ON THE NUMERICAL SOLUTION OF A NONLINEAR EIGENVALUE PROBLEM FOR THE MONGE … · 2019-03-10 · 0....
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ON THE NUMERICAL SOLUTION OF A NONLINEAR EIGENVALUE PROBLEM FOR THE MONGE-AMPÈRE OPERATOR Roland GLOWINSKI (University of Houston) & Hao LIU (Georgia Tech) Dédié à François Murat pour ses contributions à l’analyse mathématique des équations elliptiques non linéaires dont –∇ 2 u = λe u JL 2 LAB 50, March 2019
Transcript of ON THE NUMERICAL SOLUTION OF A NONLINEAR EIGENVALUE PROBLEM FOR THE MONGE … · 2019-03-10 · 0....
ONTHENUMERICALSOLUTIONOFANONLINEAREIGENVALUEPROBLEMFORTHE
MONGE-AMPÈREOPERATOR
RolandGLOWINSKI(UniversityofHouston)&HaoLIU(GeorgiaTech)
DédiéàFrançoisMuratpoursescontributionsàl’analysemathématiquedeséquationselliptiquesnonlinéairesdont
–∇2u=λeu
JL2LAB50,March2019
0.MOTIVATIONIntheearly2000,whilelookingforPDE’sfromGeometry,IcameacrossanarticleofAliceChangdiscussingtheLaplace-Beltrami-Bratu-Gelfandproblem(LBBG)
where ΣisasurfaceofRdandΔΣtheassociatedLaplace-Beltramioperator.AtthetimeIwasmoreinterestedintheMonge-Ampèreequation
whereΩisaboundedconvexdomainofRd.Forsomestrangereason,(LBBG)becameinmymind
explainingwhat’sfollows(whenIrealizedmymistakeitwastoolate).
Σ=+ΔΣ on 0ueu λ
Ω>= in )0(det 2 fuD
Ω= indet 2 ueu λD
1. FORMULATIONOFTHEEIGENVALUEPROBLEMAssumingthat ΩisaboundedconvexdomainofR2,ourgoalistosolvenumericallythefollowingnonlineareigenvalueproblem:(Monge-Ampère-Bratu-Gelfandproblem)
(MABG)Above.
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
>=−
⎩⎨⎧
Ω∂=
Ω=
>
∫Ω−
−
).0()1(
, on 0, in det
thatsuch 0 and Find2
Cde
ueu
u
u
u
x
D λ
λconvex
21dxdxd =x
REMARK1.TheconvexityofΩandu,andtheconditionu=0on∂Ω⇒u<0inΩ.REMARK2.P.L.Lions,AnnalidiMatematicaPuraedApplicada,142(1),1985containsmathematicalresultsassociatedwithacloselyrelatednonlineareigenvalueproblem(λu2insteadofλe–u).REMARK3.SupposethatΩistheunitdiskcenteredat(0,0).Lookingforradialsolutionsto(MABG)wesolve
(1)(byashootingmethodforexample).Therelatedbifurcationdiagramhasbeenvisualizedon
Figure1,below,showingaturningpointatλ≈3.7617,theassociatedfunctionutakingitsminimalvalueat(0,0)withu(0,0)≈–2.5950.
⎪⎩
⎪⎨
⎧
=ʹ=
=ʹ́ʹ
≥≤−
,0)0(,0)1((0,1), on
,0,0
uureuu
uuλ
λ
Figure1:MABGproblemontheunitdisk:Exactsolutionbifurcationdiagram
2.DIVERGENCEFORMULATIONSOF(MABG)Analternativeformulationto(MABG)isgivenby(MABG.DIV.1)Inordertotakeadvantage(viaoperator-splitting)ofthemethodologydevelopedinR.G.,H.Liu,S.Leung&J.Qian,J.Scient.Comp.,2019,forthenumericalsolutionofthecanonicalellipticMonge-Ampèreequation(E-MA)detD2u=f(>0)inΩ,u=gon∂Ω,wereformulate(MABG.DIV1)as
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
=−
⎩⎨⎧
Ω∂=
Ω−=∇∇−
>
∫Ω−
−
.)1(
, on 0, in 2) cof(
thatsuch 0 and Find2
Cdxe
ueuu
u
u
uλ
λ
D. convex
(MABG.DIV.2)
3.ANASSOCIATEDINITIALVALUEPROBLEMWearegoingtoassociatewith(MABG.DIV.2)aninitialvalueproblemwhosesteadystatesolutionssolve(MABG).Aftertime-discretization,theresultingalgorithmcanbeviewedasanonlinearvariantoftheinversepowermethodwithshift(theshiftbeinghereassociatedwiththeoperatorI/τ,τbeingatime-discretizationstep).
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
=−
=−
⎩⎨⎧
Ω∂=
Ω−=∇∇−
>
∫Ω−
−
.)1(
,
, on 0, in 2) cof(
s.t. 0 and Find
2
Cde
u
ueu
u
u
u
x
0Dp
p.
p
λ
λ
definite-semipositive symmetric convex,
(MABG.DIV.2)-→(MABG.IVP)(MABG.IVP)Above:(i)ϕ(t):x→ϕ(x,t).(ii)ε>0(ε≈h2inpractice).(iii)u0≤0.(iv)γ>0.(v)p0SPSD(pointwise).
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
=
>∀=−
+∞×Ω=−+∂∂
⎪⎩
⎪⎨
⎧
+∞×Ω∂=
+∞×Ω−=∇+∇−∂∂
>≤
∫Ω−
−
).,())0(),0((
,0,1)1(
),(0, in )(
),(0, on 0
),(0, in2] cof[
: thatso 0)( e), (point wis )(,0)( Find
00
2
pp
x
0Dpp
pI.
SPSDp
uu
tde
ut
u
eutu
tttu
u
u
γ
λε
λ
4.TIME-DISCRETIZATIONBYOPERATOR-SPLITTINGOF(MABG.IVP)Withτ(>0)atime-discretizationstep(fixedforsimplicity)weapproximate(MABG.IVP)by:(0)Forn≥0,(un,pn)→un+1/3→(un+2/3,pn+1)→un+1asfollows:FirstStep:Solvethefollowing(well-posed)linearellipticproblem(1)SecondStep:(2)1(2)2
).,(),( 0000 pp uu =
[ ]⎩⎨⎧
Ω∂=
Ω=∇+∇−+
++
. on 0 , in cof
3/1
3/13/1
n
nnnn
uuuu pI. ετ
[ ]
⎪⎩
⎪⎨⎧
=−=∈⇔=−
−=−
Ω∈−+=
∫ ∫Ω Ω
−+−
−+++
+−−+
+
+
+
}).1)(,measurable {()1(
,2
, a.e.,)()1()()(
3/2
13/13/2
3/121
3/2
3/2
CdeSuCde
euu
ueeP
Cnu
unnn
nnn
n
n
xx
xxDxpxp
ϕ
γτγτ
ϕ
τλ
ThirdStep:(3)un+1=inf(0,un+2/3).Above:♦Problem(1)isaveryclassicallinearellipticproblem.Ithasthefollowingproperty:
♦P+isanoperator,mappingthespaceofthe2×2realsymmetricmatricesontotheclosedconvexconeoftherealSPSD2×2matrices(ifqisa2×2realsymmetricmatrixwitheigenvaluesµ1andµ2,∃S∈O(2)s.t.then,operatorP+isdefinedby
.00 3/1 ≤⇒≤ +nn uu
;0
0 1
2
1 −⎟⎟⎠
⎞⎜⎜⎝
⎛= SSq
µ
µ
).),0max(0
0),0max()( 1
2
1 −+ ⎟⎟
⎠
⎞⎜⎜⎝
⎛= SSq
µ
µP
♦Weconsidersystem(2)2asanoptimalitysystemassociatedwiththefollowingminimizationproblem
(MIN)Itfollowsfrom(MIN)thatun+2/3istheL2-projectionofun+1/3onto SC.Unlessun+1/3∈ SCproblem(MINP)mayhavenosolutionsince SCisnotweaklyclosedinL2(Ω).Fortunatelyitsdiscreteanalogueshavesolutions.
♦ Algorithm(0)–(3)hasclearlytheflavorofaninversepowermethod(withtruncation).
♦Step3hasbeenincludedtobeonthesafeside.Numericalexperimentssuggestthatifalgorithm(0)–(3)isproperlyinitialized,(3)isuseless.
. ⎥⎦⎤
⎢⎣
⎡ −= ∫∫ Ω
+
Ω∈+ xx dudu n
Sn
C vvv
3/123/2
21minarg
Twoimportantremarksareinorder:Remark4:Algorithm(0)–(3)‘enjoys’asplittingerrorforcingustouseasmalltime–discretizationstepτ.Remark5:Thereisnoneedtocomputeλn+1ateachtimestep.Indeed,
i.e.,λn+1isobtainedbytheratiooftwosmallnumbers.Itissafer(?)toproceedasfollows:
Denoteby(uτ,pτ)thelimitof(un+1/3,pn)n.Itmakessensetoapproximatethe(nonlinear)eigenvalueλbya(kindof)generalizedRayleighquotient.
,)(2
)() since(2
3/23/113/21
3/13/23/2
Ω+
−=∈⇒−=
− ∫Ω++
++−+++
+
C
duuSueuu
nnn
Cnun
nnn
τλλ
τ
x
∫∫
Ω
−Ω
∇∇+−=
x
x.pI
deu
duuuτ
τ
τττ
τ
ελ
2
)cof(
5.FINITEELEMENTAPPROXIMATIONOF(MABG)AssumingthatΩisaboundedconvexpolygonaldomainofR2(orhasbeenapproximatedbyafamilyofsuchdomains)weintroduceafamily(Th)hoftriangulationsofΩlikethoseinFigure2(his,typically,thelengthofthelargestedge(s)ofTh).Next,weapproximatethefunctionalspacesH1(Ω)andH0
1(Ω)byand
respectively,P1beingthespaceofthepolynomialfunctionsoftwovariablesofdegree≤1.Figure2Sometriangulations
},,),({ 10
hTh TPCV T∈∀∈Ω∈= ϕϕ
)),(}(0,{ 100 Ω∩==∈= Ω∂ HVVV hhh ϕϕ
LetusdenotebyΣh(resp.,Σ0h)thesetoftheverticesofTh(resp.,thesetΣh\Σh∩∂Ω).WehavethendimVh=CardΣh(:=Nh)anddimV0h=CardΣ0h(:=N0h).WeassumethattheverticesofThhavebeennumberedsothatFork=1,…,N0h, we define ωkastheunionofthosetrianglesofThwhichhaveQkasacommonvertex.Wedenoteby|ωk|themeasure(area)ofωk.ApproximatingtheMonge-Ampèrepartofthesplittingscheme(0)–(3)isa(boring)andtimeconsumingrepetitionofRG-HL-TL&JQ,J.Scient.Comp.,2019.Focusingontheeigenvaluepart,weapproximateSCbySChdefined(trapezoidalrule)by:
∑ ==h
Nkk
hQ0 1 .}{ 0
}.3)1(,{ )(100 CeVS kh QN
k khCh =−∈= −
=∑ ϕωϕ
ThediscreteanalogueoftheL2–projectionontoSCisdonebyaSQPmethodveryeasytoimplement,showing(asexpected)fastconvergenceproperties.
Remark6:Wecantakeadvantageofthefactthatτissmallbyreplacing(MIN)by(MIN.LIN)obtainedbylinearization(MIN.LIN)where
Theclosedformsolutionof(MJN.LIN)isgivenby
, ⎥⎦⎤
⎢⎣
⎡ −= ∫∫ Ω
+
Ω∈
++ xx dudu n
DSn
nC
vvv3/123/2
21minarg 3/1
}.)1(|{ 3/13/1 3/1
∫Ω+−+ Ω+=−+=
+
CdueDS nunC
n
xvv
Theabovefunctioncoincideswiththe1stiterateoftheSQPmethodinitializedbyun+1/3.
3/1
3/1
3/1
23/13/2
)( +
+
+
−
Ω
−Ω
−
++
∫∫ Ω+−
+=n
n
n
uu
unn e
de
Cdeuu
x
x
6.NUMERICALRESULTS6.1.TESTPROBLEMFORTHEUNITDISK
Figure3:MABGproblemontheunitdisk:Exactsolutionbifurcationdiagram
Figure4:Bifurcationdiagramcomparisonfortheunitdisk
ThediscrepancyshownonFigure4forsmallvaluesofCistheresultofanerroneousinitialization.Thismistakehasbeencorrected.Thereasonweexhibitthesepartiallywrongresultsistoshowtherobustnessofourmethodology.Indeed,withouthumanintervention,ourmethod‘returnsquicklybyitself’onthecorrectapproximatebifurcationdiagramsasCincreases.
Figure5.Resultsattheexactturningpoint
(C=10.5)
_________________________________________________________________________________________________________________________________________
λh–λ0=quasi-textbookO(h)
6.2.TESTPROBLEMSFORREGULARIZEDSQUARESq=4FIGURE6
« il faut essayer, vas-y »(1)
CEDRICVILLANIDEAd’ANALYSENUMERIQUE(UniversitéP.&M.Curie)
FieldsMedal2010
MERCIPOURVOTREATTENTION_______________________________________________________(1)“Onehastotry,let’sgo”
