on Approximationsand Fast ReanalysisinEngineeringOptimization
May25 {June 2,2000
Redued-Basis Output-Bound
Methods for Ellipti Partial
Dierential Equations 1
DimitriosV. Rovas
DepartmentofMehanialEngineering,Room3-243,
Massahusetts InstituteofTehnology,Cambridge,MA,
02139-4307,rovasmit.edu
Anthony T.Patera
DepartmentofMehanialEngineering,Room3-264,
Massahusetts InstituteofTehnology,Cambridge,MA,
02139-4307,pateramit.edu
Abstrat 2
We present a two-stage
o-line/on-line blakbox redued-basis output bound method
for the predition of outputs of interest assoiated
withelliptipartialdierentialequationswithaÆne
parameter dependene. The method is
harater-ized by(i) Galerkinprojetion onto aredued-basis
spae omprisingsolutions at seleted points in
pa-rameterspae,and(ii)arigorousoutputerrorbound
based on the dual norm of the resulting residual.
Theomputationalomplexityoftheon-linestageof
the proedure sales onlywiththedimensionof the
redued-basis spae and the parametri omplexity
of the partial dierential operator. The method is
thusbotheÆient and ertain: thankstothea
pos-teriori error bounds, we may safely retain only the
minimal numberof modes neessary to ahieve the
presribed auray in the output of interest. The
tehniqueispartiularlyappropriateforappliations
suh as design, optimization, and ontrol, in whih
repeated and rapid evaluation of the output is
re-quired;inthelimitofmanyevaluations,themethod
an beseveral orders ofmagnitude faster than
stan-dard (nite element) approximation. To illustrate
themethod,we onsiderthedesignof athermaln.
1. Motivation
To motivate and illustrateourmethodswe onsider
a spei example, a thermal n. The n, shown
in Figure 1, onsists of a entral \post" and four
horizontal plateswhihwe denote \subns;" then
ondutsheatfromapresribedux\soure"at the
root through the large-surfae-area subns to
sur-rounding owing air. The n is haraterized by
seven design parameters, or \inputs," 2 D
IR P=7
, where i
=k i
;i =1;:::;4; 5
=Bi; 6
= L;
and 7
= t. Here k i
is thethermal ondutivity of
the i th
subn (normalized relative to the post
on-dutivity);Bi istheBiotnumber,anondimensional
heat transfer oeÆient reeting onvetive
trans-port to the air at the n surfaes; and L and t are
the lengthand thiknessof the subns(normalized
relativetothepostwidth). Theperformanemetri,
or\output,"s2IR,ishosentobetheaverage
tem-peratureofthenrootnormalizedbythepresribed
heatux into then root,
root .
Figure 1
Wean expressourinput-output relationshipas
s = ` O
(u()), where ` O
(v) is a (ontinuous)
lin-ear funtional | ` O
(v) = R
root
v | and u() is
the temperature distribution within the n. (The
1
ThematerialpresentedinthisartileisanexpositoryversionofworkperformedinollaborationwithDr.LuMahielsof
LawreneLivermoreNationalLaboratoryandProfessorYvonMadayofUniversityofParisVIandreportedingreaterdetail
inreferenes[1 ,2,3,4 ℄. WealsothankProfessor JaimePeraire ofMIT, ProfessorEinar Rnquistof NorwegianUniversity
of Siene and Tehnology, Mr. Roland VonKaenel of EPFL, and Ms. Shidrati Ali of NationalUniversity of Singapore{
Singapore-MITAllianeforhelpfulomments. TheworkissupportedbyAFOSR,NASALangleyResearhCenter,andthe
Singapore-MITAlliane.
2
tial oordinate, x;we expliitlyindiatethis
depen-deneonlyasneeded.) Thetemperaturedistribution
u()2Y satisesthe weak form of theellipti
par-tial dierentialequation desribingheat ondution
inthe n,
a(u;v;)=`(v);8v 2Y; (1)
a(u;v;)istheweakrepresentationoftheLaplaian,
and`(v)reetsthepresribedheatuxattheroot.
Here Y istheappropriateHilbertspaewith
assoi-atedinnerprodut(;)
Y
andinduednormkk
Y 3
.
Thebilinearforma(;;)issymmetri,a(w;v;)=
a(v;w;);8w;v 2 Y 2
;8 2 D; uniformly
ontinu-ous, ja(w;v;)j kwk
Y kvk
Y
;8w;v 2Y 2
;8 2D;
and oerive, kvk 2
Y
a(v;v;);8v 2 Y;8 2 D.
Here and are positive real onstants. Finally,
theform `(v) isalinear boundedfuntional;forour
hoieofsalingandoutput,` O
(v)=`(v),whihwe
willexploitto simplifythe exposition.
It is readily shown that our form a an be
ex-pressed as
a(w;v;)= Q
X
q=1
q
()a q
(w;v);8w;v 2Y 2
;82D;
(2)
for appropriately hosen funtions q
:D ! IR and
assoiated-independentbilinearformsa q
:YY !
IR, q =1;:::;Q: theparameter dependene is thus
\aÆne" or\separable."Notethatweposeour
prob-lem on a xed n referene domain in order to
ensurethattheparametri dependeneongeometry
|Landt|entersthrougha(;;)andultimately
the q
(). Forourpartiularproblem,Q=15;ifwe
freeze (x) allparametersexeptLand t(suhthat
P
e
=2),Q=8;ifwefreezeonlyLandt(suhthat
P
e
=5),Q=6.
Our goal isto onstrut an approximation to u,
~
u,andheneapproximationtos,s~=` O
(u ),~ whihis
(i) ertiably aurate, and (ii) very eÆient inthe
limit of many evaluations. By the latter we mean
that, followingan initialxed investment,the
addi-tionalinremental ost toevaluate~s()foranynew
2Dismuhlessthantheeortrequiredtodiretly
ompute s() = ` O
(u()) by (say) standard nite
element approximation. Thisdenitionof eÆieny
optimization,and ontrol, in whih we require very
rapidresponseand manyoutput evaluations.
2. Redued-Basis Approximation
Redued-basis methods (e.g., [5 , 6, 7℄) are a
now-lassial approah that are a speial
\parameter-spae" version of weighted{residual(here Galerkin)
approximation. To dene the (or a) redued{basis
proedure, we rst introdue a sample in
param-eter spae, S N
= f
1 ;:::;
N
g, and assoiated
redued-basis spae W N
= spanf
1
u(
1 );
2
u(
2 );:::;
N
u(
N
)g;whereu(
i
)satises(1)for
=
i
2D (note i
refers to the i th
omponent of
the P{tuple , whereas
i
refers to thei th
P{tuple
inS N
). We thenrequireourredued-basis
approxi-mation to u() forany given , u N
() 2W N
Y,
to satisfy
a(u N
();v;)=`(v);8v2W N
; (3)
theredued-basis approximation to s() an
subse-quentlybe evaluatedass N
()=` O
(u N
()).
It is asimplematter to showthat
ku() u N
()k
Y
r
min
w N
2W N
ku() w N
k
Y ; (4)
whihstatesthatourapproximationisinsomesense
optimalintheY norm. Itan alsobereadilyshown
forourpartiular problemthat
s()=s N
()+a(e N
();e N
();); (5)
wheree N
=u u N
. It followsfrom (4),(5), and the
ontinuityof athat
js() s N
()j
2
( min
w N
2W N
ku() w N
k
Y )
2
; (6)
thus our output approximation is also optimal in
some sense.
We must, of ourse, also understand the extent
to whih the best w N
in W N
an, indeed,
approx-imate the requisite temperature distribution. The
essential point is that, although W N
learly does
not have any approximation properties for general
3
HereY =H 1
(),thespae offuntionsthatare squareintegrableandthat havesquareintegrablerst(distributional)
derivativesoverthenreferenedomain. Theinnerprodut(w;v)Y maybehosentobe R
funtions in Y, simple interpolation arguments in
parameter spae suggest that W N
should
approxi-mate wellu()even for very modest N;indeed,
ex-ponential onverge is obtained in N for suÆiently
smooth -dependene (e.g., [6 , 7℄). It is for this
reason that, even inhigh-dimensional(large P)
pa-rameter spaes, redued-basis methods ontinue to
performwell| muhbetter than\non-state-spae"
diret interpolation of (;s()) input-output pairs.
In somesense, redued-basismethodstransform
ex-tensive parameter-spaeexplorationfrom aproblem
into anopportunity.
We now turn to the omputational issues. We
rst expresstheredued-basis approximation as
u N
(x;)= N
X
j=1 u
N
j ()
j
(x )=(u N
()) T
(x); (7)
andhoosefortestfuntionsv=
i
(x );i=1;:::;N.
Wetheninserttheserepresentationsinto(3)toyield
the desiredalgebraiequations foru N
()2IR N
,
N
X
j=1 a(
j ;
i ;)u
N
j =`(
i
); i=1;:::;N: (8)
Equation (8)an be writteninmatrixform as
A()u N
()=L; (9)
whereA()2IR NN
istheSPDmatrixwithentries
A
i;j
() = a(
j ;
i
;);1 i;j N, and L 2 IR N
is
the\load"vetorwithentriesL
i =`(
i
);1iN.
We nowevoke (2)to notethat
A
i;j
()=a(
j ;
i ;)=
Q
X
q=1
q
()a q
(
j ;
i )
= Q
X
q=1
q
()A q
i;j ; (10)
where thematries A q
2IR NN
aregiven byA q
i;j =
a q
(
j ;
i
);1 i;j N;q = 1;:::;Q. The
o-line/on-line deompositionis now lear. In the
o-line stage,we onstrut theA q
;q=1;:::;Q. Inthe
on-line stage,for any given , we rst form A from
the A q
aording to (10); we next invert (9)to nd
u N
(); and we thenompute s N
() =` O
(u N
())=
`(u N
())=(u N
()) T
L. Aswe shallsee,N will
typ-ially be O(10) for our partiular problem. Thus,
as required,the inremental ost to evaluate s ()
foranygiven new is very small: O(N 2
Q) to form
A(); O(N 3
) to invert (the typially dense) A ()
system; andO(N) to evaluate s N
() fromu N
().
Butallisnotwell. Theaboveapriori resultstell
usonlythatwearedoingaswellaspossible;itdoes
nottellushow wellweare doing. In partiular,the
errorinouroutputisnotknown,andhenethe
min-imal number of basis funtions required to satisfy
the desired error tolerane an not be asertained.
Asaresult,eithertoomanyortoofewfuntionsare
retained; the former results in omputational
inef-ieny,the latter inunertainty and unaeptably
inaurate preditions. We thus need a posteriori
errorboundsaswell.
3. Output Bounds
To begin, we assume that we may nd a funtion
g():D ! IR
+
and symmetri ontinuous oerive
bilinearform ^a:Y Y !IR suhthat
kvk 2
Y
g() ^a (v;v) a(v;v;);8v 2Y;82D;
(11)
forsome realpositiveonstant ;forourthermaln
problemwean readilynda g() anda(w;^ v) suh
that(11) is satised. The proedureis thensimple:
we rst omputee()^ 2Y solutionof
g() ^a (^e ();v) =R (v;);8v2Y; (12)
whereR (v;)`(v) a(u N
;v;)istheresidual;we
thenevaluateourboundsas
s N
()=s N
(); s N
+
()=s N
()+ N
(); (13)
where
N
()=g() ^a (^e ();e ())^ (14)
is the bound gap. The notion of output bounds is
not restrited to redued{basis approximations: it
an also be applied within the ontext of standard
(adaptive)niteelementdisretizationaswellas
it-erative (Krylov)solutionstrategies [8 , 9℄.
Wean then showthat
s N
() s()s N
+
(); 8N; (15)
we thus have a ertiate of delity for s N
| it is
within N
() of s(). The proof of (15) is
sim-ple. To prove the left inequality we need only
therightinequalitywerst notethatR (e N ();)= `(e N ()) a(u N ();e N
();)=a(e N ();e N ();), sine `(e N
()) = a(u;e N
();) from (1) for v =
e N
(); we next hoose v =e N
() in(12) to obtain
g() ^a (^e ();e N
()) = a(e N
();e N
();); from this
result and theright inequalityof (11) we thenhave
N
() g() ^a (^e ;^e)
= g() ^a (^e e N
;e^ e N
)+2a(e N
;e N
)
g() ^a (e N
;e N
)
g() ^a (^e e N
;e^ e N
)+a(e N
;e N
);
(16)
from(16)andtheleftinequalityof(11)wethus
on-lude that N
() a(e N
;e N
); a omparison of (5)
and (13) thenompletes theproof.
Our a posteriori bound result indiates that,
through N
, we an now asertain the auray
of our output predition, whih will in turn
per-mitustoadaptivelymodifyourapproximationuntil
any presribed error tolerane is satised (see
be-low). However, from theperspetiveof eÆieny,it
is also ritial that N
() be a good error
estima-tor; a poor estimator will enourage us to
unne-essarily rene an approximation whih is, in fat,
adequate. To prevent the latter we would like the
eetivity N () N ()=js() s N
()j to be
or-derunity. Forourproblemitissimpletoprovethat
N
() =, and thus N
() is ertainly bounded
independentofandN;inpratie,eetivitiesare
typially less than 10, whih is adequate given the
rapidonvergene ofredued-basisapproximations.
We now turn to the omputational issues. We
rst note that, from (2) and (7), (12) an be
re-written as
^
a(^e ();v)=
1 g() `(v) Q X q=1 N X j=1 q ()u N j ()a q ( j ;v) ! ;
8v2Y: (17)
We thus see from simple linear superposition that
^
e() an beexpressed as
^ e()= 1 g() (^z 0 + Q X q=1 N X j=1 q ()u N j ()^z q j ); (18)
where z^
0
2 Y satises ^a(^z
0
;v) = `(v);8v 2 Y;
and z^ q
j
2 Y;j = 1;:::;N;q = 1;:::;Q, satises ^ a(^z
j
;v) = a q
(
j
;v);8v 2 Y: It then follows that
we an express N
() of(14) as
N ()= 1 g() " ^ a(^z 0 ;z^
0 )
| {z }
0 + 2 Q X q=1 N X j=1 q ()u N j
()^a(^z
0 ;z^
q
j )
| {z }
q j + Q X q=1 Q X q 0 =1 N X j=1 N X j 0 =1 q () q 0 ()u N j ()u N j 0
()^a (^z q
j ;z^
q 0
j 0
)
| {z }
qq 0 jj 0 # ; (19) s N +
() thendiretlyfollowsfrom (13).
The o-line/on-line deomposition is now lear.
In the o-line stage we ompute z^
0
and z^ q
j ;j =
1;:::;N;q = 1;:::;Q, and thenthe inner produts
0 ; q j ,and qq 0 jj 0
denedin(19). Intheon-line stage,
for any given new , and given s N
() and u N
()
asomputed in theon-line stage of theoutput
pre-dition proess (Setion2), we rst evaluate N () as N ()= 1 g() " 0 +2 Q X q=1 N X j=1 q ()u N j () q j + Q X q=1 Q X q 0 =1 N X j=1 N X j 0 =1 q () q 0 ()u N j ()u N j 0 () qq 0 jj 0 # ; (20)
andthen evaluates N
+
()=s N
()+ N
(). The
in-remental ost to evaluates N
+
() for anygiven new
is very small: O(N 2
Q 2
).
4. Numerial Algorithm
Inthesimplestasewetakeoureldandoutput
ap-proximationsto beu()~ =u N
()ands()~ =s N
(),
respetively, for some given N, and then ompute
N
() to assess the error. However, it is very
easy to improve upon this reipe. To wit, we take
~
u()= u ~
N
() and s()~ = s ~
N
(), where u ~ N () and s ~ N
() are the redued-basis approximations
assoi-atedwithasubspaeofW N
,W ~
N
,inwhihweselet
only ~
N of ouravailablebasis funtions. Inpratie,
we inludeinW ~
N
to sample points
i
losest to thenew of interest;
we ontinue to augment ourspae until ~
N
() "
(and hene js() s ~
N
()j "), where " is the
a-eptable error in the output predition. If we
sat-isfy our riterion for ~
N N the adaptive
proe-dure is entirely ontained within the on-line stage
of the proedure; and the omplexity of this stage
is redued(roughly)from O(N 2
Q+N 3
+N 2
Q 2
)to
O( ~
N 2
Q+ ~
N 3
+ ~
N 2
Q 2
) |oftenrepresenting
onsid-erable savings. Note the ritial role that ourerror
boundplays ineeting thiseonomy.
In pratie | to ensure that the
i ;z^
0 ;z^
q
j
are atually alulable | we replae the innite{
dimensional spae Y with a very high dimensional
\truth" spae Y
T
(e.g., a nite{element spae
as-soiated with a very ne triangulation). It
fol-lows that our bounds are not in fat for s, but
rather for s
T = `
O
(u
T
), where u
T 2 Y
T
satises
a(u
T
;v;) = `(v);8v 2 Y
T
. The essential point is
that Y
T
may be hosen very onservatively | and
henethedierenebetweens
T
andsrendered
arbi-trarilysmall|sine(i)theon{lineworkandstorage
isinfatindependentofthedimensionofY
T
,N,and
(ii)theo{lineworkwillremainmodestsineN will
typiallybequitesmall. Theunertaintyintrodued
bythetruth approximationis thusminimal.
5. Results and Disussion
We rst demonstrate the auray of the redued{
basis output predition and the output bounds by
onsidering the ase P
e
= 5 in whih L = 2:5 and
t = 0:25 are xed; the remaining parameters are
permitted to vary in the domain k 1
;k 2
;k 3
;k 4
;Bi 2
D
e
[0:1;10℄ 4
[0:01;1℄. ThesamplepointsforS N
are hosen randomly(uniformly)over D
e
;the new
value of to whihwe applythe redued{basis
ap-proximation is taken to be k 1
= 0:5;k 2
=1:0;k 3
=
3:0;k 4
= 9:0;Bi = 0:1 (similar results are obtained
atotherpointsinD
e
). WepresentinTable1the
a-tualerrorjs() s N
()j; theestimatederror N
()
(our stritupperboundforjs() s N
()j); andthe
eetivity N
()(the ratiooftheestimatedand
a-tualerrors). Weobservethehighaurayandrapid
onvergeneoftheredued{basispredition,evenfor
thisrelativelyhigh{dimensionalparameterspae(10
pointsorrespondstofewerthantwopoints\ineah
diretion"); and the very good auray (low
ee-tivity)of ourerrorbound (). The ombination
of high auray and ertiable delity permits us
to proeedwithanextremelylownumberofmodes,
withorrespondinglylowomputationalost.
N js s
N
j
N
N
10 1:4810 3
2:3410 2
15.82
20 2:9410 4
2:5910 3
8.81
30 1:8010 5
3:0910 4
17.12
40 1:8710 6
2:4510 5
13.10
50 1:1710 7
2:0810 6
17.98
Table 1
As regards omputational ost, in the limit of
\innitelymany"evaluations,thealulationofs()~
towithin0.1%ofs
T
isroughly285timesfasterthan
diretalulationofs
T =`
O
(u
T
);hereu
T
isour
un-derlying \truth" nite element approximation (see
Setion 4). The breakeven point | at whih the
redued{basis approximation rst beomes less
ex-pensive than diret evaluation of s
T
| is roughly
142 evaluations. In making these omparisons we
must of oursenotbias theonlusion: our \truth"
approximation here is not overly ne; and our
solu-tionstrategyforu
T 2Y
T
| anILU{preonditioned
sparsity{exploitingonjugate{gradientproedure|
is quite eÆient. The redued{basis approah is
muh faster simply beause the dimension of W N
,
N, is muh smaller than the dimension of Y
T , N
(whih more than ompensates forthe loss of
spar-sityin A). Formore diÆultproblems that require
larger N | problems with more spatial struture,
orinmoreompliatedgeometry,orinthree
dimen-sions|orthatarenotasamenableto fastsolution
methods on Y
T
| problemswith less \nie"
math-ematialstruture | therelativeomputational
ef-ieny of the redued{basis approah willbe even
more dramati.
The obvious advantage of theredued{basis
ap-proah within thedesign, optimization, and ontrol
environment is the very rapid response. However,
the \blakbox" nature of the on{line omponent of
the proedure has other advantages. In
partiu-lar, the on{line (e.g., MATLAB) ode is very
sim-ple,non{proprietary,transportable,and ompletely
deoupled from the (often very ompliated) o{
line \truth" ode. This is partiularly important
in theontext of multidisiplinary design
also suggests new approahes to eletroni
hand-books | parameter{spae exploration through
a-tionableequationsthatproviderapidandertiably
aurate solutionsto omplexproblems.
We losethissetionwitha more applied
exam-ple. We now x all parameters exept L and t, so
that P
e
= 2; (L;t) are permitted to vary within
D
e
= [2:0;3:0℄[0:1;0:5℄. We hoose for our two
outputs the volume of the n (easily alulable of
ourse), V, and the root average temperature (as
dened above), s. Asour \design exerise" we now
onstrut theahievable set| all those (V;s)pairs
assoiated with some (L;t) in D; the result, based
on many evaluations of (V;s ~
N
+
) for dierent values
of(L;t)2D
e
,isshowninFigure2. Wepresentthe
resultsintermsofs ~
N
+
ratherthans ~
N
toensurethat
theatualtemperatures
T
willalwaysbelowerthan
ourpreditions(thatis,onservativewithinthe
on-text of the design problem); and we hoose ~
N (see
Setion 4) suh thats ~
N
+
is always within0.1% ofs
T
to ensure that the design proess is not misled by
inauratepreditions. Notethat,given theobvious
preferenes of lower volumeand lower temperature,
thedesignerwillbemostinterested inthelowerleft
boundary of the ahievable set | the Pareto
eÆ-ient frontier; althoughthisboundaryan of ourse
be foundwithoutonstruting theentireahievable
set, manyevaluationsof theoutputswillstillbe
re-quired.
19
20
21
22
23
24
25
26
27
4
6
8
10
12
14
16
Figure 2
Many (though not all) of the assumptions that we
haveintroduedareassumptionsofonveniene, not
neessity,intendedtosimplifytheexposition. First,
the output funtional ` O
need not be same as the
inhomogeneity`;withtheintrodutionofanadjoint
(or dual) problem [2℄, all of our results above
ex-tend to the more general ase. Seond, the
fun-tion g() need not be known a priori: g() is
re-lated to an eigenvalue problem whih an itself be
readilyapproximatedbyaredued{basisspae
on-struted as the span of appropriate eigenfuntions
(intheoryweannowonlyproveasymptoti
bound-ing properties as N ! 1, however in pratie the
redued{basis eigenvalue approximation onverges
veryrapidly,andthereisthuslittlelossofertainty).
Third,these same notionsextend, with some
modi-ation,tononoeriveproblems, whereg()isnow
infattheinf{supstabilityparameter[3 ,4 ℄. Finally,
nonsymmetri operators are readily treated, as are
ertain lasses of nonlinearity inthe state variables
(e.g.,eigenvalueproblems[1℄andBurgersequation).
Perhaps the most limiting assumption is (2),
aÆne dependene on the parameter. In some ases
(2)mayindeedapply,butQmayberatherlarge. In
suh ases we an perhaps redue the O(Q 2
)
om-plexityandstorageoftheo{lineandon{linestages
to O(Q)byintroduingaredued{basis
approxima-tion ofthe errorequation (12)fora suitablyhosen
\staggered"samplesetS M
err
andassoiatedredued{
basis spae onstruted as the span of appropriate
errorfuntions. These ideasmayalso extend to the
ase inwhihtheparameter dependenean notbe
expressed (or aurately approximated) as in (2);
however we would now need to at least partially
abandontheblakboxnatureof theon{linestageof
omputation,allowingevaluation(though not
inver-sion) of the truth{approximation operator, as well
asstorage of some redued{basisvetors of size N.
Thesemethodsareurrentlyunderdevelopment;the
ideasof thisnal paragraph areat present
speula-tive.
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