Application of Optimal Transport to Evolutionary PDEs 1 - Gradient flows in linear spaces and their

30 Pages·2010·0.33 MB·English

Introduction Mainestimates Infinitedimension Variationalapproximation
Application of Optimal Transport to
Evolutionary PDEs
1 - Gradient flows in linear spaces and their variational
approximation
Giuseppe Savar´e
http://www.imati.cnr.it/∼savare
Department of Mathematics, University of Pavia, Italy
2010CNASummerSchool
NewVistasinImageProcessingandPDEs
CarnegieMellonCenterforNonlinearAnalysis,Pittsburgh,June 7–12, 2010
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Introduction Mainestimates Infinitedimension Variationalapproximation
Outline
1 An informal introduction to gradient flows
2 The simplest setting and the main estimates
3 Infinite dimensional spaces
4 The Minimizing Movement Scheme
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Introduction Mainestimates Infinitedimension Variationalapproximation
The basic ingredients of gradient flows
(cid:73) A functionalΦ:X→RfunctiondefinedinsomeambientspaceX (initially
X:=Rd forsimplicity).
(cid:73) (metric) Velocity: somenorm(cid:107)·(cid:107)tomeasurethevelocity(andthe
length/energy)ofthecurvesw:t∈(a,b)(cid:55)→wt∈X.
velocity (cid:107)w˙t(cid:107),
Z b Z b
length L[w]:= (cid:107)w˙t(cid:107)dt, energy E[w]:= (cid:107)w˙t(cid:107)2dt
a a
Typically(cid:107)·(cid:107)istheeuclideannorm,butitcouldbeageneraloneandit
could also depend on the point(Riemannian/Finslerstructure).
Itisstrictlyrelatedtoadistancebytheformulae
n o
distance d(w0,w1):=inf L[w]:w(a)=w0, w(b)=w1
metric velocity |w˙t|:= lim d(wt,wt+h) =(cid:107)w˙t(cid:107).
h→0 |h|
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Introduction Mainestimates Infinitedimension Variationalapproximation
Heuristics: drection of maximal dissipation rate
LetDΦ∈X∗ denotesthedifferentialofΦ.
Dissipation along a curve and chain rule: ifw:t∈(a,b)(cid:55)→wt∈X isa
smoothcurvewithtimederivativew˙t:= ddtwt then
d
Dissipation rate of Φ along w:=− Φ(wt)=−(cid:104)DΦ(wt),w˙t(cid:105).
dt
Basic rule: choosethedirectionofmaximal dissipation rate with respect
to the given velocityamongallthecurvestroughapointw:
Slope |∂Φ|(w):=supn−ddtΦ(wt) :wt=w, w˙t(cid:54)=0o
(cid:107)w˙t(cid:107)
Bythechainrule,theslopeofΦisthedualnormofitsdifferential:
Φ(w)−Φ(z)
Slope=|∂Φ|(w)=(cid:107)−DΦ(w)(cid:107)∗=limsup .
z→w d(w,z)
Adirectionv=u˙t isof maximal slopeifitrealizesthe“sup“,i.e.
−(cid:104)DΦ(u),v(cid:105)=(cid:107)v(cid:107)·(cid:107)−DΦ(u)(cid:107)∗.
ByintroducingthedualitymapJ :=D`1(cid:107)·(cid:107)2´
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v hasthesamedirectionofJ−1(−DΦ(u)).
When(cid:107)·(cid:107)iseuclideanJ islinearandJ−1DΦ=∇Φ,
v hasthesamedirectionof −∇Φ(u).
InthiscaseweusuallyidentifyX withitsdual,andDΦwiththegradient∇Φ.
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Introduction Mainestimates Infinitedimension Variationalapproximation
The choice of the speed
(cid:7) (cid:4)
(cid:6)The velocity v=u˙ has the same direction of J−1(−DΦ(u)).(cid:5)
Toprovideacompletedescriptionthe speed(thenormofv)hastobeprescribed.
Ingeneral,onecanintroducean
increasing homeomorphism β:[0,+∞)→[0,+∞)
andaskfor
` ´
β (cid:107)v(cid:107) =slope=|∂Φ|(u)=(cid:107)DΦ(u)(cid:107)∗.
Simplest(andtypical)choice: β(r)=r,velocity=slope,(cid:107)v(cid:107)=(cid:107)−DΦ|(u)(cid:107)∗.
Moregenerally
Z r Z r
ψ(r):= β(s)ds, ψ∗(r):= β−1(s)ds,
0 0
ψ∗ isthe(dual,conjugate)Legendretransformofψ,
β`(cid:107)v(cid:107)´=(cid:107)−DΦ|(u)(cid:107)∗ ⇔ (cid:107)v(cid:107)(cid:107)−DΦ(u)(cid:107)∗=ψ`(cid:107)v(cid:107)´+ψ∗((cid:107)−DΦ(u)(cid:107)∗).
Thecompleteconditionreads
DΨ(u˙)=−DΦ(u)
where
Ψ(v):=ψ((cid:107)v(cid:107)) isthe“dissipationpotential”
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Introduction Mainestimates Infinitedimension Variationalapproximation
Doubly nonlinear evolution equation
(cid:7) (cid:4)
(cid:6)DΨ(u˙)=−DΦ(u)(cid:5)
(cid:73) FunctionalΦ.
(cid:73) Velocityofacurve(cid:107)vt(cid:107)=(cid:107)u˙t(cid:107).
(cid:73) Slope(cid:107)−DΦ(ut)(cid:107)∗=“maximaldissipationrate”=supn−dd(cid:107)twΦ˙(tw(cid:107)t)o
(cid:73) Speedfunctionβ,itsprimitiveψ,thedissipationpotentialΨ(v):=ψ((cid:107)v(cid:107))
Problem
Find a curve u starting from u0 whose direction at each time realizes the
maximal dissipation rate of Φ and whose speed is linked to the slope by the
equation β((cid:107)u˙t(cid:107))=(cid:107)−DΦ(ut)(cid:107)∗.
Alongsuchacurve
d
− Φ(ut)=(cid:107)u˙t(cid:107)(cid:107)−DΦ(ut)(cid:107)∗=ψ((cid:107)u˙t(cid:107))+ψ∗((cid:107)−DΦ(ut)(cid:107)∗)
dt
=Ψ(ut)+Ψ∗(−DΦ(ut)).
Alonganycuvew:
d
− Φ(wt)≤(cid:107)w˙t(cid:107)(cid:107)−DΦ(wt)(cid:107)∗≤ψ((cid:107)w˙t(cid:107))+ψ∗((cid:107)−DΦ(wt)(cid:107)∗)
dt
=Ψ(wt)+Ψ∗(−DΦ(wt)).
De Giorgi characterizationofcurvesofmaximalslope.
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Introduction Mainestimates Infinitedimension Variationalapproximation
The “simplest” case: gradient flows
Normvelocity(cid:107)·(cid:107)(cid:32)|·|iseuclideanlike,J isalinearisometry(cid:107)·(cid:107)∗(cid:32)|·|,
∇Φ=J−1(DΦ),β(r)=r,ψ(r)=ψ∗(r)= 1r2.
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MAIN PROBLEM: find u:[0,+∞)→X such that
d
dtut=−∇Φ(ut) t∈[0,+∞); u|t=0=u0 (GF)
(cid:73) Starting level: X≈Rd,finitedimensionaleuclideanspace
ΦisofclassC2,D2Φ≥λI.
(cid:73) Slight variants: Φ(u)(cid:32)Φt(u):=Φ(u)−(cid:104)ft,u(cid:105),timedependentforcing
term
X≈Md,smoothRiemannianmanifold.
(cid:73) Applications to PDE’s: X:=Hilbert(typicallyL2-like),
Φ:X→(−∞,+∞]λ-convexandjustlower-semicontinuous
∇Φ(cid:32)∂Φ,multivaluedsubdifferentialofΦ,differentialinclusions.
(cid:73) Relax λ-convexity assumption
(cid:73) Further step: fromlineartometricstructures...
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Introduction Mainestimates Infinitedimension Variationalapproximation
Main Estimate I: energy identity
(cid:11) (cid:8)
d
ut=−∇Φ(ut) (GF)
(cid:10)dt (cid:9)
EvaluatingthedissipationrateofΦ
d
− Φ(ut)=|u˙t|2=|∇Φ(ut)|2 (I)
dt
Integratingintime
Z t
Φ(ut)+ |u˙s|2ds=Φ(u0)
0
1 1
“DeGiorgisplitting”: |u˙|2= |u˙|2+ |∇Φ(u)|2 (recallψ(r)=ψ∗(r)= 1r2)
2 2 2
Curves of maximal slope
Z t“1 1 ”
Φ(u0)−Φ(ut)= |u˙s|2+ |∇Φ(us)|2 ds (I)
0 2 2
Alonganyothercurvew
Z t Z t“1 1 ”
Φ(w0)−Φ(wt)= −∇Φ(ws)·w˙sds≤ |w˙s|2+ |∇Φ(ws)|2 ds
0 0 2 2
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Introduction Mainestimates Infinitedimension Variationalapproximation
Energy identity and variational characterization of (GF)
Theorem
If a Lipschitz curve u:[0,+∞)→X satisfies the differential inequality
d 1 1
Φ(ut)≤− |u˙t|2− |∇Φ(ut)|2 (1)
dt 2 2
even in the weaker integrated form
Z t“1 1 ”
Φ(ut)+ |u˙s|2+ |∇Φ(us)|2 ds≤Φ(u0) (2)
0 2 2
then u is a solution of the Gradient Flow
d
ut=−∇Φ(ut). (GF)
dt
Proof: Chainrule:
Z t
˙ ¸
Φ(ut)+ −∇Φ(us),u˙s ds=Φ(u0) (3)
0
Subtracting(3)to(2)weget
Z0t“12|u˙s|2+ 21|∇Φ(us)|2−˙−∇Φ(us),u˙s¸”ds= 12Z0t˛˛˛u˙s+∇Φ(us)˛˛˛2ds≤0
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Introduction Mainestimates Infinitedimension Variationalapproximation
Exploting convexity
Convexity inequality: wθ =(1−θ)w0+θw1,
Φ(wθ)≤(1−θ)Φ(w0)+θΦ(w1) foreveryw0,w1∈X, θ∈[0,1]
Hessian inequality: D2Φ≥0
Subgradient property:
(cid:104)∇Φ(u),v−u(cid:105)≤Φ(v)−Φ(u)
Φ(v)
Φ(v)−Φ(u)
Φ(u) (cid:104)∇Φ(u),v−u(cid:105)
u v
Gradient monotonicity:
˙ ¸
∇Φ(u)−∇Φ(v),u−v ≥0
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