6>@?BA;CED u'F

A z C

œ5qaJeB\J

?BA;CHGIKJMLNI-J

XfUGŸZTCFZ?“”PYXfZTJG'B  STZQ[\XYCZC

;

—aˆ{J#QCZaU\XRc?JBnJXY[\qaJeB[\qTJ#U\[G'[\XYCZG'BDšCFB

ZaCZ?“”U\[WG[\XYCFZpG'B\šŸB\JeU\CFPYSa[DXYCFZC

A z C

—?m5JoaPRGQXYZaE

;O<=

‘?šGHSaPY[\X^“”U\[DJo§UQDqaJHJb?J_SG'[DXfCZ

A z C

PYJGFc?U5G'[

[DXfHIJQPSRI[\C

;OATU>

z

C6>WV P R'XZY

A;OATB>

z

C[C\DHV P R'XU]

A;OAT6C

F

;OAT_^

z C F

;6ATU^

t C F`*``

C ` At C

ˆ{JZaC[\J[\qG[5[\qaJM‘GFQD–?œ¡GBWcI˜8SaPYJBHIJ[\qTCŒcIQCFBDB\JUDoCZc?U[\C X ]

A;CD a

b

=c

`

ˆ{J#BDJ JB[DC2$N5¶d&F|!)\b$0´ˆ |aze)8 CFBGHCB\J!QŠCFHoTPfJe[\J§c?JeUWQBDXfoT[\XYCFZ$Cf9!j˜3UQDqaJHJeU—pˆ{J#U\SToaoCU\J

[DqG'[ X Y5XYUG'[5PYJGU\[c?X^©VJB\JZ?[DX‹G‘aPYJF—

ˆ,XY[Dqhg

A;C\Di;j>WV PRkXZY

A;CE^lV P R'XU]

A;6AT6C

F

;6ATU^

z C F

;OAT_^

t C F*``*`

C b At C

‘pJQCHJU

g

A;OATj>

z

CSCED

u A} C

ˆ{JQDqaC?CFU\J[DC U\CFPY’J§[DqaJZaCFZTPfXYZaJ„G'BJ_SG'[DXYCFZ

A } C

‘?š›l´Jœ5[\CZ?“ µM¶$m˜•m$в•T|Fu*)\bpœ5qaJBDJl5Jœ5[DCFZ

XY[\JeBWG'[DXYCFZaU§GB\JGoaoaPYXYJc$[DC PfXYZaJ„G'B\X8JŸ[\qTJoaB\C‘aPYJH G'Zc[\qaJXYZcŒSQJcPYXYZaJGB!UDš?U\[\JeHXYU#UDCFPY’FJ„c$‘?š

[DqaJXY[\JBG'[\XY’Jµ#¶$m˜·•$UQ\qTJHJŸœ5qaXRQDq›CZaPYš$B\J_SaXYBDJU§[DqaJx1GQCF‘TX‹GZ&GoaoaPYXYJc [DC›GI’FJQ[DCFB§GZcXfU

œ JPYP^“½UDSaXY[\Jc[DC$ZaCFZ{UDš?HHJe[\B\XRQPRGB\EFJH G'[DB\XRQJU„—V¥5qaJGPYEFCFBDXY[\qaH‰QGZ™‘pJU\SaHIHGB\X8JcG'U£CFPYPYC'œ

œ5XY[\q%n

Dpo*qsr a(t

o a —

(7)

z G

l5Jeœ5[\CFZŒ“µ#¶m˜•

;

EFXY’FJZ b

D ^ z

mn˜A˜8h¥

D > z

•?CPY’FJjn

A;C;sD ^ g

A;C

‘1š µ#¶$mn˜•

;!

Y D ;>";

#nl5¥5k\i{QCZ1’JB\EJZQJ

ˆ,XY[DqaXYZ™µ#¶$m˜·•Vb:J„GFQDq™x?GFQC‘aXRG'Z{’FJ„Q[\CB§oaBDCac?SQ[XYU§GoaoaB\C„¢aXYHG[\Jc‘?š›JeXf[DqaJBGI]pB\U\[ŸCFBG UDJ“

QCZcCFBc?JB]pZaXY[\Jc?X^©VJB\JZpQJUQ\qTJHJF— jXQSTPf[DXYJU#G'B\XYUDXfZTE BDCFH‰UDSQDq™G'Z{G'oaoTB\C1GQDq{XfZpQPYSc?JŸ[\qaJ

 CPYPfC'œ5XYZaEpb

zF—5³ XfZTXf[DJ¡cŒXÒ©VJBDJZQJ5UQ\qTJHJU8XYZ?[\BDCŒcŒSQJ5JBDB\CFBDUœ5qaXRQDq§c?J[DJB\XYCBWG'[DJ5‘C[\qµ#¶$mn˜•#G'Zpc#l5Jœ5[DCFZ

QCFZ?’JB\EJZQJF— ˆ{J#BDJ JB#[DC $ÐN5˜|}*)8 CFBGUD[\Sc?šC'[DqaJŸ]ZaXY[\JŸc?X^©`JBDJZQJIUWQDqaJHIJ¤µM¶$m˜•1“

G'PYEFCB\XY[\qTHU—

ta—5m5JUD[WGB\[\XYZaEŽXYUCFZaJ$œ5G„š™[DC™BDJc?SQJ[\qTJ›UD[\CFBG'EJ›QCFUD[b ‘aSa[I[\qaXYUGoaoaB\C?GFQDq c?J„QB\JGU\JU[\qaJ

QCFZ?’JB\EJZQJŸBWG[\JF—

}a— hZaC[\qaJeB5œ¡G„šXYU [\CM]Zc G#oTB\JQCZc?XY[\XYCZaJB5‘aSa[ [\BGFc?XY[DXfCZG'PVG'oToaB\C?GFQDqaJU5PYXY–FJG'ZXYZQŠCFHoTPfJe[\J

i$# ½GQ[\CB\XYUWG[\XYCFZ G'BDJ!HIJHCB\šQCFZTU\SaHIXfZTE—

ˆ{JZaC[\J[\qG[5œJ#GB\JXYZ[DqaJ#QCFZ?[DJ¢a[œ5qaJBDJ#U\SpQQJUDU\XY’FJŸPYXfZTJG'B5UDš1UD[\JHIU5ZaCF[DJc

n rR t

; rR t

D ^ g rR t

qG„’J§[DC ‘JUDCFPY’FJc‘?š›µ#¶$mn˜•V—pˆ,XY[\qmn

rR t

QŠCFZaUD[WG'Z?[§GZcU\š?HHIJ[\BDX‹Qb•aGGFc6$ЕaGG(&&% )UD[\Sc?š[\qaJ

BDJSaU\JC'5[\qaJ$¦nB\š?PYC'’›UDSa‘aUDoGFQJ [DC›GFQQŠJPYJBWG[\J$[DqaJQCZ?’FJBDEFJZQŠJC[\qaJ i:GZQ8CU§oaBDCaQJU\U„—¶ CB\J

BDJQJZ?[\PYš?b'SaXY–6$'SaXR|F}[)5SaUDJc{[DqaJUG'HJŸXRc?JG XYZ{[DqaJQCZ?[\J¢a[ŸC'[\qTJU\CPYSa[\XYCFZC'[DqaJXYZQŠCFHoTB\JU “

UDXf‘TPfJlG„’?XYJB”“\•?[DCF–FJeUJ_SG[\XYCFZTU‘?šµ#¶$m˜·•amj$'!c(''!|}!)\—50´JU\qTC'œJcIGEFC?CacGQQJPYJBG'[DXfCZC:[\qaJ

QCHoaST[WG'[DXYCFZC[\qaJ#oTB\JUDU\SaBDJ‘1šI[\qaJMPfXYZaJ„G'B5U\CPY’FJB„—ak”Z $ÐN qaC1|3*)bTœJoaBDCFopCFUDJc›GŸZTJœ G'PYEFCB\XY[\qTH

[DCGQQJPYJBG'[DJ[\qaJl5Jœ5[DCFZ?“\µ#¶$mn˜•G'PYEFCB\XY[\qaH—F¥5qTJ§H G'XYZ X‹cŒJGœ5G'U[\C G'oaoTB\C„¢aXYHG'[DJ[\qaJŸZaJ¢a[

l´Jœ5[\CZªUD[\JoŽ BDCFH [\qTJ¦nB\š?PYC'’›‘pG'U\J$GZc{[DqaJ 0´JU\UDJZ?‘JBDE™H G'[\BDX^¢&QCFHIoaSa[DJc{‘?š[\qaJ hBDZaCFPRc?X

oaBDCaQJU\U„—:hnZ›GFQeQJPYJBWG[\XYCFZC' [\qaJŸC'’FJBG'PYP:oTB\CaQJUDU#œ¡GUC‘aU\JBD’FJc<—Vk”Z›GBDJQJZ?[#opG'opJBU$˜·²A|v*)\bG

oaBDJQCZc?XY[\XYCFZTJB§œ5G'UŸ‘aSaXYPY[[\C$XYHIoaB\C'’J§[DqaJBDJU\[G'B\[DJc“\µ#¶$mn˜•™GPfECFBDXf[DqaH›—T¥5qaJoTB\JQCZc?XY[\XYCZaJB

œ5G'U#‘pG'U\J„c{CFZ™GZ{JU\[DXYHG'[DXYCFZ$C¡[DqaJUDHG'PYP8JXYEFJeZ1’FGPYSaJ¤GZcXY[\U§GU\UDCŒQŠX‹G[\Jc JXYEFJZTU\oGQJF—8k”Z{[\qTXfU

oGoJeBbpœ5XY[\q›GŸUDXfHIXYP‹GBG'oToaB\C?GFQDq:baœ J#‘aSTXfPRc GQDqaJGo$oaB\J„QCFZcŒXf[DXYCFZaJBn BDCFH"[\qaJŸ¦BDš?PfC'’I‘G'UDJ§G'Zpc

[DqaJ405JeU\U\JeZ1‘pJBDEH G'[DB\X^¢QŠCFHoTSa[\JcI‘?šµ#¶$mn˜•V—

¥5qaXYUIoG'opJBXYU QCFHIoCU\Jc3G'U CFPYPYC'œ5U—³ XYB\UD[XYZ3U\JQŠ[\XYCFZ tab GU\qaCB\[BDJQGPfPCw[DqaJ{µ#¶$m˜·•ªGP^“

ECFB\XY[DqaH XYU§EXY’FJZŽœ5qaXRQ\qŽCFSa[DPYXfZTJU§UDCFHIJZaCF[G'[\XYCZaU—8¥5qaJeZ:b8œJ c?JUQB\XY‘pJ$[\qaJ oaBDJQCZc?XY[\XYCFZTJBXYZ

UDJQ[\XYCZ™}G'Zpc$XY[\Un[\qaJeCFB\Je[\XRQG'P QCFZ?[\BDXY‘aSa[\XYCZ:—V³:XYZGPYPfš?b?XYZ$U\J„Q[\XYCFZ 3pbpZ1STHJBDX‹QeG'P:BDJU\STPf[DU#U\qTC'œ [\qaJ

oaBGFQ[DXRQG'P ’FG'PYSaJC[\qaXYUG'oToaB\C?GFQDqXYZ [\qaJ#QCZ?[\J¢a[CN5³8jŸ—

(8)

l´CF[WG[\XYCFZTU

G

ˆ{Jc?JeZaCF[DJ‘?š

R

[\qTJU\Je[C'opCFPYš?ZaCFHIX‹GPYUC'!cŒJEFBDJJ Sao™[DC

T

qG„’?XYZaE$[DqaJ ’FG'PYSaJ

z#G[uabaG'Zpc‘?šŸ[DqaJXRc?JZ?[\XY[”šH G'[DB\X^¢¤C=BG'Za–V—

k”ZŽ[\qaXYU#UDJQ[\XYCZ:b8œJIoaB\JeU\JZ?[‘aBDXfJš{[\qaJ µ#¶$mn˜•{G'PYEFCB\XY[\qTH›baXYZ?[\BDCac?SQXYZaE$UDCFHIJZaCF[G'[\XYCZaU#œ J

œ5XYPYPVZaJJcP‹G[\JB„—

N CZaU\XRc?JBDXYZaE[\qaJMU\š?U\[DJH

n

KD ^ gUF

A3 C

œ5qaJeB\J n D n

A;C b6g

D g A;C

GZc%n{XYU!ZTCFZ?“”U\XYZaESaPRG'Bb?[DqaJŸoaB\XYZQXYoaPYJ#C¡µ#¶$mn˜• XYU[DCHIXfZTXfHIX8J

^ n A

> Cv^

g ]nœ5XY[\q

XfZI[\qaJŸ¦BDš?PfC'’UDSa‘aUDoGFQJ

A

nkF

F

C

c?J]ZaJ„c ZaJ¢a[—

I¬! £‚"  °#%$'&)(+* '-,j.0/0132 542k 6

A

n'F

F

C

"22* e.e.k-1S6

D ^ g ^ n

kBU* Bn 72 '82 542'(92k':4;.#•?oGZ=<>

F!n'

F*``*`Fn

@?sY

0AB k!

(!*S2U-C2

2 U*

¥5q?SaU„b

A3 C

XYU5B\JoaPRGQJc ‘?š[\qTJ CFPYPYC'œ5XYZaEHXYZaXYHX8G'[DXYC'Z§oaBDCF‘aPYJH

HXYZ

D"E0F

r)G

<HJI<

t ^ n ](`

Av C

¥5qaJŸ¦BDš?PfC'’IU\Sa‘TU\oGQJ§XYUnEFJZaJeBWG'[DJc‘1šI[\qaJMhB\ZTCFPRc?X:oTB\CaQJUDU!PYJGc?XYZaE[\C[DqaJM CFPYPYC'œ5XYZaE#µ#¶$m˜·•

GPfECFBDXf[DqaH›—

t G

µ#¶$mn˜•

ALK:C

EXf’JZNMPO,uŸ[\qaJ[DCFPYJBWGZQJ#C=[DqaJ#U\[DCFoaoTXfZTE¤QBDXY[\JBDXfCZ›G'Zpc

RQ IKJ

[\qaJMXfZTXf[DXRG'PpEFSaJUDUb

S

€0(#T'—U

D ^ g ^ n b

V D ] b5W Y D

V

b#

D u

S

€X$'T'—Fƒ'¬ZY¬?…a‚

D > z

AG C-[

Y D n\W

^

]^

_

Y`

^< W ^

œ5XY[\q

` ^<

DHA n\W0'FW

^C ALa6D

z(F*`*``F

C

` Y < D [ Y ]

W Y D [ Y

` Y <

A‘ C

QŠCFHoTSa[\J8bU

D

HIXYZ

c0E Ied

V\f Y ^ X hg ]

i:‚" "=bU9jM g

]n°pƒP

DkK

S

€Xl'T'—1QCHoaST[\Jmg0?b

Don

0g0G'Zc

D >p

(9)

bU9O M g ]n‚8¬Z

D

µj9n¥-9

S

€0(#T

¬Z±

œ5qaJBDJ

A

X:

C <^ D ` <^ z jNj >

z(F j

a j

u CF[\qTJB\œ5XYU\J

n D A W YF*``*`FW0 C F

GZc

V\f Y ^ X g0

] D

HIXfZ

c0E Imd

V\f Y ^ X 0g

](`

¥5qaJŸµBWGH“\•aQDqaHXYc?[8oaB\CaQJUDUEFJZaJeBWG'[DJU#G'ZICFB\[DqaCFECFZGPV‘G'UDXYU5U\SQDq [\qG[

n n D n YX: F AK C

XYHoTPfš?XYZaE

HIXfZ

D"E0F

r)G

<HJI*<

t ^ n ] D HXYZ

c0E Imd

V\f

Y ^ X 0g

] ` A% C

$

k”Z CFSTBQCFZ?[\JŠ¢Œ[„baœ JqG„’FJ[\CUDCFPY’FJUDSQJUDU\XY’FJ#PYXYZaJGB5U\š?U\[DJHU

n rR t

rR t

DH^

g rR t

`

m´CFSaEqaPYš›UDoJG–?XfZTEb CFSaBGFQQŠJPYJBWG[\XYCFZ3G'PYEFCB\XY[\qaH G'BDJ‘GU\Jc CFZ™[DqaJ£CFPYPYC'œ5XfZTEXRc?JG

G

N CHoaST[\J

G[!CZaJ#UD[\Jo

AT GZc

EXf’JZ

C

GoaBDJQCFZpc?XY[\XYCFZaJeB!STU\XYZaE X Q I r Y

t

G'Zpc

n Y Q I-J r Yt

UG'[\XYU  š?XfZTE

A K CKA

œ5XY[Dq n r R

t

D n C

GZcI[\CBDJSaU\JMXf[ £CFB

TWT

G'Zc —

k”Z{UDJQ[\XYCZª}a—Yzbœ JŸ]BDU\[ŸCFSa[DPfXYZaJG ‘GU\XRQGFQeQJPYJBWG[\XYCFZ™GPYEFCFBDXY[\qaH—a¥5qaJZ£B\CFH [\qaJ µ#¶$mn˜•›GP^“

ECFB\XY[DqaH‰oaBDJU\JeZ1[DJc{XYZ[\qaJoTB\J’?XYCFSaUMU\JQ[DXYCFZ:b œJŸ‘aSaXYPRcGoTB\JQCZc?XY[\XYCZaJB#XYZU\Sa‘aUDJQ[DXfCZª}T—tT—pk”Z

UDSa‘aU\J„Q[\XYCFZ{}a—Ð}TbFœ J#U\[DSc?šI[\qaJ§QŠCFZ?’FJBDEFJZpQJ§Cµ#¶m˜• G'oToaPYXfJ„c[\C[DqaJ#oaBDJQCZc?XY[\XYCFZTJcIHG'[DB\X^¢:—

hnZc™GU\qaCB\[’FGB\XRG'[DXfCZ›[G'–?XYZaE$XYZ?[\C›GQQCSaZ?[§[DqaJ J’FCPfST[\XYCFZ{C[\qaJIx1GFQŠCF‘aXRG'ZŽœ5XYPfP·‘J EFXY’JZ{XYZ

UDSa‘aU\J„Q[\XYCFZ}a—ÐK#STU\XYZaEGŸBWGZa–“½CZaJSaoVcaG'[DJF—

!#"%$#&(')+*,$-!.+)/"('0&(132

ˆ{Jn]BDU\[EXY’FJ#GŸ‘G'UDXRQ!GPYEFCFBDXY[\qaH C'GQQJPYJBG'[\XYCZœ5qTX‹QDq›GXYHUG[QCFHIoaSa[\XYZaEM[\qaJ#oTB\JQCZc?XY[\XYCZaJB

G[5CFZaJU\[DJo›GZcI[\CB\JeSaU\J#XY[· CFB5UDCFPY’?XYZaE§UDJ’FJeBWG'P PYXfZTJG'B5UDš1UD[\JHIU—

(10)

} G

h!QeQJPYJBWG[\XYCFZ

T D u'F*``*`

;QDM;OAT6C

TQDAT6C

D u F`*``

QD A C

CFo

A F T F SF T6CSD

—[\BDSaJF—?‚ =¬Z

•?CFPY’Jjn

rR t

? R

A?

R

C

?vY

; D ^ g rR t

¬ "”€¬

•?CFPY’Jjn

rR t

;

D ^ g rR t

¬Z±

CŒ_

A F T F SF T6C

‚ 8¬Z›‘aSTXfPRc

? R

;!

Y D ;>";

;OATB>

z

CEDM;!

Y

* '0&('!)/* ) &(1 3"( )+* '0&('!)+* #"

ˆ{JQCZaU\XRc?JeB[DqaJ5PYXfZTJG'BUDš?U\[\JeH n

-D ^ g GZcŸG'U\UDSaHJ5[DqG'[·œJGoaoaPYXYJcPŸXY[DJBWG[\XYCFZaU·C':µ#¶$m˜·•

UD[WG'BD[\XYZaEIœ5Xf[Dq:

D ^ g ^ n

bTU\C[\qpG'[œ JEFJ[G¦nB\š?PYC'’‘GU\XYU

n Y!GZc GU05JUDU\JZ?‘pJB\E$H G'[DB\X^¢

X U\SQDq [\qG[ n n D n YX: œ5XY[Dq[DqaJn CFPYPYC'œ5XYZaEm ½GFQ[DCFBDXfUG'[DXfCZ

X

D

e F

D D A

F Y

C 8

u

`

l´C'œ#bFœ J!c?JŠ]ZaJ#GoaBDJQCFZpc?XY[\XYCFZaJeB

A

n'Fg

C

b'ZaC[\Jc qaJBDJ5 CB[DqaJH G'[DB\X^¢Un—¥5qaJCF‘FgŠJQ[\XY’JXYU[\C

UDCFPY’FJBn›JŠ¢TGQ[\PYšICFZ [\qaJUDSa‘aU\opGFQJ#UDoG'ZTZaJc ‘?š

n Y

I¬! £‚"  °#%l'&)(s

Q I F

D

u"!

D n Y

?sY

u

u z = ^

Y$#

n =

Y > J `

ˆ,XY[Dq›[DqaXYU§c?J]ZTXf[DXYCFZ:bpœ JU\qaC'œ¨[\qG[#[\qaJoTB\JQCZc?XY[\XYCZaJcHG[\BDXÒ¢%n% qGUPJXYEFJZ?’FG'PYSaJeU!J„_FSpG'P

[DC&™GU\UDCŒQŠX‹G[\Jc[\C[DqaJ#XYZ?’FG'BDX‹GZ?[UDSa‘aU\opGFQJŸU\oGZaZaJc‘?š

n Y

?—p¶ CFBDJ#oaB\J„QXYU\JPYš?bœ J#qG„’J[\qaJ

 CPYPfC'œ5XYZaEM[\qaJCB\JH

G

' 8¬?°pƒ'¬$ l'&)(s kj '((0/4"2 *)

I J 4

+)

D AJn

Y

F-,

C

D AJn

Y F n Y Y F-,

C

D AJn

Y F-, ] C

(11)

k!

n%

D )

i n'YS]

u n5]] )= F A& C

B

' n Ye]

D AJn

Y C=

n,%Y_Bn ]]

D , = ] n ,%Y

B , Y

DHA W0 YF-,

C

ƒ'°V°%&

A

[\qaJeCFB\JeH

C

iVJe[[\qTJ!CB\[DqaCFECFZGPV‘G'UDJR¨C

I J

‘J

D AJn Y F-,

C F

D AJn F-, Y C `

m´Jœ5B\XY[\XYZaEZn›XYZ [\qTJU\J#ZTJœ ‘pG'U\XYU G'Zpc+)£baPYJGFc[\C[DqaJ CPfPYC'œ5XYZaEPYJHH Ga—

·¬ {…%l'&)( +)

=

n\

D

8 n Ye]

u n ]]

ƒ'°V°%&

A

PYJHH G

C

³aBDCFH

AK C

b?œ J!qpG„’FJ

AJn

Y C = n n D ` A| C

¥5q?SaU+)

=

n\

D AJn

Y F-,2]

C = n AJn

'F-,%Y

C

D

8

AJn

Y C= n,%Y

, =

] n n , =

]

n, Y

D

8 n'YS]

u n ]]

³TSaB\[DqaJBDHCFBDJFb ¨QG'Z ‘pJ#B\Jeœ5B\XY[\JZG'U

·¬ {…%l'&$

D

m+)

= B

D

?sY

u

u J ?

ƒ'°V°%&›³TB\CFH [\qaJ§cŒJ]ZaXY[\XYCZ}a—Yzb

Don

Y

?vY

u

u z = ^

Y # n =

Y > J

Don

Y

?sY

u

u z A n Y C =

^pn Y n =

Y > J

Don

Y

?sY

u

u z A n Y C = >

,,

= `

³TB\CH PYJHH G!}T—YzFb?œ J!qpG„’FJ

)= n\

D

i n'YS]

u n5]]

GZcI[\qaJ#[DqaJCB\JH  CFPYPYC'œ5U—

k”Z$[DqaJ#ZaJŠ¢Œ[UDSa‘aU\J„Q[\XYCFZ bœ J#UD[\Sc?š [DqaJ#BWG[\JŸC'QCZ1’JB\EJZQJŸC'µ#¶m˜• G'oToaPYXfJ„c[DC

D

n%

D

+) m

=

—?ˆ{JU\qaC'œ qaC'œ,[DCQ\qTC1CU\J{[DCCFoa[DXYHX8J5[\qTJ#oaB\J„QCFZcŒXf[DXYCFZaJB„baXYZBDJPRG'[DXfCZ [\C,

=

] n ,%Y'—

(12)

"() "&('!

hnU\U\STHXYZaE$[DqG'[ŸœJIœ¡GZ?[§[DC$U\CPf’J2n

D

U\[G'BD[\XYZaEœ5Xf[Dq™G'ZŽXfZTXf[DXRG'PBDJU\XRc?SGP

b:œ JUD[\ScŒš

[DqaJB\JUDXRc?SG'P ^ G„ [DJB

a

XY[\JeBWG'[DXYCFZaUC'5µ#¶$m˜·•›GoaoaPYXYJc [\C%n%

D

§—:i<J[ D =

—V²š[\qaJ

CB\[\qTCFEFCZG'PYXY[”šŸC' +) b?œ Jn]B\UD[qG„’J

] D ] A

zu

C

¥5qaJMHG'XYZŸB\JUDSaPY[C=[\qTXfU5UDSa‘aUDJQ[\XYCZ$XYU5‘G'UDJc CFZI[\qaJw CFPYPYC'œ5XYZaEPfJeHHGT—

·¬ {…%l'&l A

!

J

?=

^ n ]]

C

72 2 "( U/ k!

Q I J J

:2 "( Z//82 k :k

?sY

D

i u

u n5]]

D

ƒ'°V°%& ˆ{J[G'–FJ

D

:YS]

u J ?

œ5XY[\qZYS]

DN^

n Ye]

A

i

J ? ^ n5]]

C

?sY

—:¥5qTJZ:b8œ JQGZ

JGU\XYPYšQDqaJQD– [\qpG'[

?sY

D

! u

u n5]]

l´C'œ#b?œJœ5XYPYPHIJG'UDSaB\J[\qTJ!UDoJeJc“”Sao$C'µ#¶$mn˜•[DqG'[5œ J#QG'Z CF‘T[WG'XYZIœ5Xf[DqGZG'oToaB\CoaB\XRG'[DJSaU\J

C¡[DqaJoaB\J„QCFZcŒXf[DXYCFZaJB„—:³ XYB\U\[„bVœ JZaJJcUDCFHJŸGFcac?XY[DXfCZG'P8ZaC[WG[\XYCFZaU„—pk”Z›[DqaJ ‘pG'U\Jb

4D

œ5qaJeB\J

DNA Y F ] C =

XYUŸoG'BD[\XY[\XYCZaJc{QCB\B\JeU\opCFZc?XYZaEPYš{[\C$[DqaJoGB\[DXf[DXYCFZ{C'

DNA

Y F ] C

—=i<J[

) D

)nœ5qaJBDJ )

DNA u'F )] C=

œ5XY[\q )] P‹G[\JBc?J]pZaJcV—8ˆ{JQCZaU\XRc?JeB[”œC›µM¶$m˜•oTB\CaQJUDU\JU

GoaoaPYXYJc[\C U\[WGB\[DXfZTE›BDJU\opJQ[DXf’JPYš B\CH

GZc ) — ˆ{JZaCF[DJ‘1šJ¢a[\JeZaU\XYCFZ ^

G'Zc )^ [\qaJ

BDJU\XRc?SGP=G  [\JB

K

XY[\JBG'[\XYCZaUCµ#¶$m˜·•V—?¥5qaJZ:bTœJEJ[5[\qaJoaBDCFopCFUDXf[DXYCFZ:—

ƒ'°#Y°V€ £‚"  °#%l'&)( A i

J ? ^ n5]]

C

72 :2 ( Z// k!

^ Y ] j )^ ] j A C ) ]

j A

C ) ]

40// Q ^

B

) D Au F A J

?=

^

?sY

n5]]

C

5]

C =

ƒ'°V°%&$ˆ{JqG„’FJ

^ Y ] D

HIXYZ

E

I A C ]

j A C ` Y A C ]nœ5Xf[Dq

` Y Q Y

j A C

?vY

` Y A C ]

hnU

?sY

D

Y >

n5]]5] F

œ J[WG'–J

` Y A CD

z ^

?sY

—?ˆ{JCF‘a[G'XYZ

?vY

` Y A C D

?sY

^

?vY

?sY

D

?vY

U]

)] D

?sY

) D ) F

(13)

œ5XY[\q )]

DHA J

?=

^

?vY

n ]]

C

5]—

•?C

^ Y ] j A

C ) ]

j A C ) ] `

k”Z oGB\[\XRQSTP‹GBb

^ Y ] j

)

^ ] `

œ5qaXRQDq JZQPYCU\JUn[\qaJ#oTB\C?C'W—

ˆ{J#c?JcŒSQJ#JGU\XYPYš BDCFH

A

zu

C

GQCB\CPfPRGB\šC'8[\qaJMoaB\CoCU\XY[\XYCFZ

A

}T—fz

C —

°ƒ'° "" …aƒ'­l'&)( A i

J

?=

^

n5]]

C

72 2 "( Z//.' ^ Y D +) ^

YP2* 72. Q ^

^ Y ] j )^ ] j

)) ]

B )) D A u'F ))

] C = F ))

] D A n ]]

C )]

l´C'œ#b?œJ§GB\J#G'‘TPfJ[DC§PYXYZa–[\C , =

]

n, Yn[\qaJ§QŠCFZ?’FJBDEFJZpQJ#C'8[\qaJŸµ#¶$m˜·•HJe[\qaCacGoaoaPYXYJcI[\C §—

' 8¬?°pƒ'¬$ l'&$ R22 B "(B'*

A

i

J ? ^ n ]]

C 2 :2 "( Z//-k!

^ Y ] j A YS]

]] > z C

HXYZ

E

I A n ]]

C )] ]

B

%n'YS]

D AJn

Y C=

n, Y !-n ]]

D , =

]

n,%Y ! )

]

DNA

^

?vY

n5]]

C ]$! ZYS]

DN^

n'YS]

A

i

J

?=

^

n5]]

C

?vY

^ Y ] D

HIXYZ

E

I A C )

] `

ƒ'°V°%&$ˆ{JqG„’FJb

)) ]] D A n C ) ]]

D YS] A n ]]

C )

]

A n ]]

C )] ]]

j A :YS]

]] > z C A

n5]]

C ) ] ]]

¥5qaJM[\qaJCB\JH  CPfPYC'œ5U œ5qaJZ œ JwGoaoaPYš[\qTJ§N CFBDCFPYPRG'B\š}a—Yz—

l´C'œ#bVœJQGZ™c?XYUWQSaUDU#[\qTJ[\qaJCB\J[DXRQG'P B\JUDSaPY[\U#C‘a[WGXYZaJcG'Zc[\qaJQDqaCXRQJC'[DqaJŸ BDJJoG'BG'HIJ[\JB

—a³ XYB\U\[„b?œJ#QCZaU\XRc?JeB5[\qaJBWG[\JC'QCFZ?’JB\EJZQJ#Cµ#¶m˜•œ5Xf[Dq[DqaJoaB\JQŠCFZc?XY[DXfCZaJcIHG[\BDXÒ¢ §—

ˆ{J§QDqaC?CU\J ŽU\SQDq[\qG[

n'YS]

]9j

]

GZc

n5]]

]9j

]

—p¥5q?SaU„b

A

i

^

n5]]

C

XYU!ZTCFZ?“”U\XYZaESaPRG'BG'Zpc

:YS]

] j z—1³TSaB\[DqaJBDHCFBDJFb ]5XYUXYZc?JopJZcTG'Z?[C'% —F¥5q?SaU„b'XYZ¥5qaJCB\JH }a—ÐtTb„ CFBPRG'B\EJJeZaCFSaEq:b

œ J5qG„’FJ5UDqaC'œ5Z[\qpG'[[DqaJQCFZ?’JB\EJZQJXYUB\JeP‹G[\JcŸ[DCUn5]]

D , =

]

n,%Y —F²·Sa[GU ŸXYUXYZ§EJZaJBG'PpU\H G'PYP

QCHoGB\J[\C ba[\qTJ§QCFZpc?XY[\XYCFZ Z?SaH‘pJB5C'Qn5]]

D , =

]

n, YnXfU5ZTCF[ESG'BG'Z?[\JeJ#[\C‘pJ#U\H G'PYPYJB [\qGZ

[DqaJ§QCZc?XY[\XYCFZIZ?SaH‘JeBC n—al´J’FJBD[\qaJePfJeU\Ub<Xҏ

XYU5QPYCFUDJ#[\C

[\qaJZ œ JJ¢aoJQŠ[

)] j }t

5]

o

`

k”Z [\qTJ#ZaJ¢a[5U\J„Q[\XYCFZ baœ J!œ5XYPYP:QCHoGB\J[DqaJ#µ#¶$mn˜•IhPYEFCB\XY[\qaH GoaoaPYXYJcI[\CZn&G'ZpcŸ—

(14)

)/2 3$-"(' )/*

l´C'œ#b:œ JU\[DSc?š{[\qTJµ#¶$mn˜•{G'PYECFB\XY[DqaHGoaoaPYXYJc$[DC@nŽ CB#[\qaJIUWGHJŸPfXYZaJ„G'B#UDš1UD[\JH n D

œ5XY[\q [DqaJ#UWGHJnXfZTXf[DXRG'PpB\JUDX‹cŒSG'P

—?hZcIœ J§QCFHIoG'BDJ[DqaXYU5oaB\CaQJUDUœ5Xf[Dqµ#¶m˜• G'oToaPYXfJ„c[\C

SaZpc?JB[DqaJ CPYPfC'œ5XYZaEBDJU\[DB\XRQ[\XY’J§G'UDU\SaHIoa[\XYCFZTU—

A

h§z

C

“•?oGZ=<

n A D

•?oG'Z'<>

A

nkF

F

C A b

A

h!t

C“ ^ rH I < I tH I

r

HJI!<HJI

t ] jmP a

]'b

A

h!}

C“ n

^ r

HJI<\I

t

HJI

t

r

HJI!<HJI

t

] j n

?vY

^ r

HJI <\I

t

HJI

r

HJI!<HJI

t

]—

i<J[n]B\UD[!‘pJEXfZ œ5XY[\q[\qaJŸXYZaXY[\XRG'P:GPfECFBDXf[DqaH›—ˆ{J§c?JZaC[\JŸ‘?š ^

[\qaJMB\JUDX‹cŒSG'PG„£[\JB

a

XY[\JeBWG'[DXYCFZaU5C

µ#¶m˜• G'oToaPYXfJ„c[\C

ƒ'°#Y°V€ £‚"  °#%l'&$v(!4"22 - Z2Ue$!

B k@1

^ ] j HIXfZ

E

I A n C

A n C ] >

P

aA z >

]_ ] < C ]

B D rHI < It

HJI

A n C

TvAJn

Y C ! A

CBD

z

>

_ ] <

2!@/ Z k

h4;//!

HIXfZ

E

d

I A n C ] `

ƒ'°V°%&

^ ] D

HXYZ

E d

I A n C ]

D

HXYZ

E d

I A n C A

>,A

^ =

]] C C

]

j¨HXYZ

E

I An C

An C ] > A n C A ^ C ]

j¨HXYZ

E

I An C

An C ] >

]_ Y <n A ^

C ]

j¨HXYZ

E

I An C

An C ]

>WA z >

]_ ] < C ^ ]

l´C'œ#bVœJŸoaBDC'’FJ[\qpG'[

A n C

•ŒoG'Z=<

n Y A

—Vˆ{JŸU\SaoaopCFUDJ[\qG[#XY[XfUZaC[[\B\STJFb:[Dq?SaU#[\qTJB\J

J¢aXYU\[DUeg

Q I

U\SQDq$[DqG'[

A!

A n C F n Y g C

D

uT—a³aSTB\[\qTJHCB\JFb?[DqaJB\JMJ¢aXf[DU{UDSQDq[DqG'[

An C ^ n Y

0g

]Rj

A n C ]

œ5qaXRQDq$QCFZ?[DBWGFcŒX‹QŠ[5[\qaJ#HIXYZaXYH§SaH¨oaB\CoJeB\[”šC

—

Références

Documents relatifs

Unite´ de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite´ de recherche INRIA Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 GRENOBLE Cedex

Unite´ de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite´ de recherche INRIA Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 GRENOBLE Cedex

Unite´ de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite´ de recherche INRIA Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 GRENOBLE Cedex

Unite´ de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite´ de recherche INRIA Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 GRENOBLE Cedex

Unite´ de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite´ de recherche INRIA Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 GRENOBLE Cedex

Unite´ de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite´ de recherche INRIA Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 GRENOBLE Cedex

Unite´ de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite´ de recherche INRIA Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 GRENOBLE Cedex

Unite´ de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite´ de recherche INRIA Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 GRENOBLE Cedex