Semi-Lagrangian approximation schemes for linear and Hamilton-Jacobi equations

325 Pages·2014·9.88 MB·English

by  Maurizio Falcone

Semi-Lagrangian
Approximation Schemes
for Linear and
Hamilton–Jacobi Equations
OT133_Falcone-Ferretti_FM.indd 1 10/21/2013 10:53:31 AM
Semi-Lagrangian
Approximation Schemes
for Linear and
Hamilton–Jacobi Equations
Maurizio Falcone
Sapienza – Università di Roma
Rome, Italy
Roberto Ferretti
Università di Roma Tre
Rome, Italy
Society for Industrial and Applied Mathematics
Philadelphia
OT133_Falcone-Ferretti_FM.indd 3 10/21/2013 10:53:31 AM
Copyright © 2014 by the Society for Industrial and Applied Mathematics
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Figures 3.2, 6.1, 6.4-6, 8.4-7, 9.9-11 reprinted with kind permission of Springer Science + Business
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Figures 6.2-3, 8.10-15, and right image of 8.16 reprinted with permission of World Scientific
Publishing.
Figure 7.7 reprinted with permission from Global Science Press.
Figures 7.8, 7.11, 8.2, 8.3, 9.4, 9.8 reprinted with permission from Elsevier.
Figure 7.10 reprinted with permission from John Wiley and Sons.
Figures 9.6 and 9.7 reprinted with permission from European Mathematical Society.
Left image of Figure 8.16 reprinted with permission from Antony Willits Merz.
Library of Congress Cataloging-in-Publication Data
Falcone, Maurizio, author.
Semi-Lagrangian approximation schemes for linear and Hamilton–Jacobi equations / Maurizio
Falcone, Sapienza – Università di Roma, Rome, Italy, Roberto Ferretti, Università di Roma Tre,
Rome, Italy.
pages cm. -- (Applied mathematics)
Includes bibliographical references and index.
ISBN 978-1-611973-04-4
1. Hamilton–Jacobi equations. 2. Approximation theory. 3. Differential equations,
Partial–Numerical solutions. I. Ferretti, Roberto, author. II. Title.
QA377.F35 2013
515’.353--dc23
2013033334
is a registered trademark.
OT133_Falcone-Ferretti_FM.indd 4 10/21/2013 10:53:31 AM
To our daughters
Cecilia and Flavia,
who will never read this book.
1
OT133_Falcone-Ferretti_FM.indd 5 10/21/2013 10:53:31 AM
Contents
Preface ix
Notation xi
1 Modelsandmotivations 1
1.1 Linearadvectionequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 NonlinearevolutiveproblemsofHJtype . . . . . . . . . . . . . . . . . . 3
1.3 Nonlinearstationaryproblems. . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Simpleexamplesofapproximationschemes . . . . . . . . . . . . . . . . 10
1.5 Somedifficultiesarisingintheanalysisandintheapproximation . . 12
1.6 Howtousethisbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Viscositysolutionsoffirst-orderPDEs 15
2.1 Thedefinitionofviscositysolution . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Viscositysolutionforevolutiveequations. . . . . . . . . . . . . . . . . . 24
2.3 Problemsinboundeddomains . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Viscositysolutionsandentropysolutions . . . . . . . . . . . . . . . . . . 35
2.5 Discontinuousviscositysolutions . . . . . . . . . . . . . . . . . . . . . . . 37
2.6 Commentedreferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Elementarybuildingblocks 41
3.1 AreviewofODEapproximationschemes . . . . . . . . . . . . . . . . . 41
3.2 Reconstructiontechniquesinoneandmultiplespacedimensions . . 45
3.3 Functionminimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 NumericalcomputationoftheLegendretransform . . . . . . . . . . . 71
3.5 Commentedreferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4 Convergencetheory 75
4.1 Thegeneralsetting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Convergenceresultsforlinearproblems: TheLax–Richtmeyerthe-
orem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Moreonstability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 ConvergenceresultsforHJequations . . . . . . . . . . . . . . . . . . . . 88
4.5 Numericaldiffusionanddispersion . . . . . . . . . . . . . . . . . . . . . . 98
4.6 Commentedreferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5 First-orderapproximation schemes 101
5.1 Treatingtheadvectionequation . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 TreatingtheconvexHJequation . . . . . . . . . . . . . . . . . . . . . . . 126
vii
viii Contents
5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4 Stationaryproblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.5 Commentedreferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6 High-orderSLapproximation schemes 147
6.1 SLschemesfortheadvectionequation . . . . . . . . . . . . . . . . . . . . 147
6.2 SLschemesfortheconvexHJequation . . . . . . . . . . . . . . . . . . . 172
6.3 Commentedreferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7 FluidDynamics 189
7.1 TheincompressibleEulerequationin(cid:2)2 . . . . . . . . . . . . . . . . . . 189
7.2 TheShallowWaterEquation . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.3 Someadditionaltechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.5 Commentedreferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8 Controlandgames 207
8.1 Optimalcontrolproblems: Firstexamples . . . . . . . . . . . . . . . . . 208
8.2 DynamicProgrammingforotherclassicalcontrolproblems. . . . . . 213
8.3 ThelinkbetweenBellmanequationandPontryaginMaximumPrin-
ciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
8.4 SLschemesforoptimalcontrolproblems . . . . . . . . . . . . . . . . . . 219
8.5 DynamicProgrammingfordifferentialgames . . . . . . . . . . . . . . . 246
8.6 Commentedreferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
9 Frontpropagation 269
9.1 Frontpropagationviathelevelsetmethod . . . . . . . . . . . . . . . . . 269
9.2 FrontsevolvingbyMeanCurvature . . . . . . . . . . . . . . . . . . . . . 276
9.3 Applicationstoimageprocessing . . . . . . . . . . . . . . . . . . . . . . . 279
9.4 FastMarchingMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
9.5 Commentedreferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Bibliography 299
Index 317
Preface
ThisbookisforpeopleinterestedinthenumericalsolutionofPDEs,withanempha-
sisonhigh-orderschemesforhyperbolicproblems. Infact,evolutivelinearandnonlinear
first-orderPDEsappearinavarietyofapplicationsrangingfromgeophysicalandatmo-
spheric problems to control theory, Fluid Dynamics, and image processing. Although
everyapplicationhasitsownfeatures,ausualrequirementforthenumericalsolutionis
to keep accuracy and stability without overly restrictive assumptions on the time step,
whichusuallyimplyahighcomputationalcomplexity.
WewilldescribeandanalyzebasicandrecentdevelopmentsoftheSemi-Lagrangian
techniquefortheapproximation offirst-orderPDEswithaspecialfocusonHamilton–
Jacobiequations. Themainpurposeistoobtainmethodswhichareunconditionallysta-
blewithrespecttothechoiceofthetimestep. Thebasicideadatesbacktothemethod
proposedbyCourant,Isaacson,andReesin1952[CIR52]forfirst-ordersystemsoflin-
earequationsandhasgonethroughanumberofimprovementsandextensionssince. In
particular, the main developments have been proposed within the Numerical Weather
Predictioncommunity[SC91],wherethisclassofschemeshasbecomeofcommonuse.
Asfortheadvectionequation,thedrivingforceofthismethodisthemethodofchar-
acteristicswhichaccountsfortheflowofinformationinthemodelequation. Atthenu-
mericallevel, theSemi-Lagrangian approximation mimicsthemethod ofcharacteristics
lookingforthefootofthecharacteristiccurvepassingthrougheverynodeandfollowing
thiscurveforasingletimestep.
In order to derive a numerical method from this general idea, several ingredients
should be put together, mainly a technique for ODEs to track characteristics and a re-
construction technique to recover pointwise values of the numerical solution. In this
framework,theorderofspaceandtimediscretizationsistypicallyreversedwithrespectto
moreconventionalschemes—thefirstdiscretizationisdoneintime,thesecondinspace—
althoughthefinalresultisclearlyafullydiscretescheme. Thisreversalisthekeytogetting
weakerstabilityrequirements.
Thisbookhastwomainpurposes. ThefirstistopresentSemi-Lagrangianapproxima-
tionschemes,reviewingtheirconstructionandtheoryonmodelequationswhichrange
fromthesimpleadvectionequationtoHamilton–Jacobiequations. Inthebasicsituation
offirst-orderschemes,wewillalsomakeacomparisonwithmorestandardapproximation
techniques, namelyfinitedifferenceschemesofupwindandcentered types. Thereader
willfindinformationontheconstructionoftheschemestogetherwithadetailedtheoret-
icalanalysis,atleastforthemorepopularchoicesoftheirelementarybuildingblocks. We
haveincludedanumberofpracticalalgorithmicrecipes,andwebelievethatafterreading
thisbook it should bepossible toprogram and apply themethods described herefrom
scratch.
The second goal of the book is to present some applications and show the Semi-
Lagrangian approximations at work on a variety of nonlinear problems of applicative
ix
x Preface
interest. Wealsoaimatshowingthatthistechniqueiswellsuitedfortheapproximation
ofproblemswithnonsmoothsolutions,likeweaksolutionsintheviscositysenseinthe
caseofHamilton–Jacobiequations.
The book is intended to be accessible to a reader with a basic knowledge of linear
PDEsandstandardnumericalmethods. TheanalysisofHamilton–Jacobiequationswill,
ofcourse,requirespecificanalyticaltools,whichhavebeenincludedinthesectiondevoted
toviscositysolutions. Thepresentationisconceivedtobereasonablyself-contained,but
atthesametimeitwillcollectinaunifiedframeworkresultswhicharespreadoverthelit-
erature. Clearly,moreinformationonthetheoryofviscositysolutionscanbeobtained
looking at thespecific references, and in particular at thebooks [Ba98, BCD97, FS93].
Some standard finite difference schemes for Hamilton–Jacobi equations can be found
in [Se99, OF03], whereas other schemes based on control techniques are presented in
[KuD01].
AlthoughHamilton–Jacobi equationsandtheirapplicationsaretheobject ofanex-
tensiveliterature,wewillmostlylimitourpresentationtotheSemi-Lagrangianapproach
and toitsrelationship withDynamic Programming. A detailedoverview of numerical
methodsforHamilton–Jacobi equationsgoesbeyond thescopesofthisbook, although
wehopetodevelopthistopicingreaterdetailinthefuture.
Acknowledgments Wewishtothankallcolleaguesandstudentswhocontributedto
the development of the results reviewed in this book or took part in enlightening dis-
cussionsonalgorithmicandtheoreticalissues. WearealsoindebtedtoL.Bonaventura,
E.Carlini, J.-M.Qiu, M.Restelli, G.Rosatti, C.-W. Shu, andG.Tumolo forproviding
someofthenumericaltestspresentedhere. SpecialthanksgotoourAcquisitionsEditor,
ElizabethGreenspan,forherendlesspatienceduringthewritingofthemanuscript,and
toourfamiliesfortheirconstantsupport.
MaurizioFalconeandRobertoFerretti
Rome,May20th,2013
Notation
a.e. almosteverywhere(withrespecttotheLebesguemeasure)
(cid:2)d euclideand-dim(cid:2)ensionalspace
x·y scalarproduct d x y oftwovectorsx,y∈(cid:2)d
i=1 i i
|x| euclideannormofx∈(cid:2)d,|x|=(x·x)1/2
B(x ,r) openballofcenterx andradiusr in(cid:2)d,{x:|x−x |<r}
0 0 0
B(x ,r) closedballofcenterx andradiusr in(cid:2)d,{x:|x−x |≤r}
0 0 0
∂E boundaryofthesetE
intE interiorofthesetE
E closureofthesetE
coE convexhullofthesetE
a∨b min{a,b}fora,b∈(cid:2)
a∧b max{a,b}fora,b∈(cid:2)
Du(x) gradientofafunctionu:(cid:2)d →(cid:2)atthepointx
d(x,C) distancefromthepointxtotheclosedsetC
Δt,Δx discretizationsteps,⎧incompactformΔ=(Δx,Δt)
⎪⎨+1 ifx>0,
sgn(x) signofx:sgn(x):= 0 ifx=0,
⎪
⎩
−1 ifx<0.
i imaginaryunit (cid:7)
1 ifx∈Ω,
1Ω(x) characteristicfunctionofthe(cid:7)setΩ:1Ω(x):= 0 ifx(cid:7)∈Ω.
1 ifi=j,
δ Kronecker’ssymbol:δ :=
ij ij 0 ifi(cid:7)=j.
(cid:8)x(cid:9) lowerintegerpartofx:(cid:8)x(cid:9):=max{m∈(cid:3):m≤x}
I [G](x) valueatthepointxoftheinterpolateofdegreekofthefunctiong
k
basedonthevectorG=(g(x))ofgridvaluesof g
i
ν(x) exteriornormaltoasurfaceatthepointx
y(x,t;s) positionattimes ofthetrajectorystartingatxattimet
y (s;α) positionattimes ofthetrajectorystartingatxattimet
x,t
anddrivenbythecontrolα
y (s;α) positionattimes ofthetrajectorystartingatxattimet=0
x
anddrivenbythecontrolα (cid:8)
TV(w) totalvariationoftheunivariatefunctionw:TV(w)= |w(cid:10)(x)|dx
D
suppw supportofthefunctionw
ω(δ) amodulusofcontinuity
BUC(Ω) spaceofboundeduniformlycontinuousfunctionsontheopensetΩ
Ck(Ω) spaceoffunctionsu:Ω→(cid:2)withcontinuouskthderivative
Ck(Ω) spaceoffunctionswithcompactsuppo(cid:8)rtbelongingtoCk(Ω)
0
Lp(Ω) spaceoffunctionsu:Ω→(cid:2)suchthat |u|p(x)dx<+∞,1≤p<+∞
Ω
L∞(Ω) spaceoffunctionsu:Ω→(cid:2)suchthatesssup|u(x)|<+∞
xi
xii Notation
Lp (Ω) spaceoffunctionsu∈Lp(K)foranycompactsetK⊂Ω,1≤p≤+∞
loc
(cid:13)u(cid:13) normofuinLp(Ω),1≤p≤∞
p
Wm,p(Ω) Sobolevspaceoffunctionsu:Ω→(cid:2)withthefirstmderivativesinLp(Ω)
Wm,p(Ω) SobolevspaceoffunctionsuwiththefirstmderivativesinLp(K)
loc
foranycompactsetK⊂Ω
C([0,T];X) spaceoffunctionsu:[0,T]×(cid:2)→(cid:2)suchthat(cid:13)u(t,·)(cid:13) ∈C([0,T])
X
L∞([0,T];X) spaceoffunctionsu:[0,T]×(cid:2)→(cid:2)suchthat(cid:13)u(t,·)(cid:13) ∈L∞([0,T])
X
lp spaceof pthpowersummablevectorsorsequences
l∞ spaceofboundedvectorsorsequences
V,v numericalsolutionasavectorandasavalueatthenodex
j j
Vn,vn numericalsolutionasavectorandasavalueatthenode(x ,t )
j j n
V(k),v(k) numericalsolutionattheiterationkasavectorandasavalue
j
atthenodex
j
(cid:13)V(cid:13)p norminl(cid:9)p(cid:10),Δrexno·r·m·Δalxize(cid:2)dif|vref|eαr(cid:11)r1e/αdtoifaαn<um∞er,icalsolution:
(cid:13)V(cid:13) := 1 d j j
α max |v | ifα=∞.
j j
Bt transposeofamatrixB
tr(B) traceofamatrixB
ρ(B) spectralradiusofasquarematrixB∈(cid:2)m×m
(cid:13)B(cid:13) fora(possiblyinfinite)matrixB,naturaloperatornorminlp:
p
(cid:13)BV(cid:13)
(cid:13)B(cid:13) :=sup p
p V(cid:7)=0 (cid:13)V(cid:13)p (cid:2)
(cid:13)B(cid:13)∞ ∞-normofamatrixB:(cid:13)B(cid:13)∞=ma(cid:2)xi j|bij|
(cid:13)B(cid:13)1 1-normofamatrixB:(cid:13)B(cid:13)1=m(cid:12)axj i|bij|
(cid:13)B(cid:13) 2-normofamatrixB:(cid:13)B(cid:13) = ρ(BtB)
2 2
w,w l(cid:7)oweranduppersemicontinuousenvelopes:
w(x):=liminf w(y),
y→x
w(x):=limsup w(y).
y→x