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A parareal in time procedure for the control of partial
differential equations
Yvon Maday, Gabriel Turinici
To cite this version:
Yvon Maday, Gabriel Turinici. A parareal in time procedure for the control of partial differential
equa-tions. Comptes Rendus Mathématique, Elsevier Masson, 2002, 335 (4), pp.387 - 392.
�10.1016/S1631-073X(02)02467-6�. �hal-00798362�
A parareal in time pro edure for the ontrol of partial di er-ential equations Yvon MADAY a , GabrielTURINICI b . a
LaboratoireJa ques-LouisLions,UniversitePierreetMarieCurie,Bo^te ourrier187,75252Paris Cedex05,Fran e
b
INRIARo quen ourt,B.P.105,78153Le ChesnayCedexFran eand CERMICS,ENPC,Marne laValle,Fran e.
(Proposeeen?? 2002)
Abstra t. Wehaveproposedinapreviousnoteatimedis retizationforpartialdi erentialevolution equationthatallowsforparallelimplementations.Thiss hemeisherereinterpretedasa pre onditioningpro edureonanalgebrai settingofthetimedis retization.Thisallows for extending the parallel methodology to the problem of optimal ontrol for partial di erentialequations. Wereport a rstnumeri alimplementationthatrevealsa large interest.
Une te hnique de ontr^ole d'equations aux derivees partielles en temps parareel.
Resume. On a propose dans unepre edente note, un s hema permettant de pro ter d'une ar- hite ture parallele pour ladis retisation entemps d'equationd'evolutionaux derivees partielles. Ces hemaest i iinterpretesousunangledi erentetmatri iel,permettant d'etendrele on eptpourle ontr^oled'EDP.Lespremieresexperien esnumeriquessont extremementen ourageantes.
Version fran aise abregee(Lesreferen esfontappelalaversionanglaise)
PourunoperateurA lineaire oupas, agissantd'un espa ede Hilbert V dansV 0
on onsidere le problemede ontr^ole(1)ouv estun ontr^oled'unespa eU etB estunoperateurdeL(U;V
0 ). Ce problemeest ompleteparladonneedelafon tionnellede o^ut(2)ouy
T
estune ible. Onsouhaite etendreles hemaparareela e adreetonintroduitdon unesuited'instantsT
n veri ant(3)puis lesfon tionsfy 0 ;y 1 ;:::;y n ;:::;y N 1 gtellesquey n
estlasolutionde(4)pourtoutn=0;:::;N 1. Ilest lairquel'ensemblefy
0 ;y 1 ;:::;y n ;:::;y N 1
gde(4) estliealasolutiony duproblemeinitial (1)sipourtoutn=0;:::;N 1ona(6). Onpeutainsiinterpreterdans(4)
n
ommeun ontr^ole virtuel(alaLions .f. [2℄) e quiameneaintroduire une nouvellefon tionnellede o^ut J
" (v;) ommedans (7) ou = f 0 = y 0 ;:::; n ;:::; N 1
get " >0est petit. Lamethodedu gradient pour minimiserJ
"
onduit aintroduirelesetatsadjointsp n
, n=N 1;:::;0solutionsde(9) et (10). Onaalors(13)etdon lapro eduredegradient(14). Ilest lairquela onvergen ede ette pro eduredepend de N (souventassezgrand) |et e, independammentduveritable ontr^ole v |etdon qu'il onvientdel'a elerersil'onsouhaitepro terdelade ompositionentemps.
C'estlaqu'intervientlapro edureentempsparareelleintroduitedans[3℄quel'onpeutexpliquer deja sur le probleme sans ontr^ole (15) pour lequel il ne reste que le ontr^ole virtuel dans la fon tionnellede o^ut qui s'e ritsimplement selon(16). Parexemple, sil'on onsidere queA est independantdutemps,onpeutintroduirelepropagateurF
T
telquepourtoutdonne,F T
() estlasolutionautempsT duprobleme(17). Onnotealorsquelasu essiondelaresolutiondes problemes(4),ave B=0estequivalentealaresolutionduproblemeinitial(15)sietseulementsi n =F T ( n 1
)ouen ore,sousformematri ielle omme(18)ou en ore(19)ave desnotations evidentes. Ce systeme, sous forme d'une matri e triangulaire inferieure, peut ^etre inverse en O(N)operations. Les hemaparareelpermetd'a elerer etteresolutionen onstruisantunesuite
k
qui onverge vers la solution de (19). On introduit pour ela un propagateur grossierG T
, approximationdeF
T
,parexempleparles hemadetypeEulerimpli ite(20)etonpose(21)qui s'e riten ore,sousformematri ielle(23)ave lamatri e(22)etouleresiduR es
k
=F M k
. On a montre dans [3℄ et [1℄ que ette pro edure iterative onverge en peu d'iterations, ind e-pendamment de N et que les al uls les plus o^uteux (reposant sur F
T
) peuvent^etre faits en parallele.Onendeduitque
f M
1
peutetre onsideree ommepro hedeM 1 ,ausensoulamatri e d'ampli ation f M 1
M est pro hedel'identite.
Deretourauproblemede ontr^olevirtuel,onremarquequelaresolutionde(19)s'e ritÆ e J()=0 quipeutaussiprendrelaforme(24). Onremarqueegalementquelesautp
k n (T + n ) p k n 1 (T n )dans leproblemeadjointestegalauve teurM
R es
k
'estadireauresidude(24)aupask. Onpretend ainsimaintenantque
f M 1 ( f M 1 )
estunbonpre onditionneurdeM
Metonproposelamethode degradientpre onditionnee(25).
A ndeveri erlapertinen ede etteproposition,onl'atesteesuruneequationd'evolutiontres simpleposeesurl'intervalle℄0;1[: Trouvery solutionde
y t
y 00
=v ouv estle ontr^oleet est lafon tionindi atri ede℄1=2;2=3[. Ona ontr^ole etteequationsurl'intervalledetemps℄0;100[ partantdela onditioninitialey
0
=10x(1 x)versla ibley T
=sin(2x). Lessimulations nes ( orrespondantaF
T
)sontrealiseesave lepasdetemps2:10 2
soitsansde ompositionentemps, soit ave lapro edurede ontr^ole paralleliseeave T =1. Ilest interessantde noter qu'apres seulement25iterationsdus hemapre onditionne,lavaleurdelafon tionnellede o^utestdum^eme ordredegrandeurqu'apres100iterationsdel'algorithmede ontr^ole seul(sansde ompositionen temps); e i onduitaune a elerationd'un fa teur environ400! Nous nenous attendons pasa e que e soit le as pour tousles problemesneanmoins, m^eme ave undoublementdu nombre d'iterations du gradient ave pas optimal, la pro edure parareelle donne une a eleration d'un fa teurproportionelaT=T.
REMERCIEMENTS.Cetravailaeteinitialiseen ollaborationave J.L.Lionset 'estave unetresgrande emotionquenoussouhaitonsl'asso iera etravailquis'inspireenparti ulierde[2℄.
1. Introdu tion
LetAbealinearornonlinearoperatorfrom aHilbert spa e V into V 0
and letus onsider the followingstateequation:
( y t +Ay=Bv y(0)=y 0 (1)
wherethe boundary onditionsare imposed impli itly, where the ontrol v(= v(t)) (boundaryor distributed)belongstosomespa eU andwhereB isanoperatorinL(U;V
0
). Weassumethatfor anygivenv2U,thisproblemiswellposed.
We omplementthisproblemwitha ostfun tionalto beminimized
J(v)= 1 2 Z T 0 kv(t)k 2 U dt+ 2 ky(T) y T k 2 ; (2)
where >0,y isagiventargetandthenormistheH normif,forinstan eV H V (where H isapivotHilbert spa e).
Letusin reasealittlebitthe omplexityofthisproblem byde omposing timeasfollows
0=T 0 <T 1 <:::<T n =nT <T n+1 <:::<T N =T: (3)
Thenwede nethefun tionsfy 0 ;y 1 ;:::;y n ;:::;y N 1 gsu hthat y n isthesolutionof ( y n t +Ay n =Bv n ; over(T n ;T n+1 ) y n (T + n )= n ; (4)
foranyn=0;:::;N 1. Itisatrivialitytonotethatthe olle tionofsolutionsfy 0 ;y 1 ;:::;y n ;:::;y N 1 g of(4)is onne tedwiththesolutionyoftheoriginalproblem(1)(inthesensethaty
n =y j(T n ;T n+1 ) ) ifandonlyif,foranyn=0;:::;N 1
v n =v j(Tn;Tn+1) and n =y(T n ); (5)
oragain,andmoreintrinsin ally,ifandonlyif,foranyn=0;:::;N 1
v n =v j(Tn;Tn+1) and n =y n 1 (T n ) with y 1 (T 0 )=y 0 : (6) Thisway,in(4), n
anbeinterpretedasa\virtual" ontrol(alaLions .f. [2℄)thatleadstothe followingdevelopment. Inorder tosatisfy(6)approximatively,weproposeto modifyslightlythe ostfun tionalJ asfollows
J " (v;)= 1 2 Z T 0 kv(t)k 2 U dt+ 2 ky N 1 (T) y T k 2 + 1 2"T N 1 X n=1 ky n 1 (T n ) n k 2 ; (7) where=f 0 =y 0 ;:::; n ;:::; N 1 gand">0issmall.
Inorderto solvethisminimizationproblem,we omputethederivativeofJ " (v;) ÆJ " (v;)(Æv;Æ)= N 1 X n=0 Z T n+1 Tn (v n ;Æv n ) U + (y N 1 (T) y T ;Æy N 1 (T)) + 1 "T N 1 X n=1 (y n 1 (T n ) n ;Æy n 1 (T n ) Æ n ): (8)
The introdu tion of an adjoint state is the standard argument for getting an expression of this derivative. Letp
N 1
bethesolutionover(T N 1 ;T N )of ( p N 1 t +A p N 1 =0 over(T N 1 ;T N ), p N 1 (T)= (y N 1 (T) y T ); (9)
andthe olle tionp n ,n=N 2;N 1;:::;0ofsolutionsof 8 > < > : p n t +A p n =0 over(T n ;T n+1 ), p n (T n+1 )= 1 (y n (T n+1 ) n+1 ): (10)
Wemultiplynow(9)byÆy N 1
andweintegrateintimebetweenT N 1
andT N
,thenintegrateby partsanduse(4)soastoobtain
(p N 1 (T );Æy N 1 (T ))+(p N 1 (T + N 1 );Æy N 1 (T + N 1 ))+ Z T N TN 1 (p N 1 ;BÆv N 1 )=0
anotheruseof (9)gives
(y N 1 (T) y T ;Æy N 1 (T ))+(p N 1 (T + N 1 );Æy N 1 (T + N 1 ))+ Z T N T N 1 (B p N 1 ;Æv N 1 )=0 (11)
similarly,startingfrom (10)over(T n
;T n+1
)weobtainforanyn=0;:::;N 2
1 "T (y n (T n+1 ) n+1 ;Æy n (T n+1 ))+(p n (T + n );Æy n (T + n ))+ Z Tn+1 T n (B p n ;Æv n )=0: (12)
Usingthisin(8)thenyields
ÆJ " (v;)(Æv;Æ)= N 1 X n=0 Z Tn+1 T n (v n +B p n ;Æv n ) U + N 1 X n=0 (p n (T + n );Æy n (T + n )) 1 "T N 1 X n=1 (y n 1 (T n ) n ;Æ n ) re alling rstthat p n 1 (T n )= 1 "T (y n 1 (T n ) n
),se ondthat Æy n (T + n )=Æ n ,weget ÆJ " (v;)(Æv;Æ)= N 1 X n=0 Z Tn+1 T n (v n +B p n ;Æv n ) U + N 1 X n=0 (p n (T + n );Æy n (T + n )) N 1 X n=1 (p n 1 (T n );Æ n ); = N 1 X n=0 Z T n+1 Tn (v n +B p n ;Æv n ) U + N 1 X n=1 (p n (T + n ) p n 1 (T n );Æ n ); (13) sin eÆ 0
=0. Fromthese omputations,we aneasily implementagradientmethod (orbettera onjugategradientmethod)aniterationofwhi hreads: Assume y
k n ;p k n ;v k n ; k n
areknown,then
v k +1 n =v k n (v k n +B p k n ) in (T n ;T n+1 ); n=0;:::;N 1 k +1 n = k n (p k n (T + n ) p k n 1 (T n )); n=1;:::;N 1: (14)
It is quite easy to realize that the speed of onvergen e of the latter algorithm depends on the numberN, of timesteps T
n
. Indeed, thetransferofinformation betweentime 0and timeT for y and between time T and time 0 for p an only be done by su essive iterations through the subintervals(T
n ;T
n+1
),andrequires atleastN steps. A tually, the(real) ontrolisnotinvolved in this N dependan y sin e for a standard ontrol problem this gradient pro edure onverges quiterapidly. The dependan y here learly omes from the fa t that we havede omposed time into pie es. In order to understand what kind of pre onditioner an be added to the previous iterativealgorithm,weshallinvestigateinthenextse tionthe(only)virtual ontrol.Thisisdone byletting B=0and =0.
2. The totallyvirtual problem
Whatwewanttosolvehereisthussimply ( y t +Ay=0 y(0)=y 0 (15)
overthetimeinterval(0;T),thatisassumedtoadmitauniquesolutiony. Themethodofresolution throughthevirtual ontrolinvolvesade omposition ofthetimeinterval. Itisinterestingtonote that the ostfun tionalbe omesafun tion of only: i.e. J
"
(v;) anbeidenti edwith g J() sin ethe ontrolv ismultipliedbyzeroand "T isjustamultipli ativefa tor
g J()= N 1 X n=1 ky n 1 (T n ) n k 2 ; (16) The minimum of e
J is zero and is obtained by the hoi e n
= y(T n
). We shall not use this informationhoweverbuttryinsteadtodevelopalinkbetweenthisframeandthepararealalgorithm developpedin[3℄. A tually,letusassumethatAistimeindependentsothatwe anintrodu ethe (timeinvariant)propagatorF
T
su hthat,for anygiven,F T
()is thesolution,at timeT oftheproblem ( y t +Ay=0 y(0)= (17)
Withthisnotation,itisimportanttonoti ethatthesu essionofresolutionofproblems(4),with B=0isequivalentto theresolution oftheinitial problem(15)ifandonlyif
n =F T ( n 1 )as notedin (6),oragain,in amatri ialform
0 B Id 0 ::: 0 F T Id 0 ::: 0 F T Id :: 0 :: F T Id 1 C A 0 B 0 1 :: N 1 1 C A = 0 B y 0 0 :: 0 1 C A (18)
that analsobewritten,withobviousnotations
M = F (19)
ThestandardinversionofthistriangularsysteminvolvesO(N)resolutionsofsystem(17). Inorder toa elerate,wehaveintrodu edaniterativepro edure|theparareals heme|that allowsto onstru tasequen e
k
that onvergestowardtheexa tsolutionof(19). Thispro edureinvolves a oarsepropagatorG
T
,thatisanapproximationofF T
,forinstan ethroughanimpli itEuler s hemeasfollows G T () T +AG T ()=0: (20)
Itisde nedas follows(see[1℄)forthisinterpretation)
k +1 n+1 =G T ( k +1 n )+F T ( k n ) G T ( k n ) (21)
whereithastobenoti edthat,assuming k
isknown,the omputationof k +1
involvesaparallel pro edure ( orresponding to the expensive omputation of the F
T (
k n
)) and a sequential one ( orrespondingtothefast omputationof G
T (
k +1 n
Byintrodu ingthematrix f M = 0 B Id 0 ::: 0 G T Id 0 ::: 0 G T Id :: 0 :: G T Id 1 C A (22)
thisiterativepro eduretakesthematri ialform k +1 = k + f M 1 R es k (23) wheretheresidualR es
k isde ned byR es k =F M k .
It is provenand numeri ally he kedthat this method onvergesrapidly, independently of N, wheneverthegoverningpartinAislinearpositivede nite. Itresultsthat
f M 1 anbe onsidered as losetoM 1
, inthesensethattheampli ationmatrix f M
1
M is losetoIdentity.
3. Ba k to the ontrol problem
Letusgoba knowtoourvirtual ontrolproblem. It anbeeasilyguessed,duetothesymmetri formoftheLagrangeequationsarisingfrom(9){(10)thattheresolutionofÆ
e J()=0 analsobe writenas M M = M F (24)
Indeed,theve torofjumps p k n (T + n ) p k n 1 (T n
)in the dualstateis exa tely equaltothe ve tor M
R es
k
i.e. totheresidualof(24)atstepk. Thereasonwhytheoriginalgradients hemeisslow omesfromthefa tthatthe onditionningofM isO(N). Thefa tthatwehaveprodu edagood pre onditionerforMallowstoforseethat
f M 1 ( f M 1 )
maybeagoodpre onditionnerforM
M so that, going ba k to the original ontrol problem, we propose the following pre onditionned gradientmethod v k +1 n =v k n (v k n +B p n ) in(T n ;T n+1 ); n=0;:::;N 1 k +1 = k [ f M 1 ( f M 1 ) ℄ p k n (T + n ) p k n 1 (T n ) N 1 n=1 ; n=1;:::;N 1 (25)
Inorder to he k thepertinen eof this approa h, wehavetested onaverysimpleexampleof aonedimensionalparaboli equationontheinterval℄0;1[: Findy su hthat
y t
y 00
=v where v is the ontrol and is the indi ator of ℄1=2;2=3[. We have simulated this equation over the time interval ℄0;100[ from the initial ondition y
0
= 10x(1 x) so as to drive it to the target y
T
=sin(2x). The ne simulations( orresponding to F T
) are performed with the time step 2:10
2
either withoutde omposition in timeorbyusing thepre onditionned ontroledproblem with T =1. It is impressive(and nottotally understood)to obtain that after 25iterationsof thepre onditionneds heme,the ostfun tionisaboutthesameasafter100iterationsoftheplain ontrolpro edure. Wehaveusedagradientmethodwithoptimalstep. Theparareals hemethus a hievesaspeedupof morethat 400! Wedonotexpe tthat theparareals hemewillprovidean improvementoftheiteration ount(thatmaybedueheretothelongtimeevolutionwithaquite vis ous term)onmore omplexproblems, neverthelesswe expe t that thenumberwill be about thesamesothat itwillallowforaspeedup proportionaltoT=T.
Referen es
[1℄ BaÆ o L., Bernard S., Maday Y., Turini i G. and G. Zerah, Parallel intimemole ular dynami ssimulations,submittedtoCEMRACS'01pro eedings.
[2℄ LionsJ.-L.,Virtualande e tive ontrolfordistributedsystemsandde ompositionofeverything,J.d'Anal. Math.,80,2000,pp257{297.
[3℄ Lions J.-L., Maday Y. and Turini i G.Resolutiond'EDPparuns hma entemps"pararel",C.R. A ad.S i. ParisSer. IMath. 332(2001),no.7,661{668.
