Numerical PDE-Constrained Optimization

129 Pages·2015·1.741 MB·English

by  Juan Carlos De los Reyes (auth.)

SpringerBriefs in Optimization
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Ja´nosD.Pinte´r
StephenRobinson
Tama´sTerlaky
MyT.Thai
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Juan Carlos De los Reyes
Numerical PDE-Constrained
Optimization
JuanCarlosDelosReyes
CentrodeModelizacio´nMatema´tica(MODEMAT)
andDepartmentofMathematics
EscuelaPolite´cnicaNacionalQuito
Quito
Ecuador
ISSN2190-8354 ISSN2191-575X(electronic)
SpringerBriefsinOptimization
ISBN978-3-319-13394-2 ISBN978-3-319-13395-9(eBook)
DOI10.1007/978-3-319-13395-9
LibraryofCongressControlNumber:2014956766
Mathematics SubjectClassification (2010):49K20,49K21,49J20,49J21,49M05,49M15,49M37,65K10,
65K15,35J25,35J60,35J86.
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© TheAuthor(s)2015
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ToMar´ıaSoledadandEsteban
Preface
Inrecentyears,thefieldofpartialdifferentialequation(PDE)-constrainedoptimization
hasreceivedasignificantimpulsewithlargeresearchprojectsbeingfundedbydifferent
national and international agencies. A key ingredient for this success is related to the
wideapplicabilitythatthedevelopedresultshave(e.g.,incrystalgrowth,fluidflow,or
heat phenomena). In return, application problems gave rise to further deep theoretical
and numerical developments. In particular, the numerical treatment of such problems
hasmotivatedthedesignofefficientcomputationalmethodsinordertoobtainoptimal
solutionsinamanageableamountoftime.
AlthoughsomebooksonoptimalcontrolofPDEshavebeeneditedinthepastyears,
they are mainly concentrated on theoretical aspects or on research-oriented results. At
themoment,thereisalackofstudentaccessibletextsdescribingthederivationofopti-
malityconditionsandtheapplicationofnumericaloptimizationtechniquesforthesolu-
tionofPDE-constrainedoptimizationproblems.Thistextisdevotedtofillthatgap.
Bypresentingnumericaloptimizationmethods,theirapplicationtoPDE-constrained
problems, the resulting algorithms and the corresponding MATLAB codes, we aim to
contribute to make the field of numerical PDE-constrained optimization accessible to
advancedundergraduatestudents,graduatestudents,andpractitioners.
Moreover,recentresultsintheemergingfieldofnonsmoothnumericalPDE-constrai-
nedoptimizationarealsopresented.Uptotheauthor’sknowledge,suchresultsarenot
partofanymonographyet.Weprovideanoverviewonthederivationofoptimalitycon-
ditionsandonsomesolutionalgorithmsforproblemsinvolvingboundconstraints,state
constraints,sparsityenhancingcostfunctionals,andvariationalinequalityconstraints.
After an introduction and some preliminaries on the theory and approximation of
partial differential equations, the theory of PDE-constrained optimization is presented.
vii
viii
Existenceofoptimalsolutionsandoptimalityconditionsareaddressed.Weuseageneral
frameworkthatallowstotreatbothlinearandnonlinearproblems.Firstorderoptimality
conditions are presented by means of both a reduced approach and a Lagrange multi-
plier methodology. The derivation is also illustrated with several examples, including
linearandnonlinearones.Alsosufficientsecond-orderconditionsaredevelopedandthe
applicationtosemilinearproblemsisexplained.
The next part of the book is devoted to numerical optimization methods. Classical
methods (descent, Newton, quasi-Newton, sequential quadratic programming (SQP))
arepresentedinageneral Hilbert-spaceframeworkandtheirapplicationtothespecial
structureofPDE-constrainedoptimizationproblemsexplained.Convergenceresultsare
presentedexplicitlyforthePDE-constrainedoptimizationstructure.Thealgorithmsare
carefullydescribedandMATLABcodes,forrepresentativeproblems,areincluded.
Thebox-constrainedcaseisaddressedthereafter.Thischapterfocusesonboundcon-
straints on the design (or control) variables. First- and second-order optimality condi-
tions are derived for this special class of problems and solution techniques are stud-
ied. Projection methods are explained on basis of the general optimization algorithms
developedinChap.4.Inaddition,thenonsmoothframeworkofprimal-dualandsemis-
mooth Newton methods is introduced and developed. Convergence proofs, algorithms,
andMATLABcodesareincluded.
In the last chapter, some representative nonsmooth PDE-constrained optimization
problems are addressed. Problems with cost functionals involving the L1-norm, with
state constraints, or with variational inequality constraints are considered. Numerical
strategiesforthesolutionofsuchproblemsarepresentedtogetherwiththecorrespond-
ingMATLABcodes.
This book is based on lectures given at the Humboldt-University of Berlin, at the
UniversityofHamburg,andatthefirstEscueladeControlyOptimizacio´n(ECOPT),a
summer school organized together by the Research Center on Mathematical Modeling
(MODEMAT) at EPN Quito and the Research Group on Analysis and Mathematical
ModelingValpara´ıso(AM2V)atUSMChile.
Contents
1 Introduction...................................................... 1
1.1 IntroductoryExamples ......................................... 1
1.1.1 OptimalHeating........................................ 1
1.1.2 OptimalFlowControl ................................... 2
1.1.3 ALeastSquaresParameterEstimationProbleminMeteorology 3
1.2 AClassofFinite-DimensionalOptimizationProblems .............. 4
2 BasicTheoryofPartialDifferentialEquationsandTheirDiscretization . 9
2.1 NotationandLebesgueSpaces .................................. 9
2.2 WeakDerivativesandSobolevSpaces ............................ 10
2.3 EllipticProblems.............................................. 13
2.3.1 PoissonEquation ....................................... 13
2.3.2 AGeneralLinearEllipticProblem......................... 15
2.3.3 NonlinearEquationsofMonotoneType .................... 16
2.4 DiscretizationbyFiniteDifferences .............................. 20
3 TheoryofPDE-ConstrainedOptimization ........................... 25
3.1 ProblemStatementandExistenceofSolutions ..................... 25
3.2 FirstOrderNecessaryConditions ................................ 27
3.2.1 DifferentiabilityinBanachSpaces......................... 27
3.2.2 OptimalityCondition.................................... 30
3.3 LagrangianApproach .......................................... 33
3.4 SecondOrderSufficientOptimalityConditions..................... 38
ix
x Contents
4 NumericalOptimizationMethods................................... 43
4.1 DescentMethods ............................................. 44
4.2 Newton’sMethod ............................................. 53
4.3 Quasi-NewtonMethods ........................................ 60
4.4 SequentialQuadraticProgramming(SQP)......................... 65
5 Box-ConstrainedProblems......................................... 69
5.1 ProblemStatementandExistenceofSolutions ..................... 69
5.2 OptimalityConditions ......................................... 70
5.3 ProjectionMethods............................................ 75
5.4 PrimalDualActiveSetAlgorithm(PDAS) ........................ 80
5.5 SemismoothNewtonMethods(SSN) ............................. 86
6 NonsmoothPDE-ConstrainedOptimization.......................... 91
6.1 SparseL1-Optimization ........................................ 91
6.2 PointwiseStateConstraints ..................................... 96
6.3 VariationalInequalityConstraints ................................ 101
6.3.1 InequalitiesoftheFirstKind.............................. 103
6.3.2 InequalitiesoftheSecondKind ........................... 110
References............................................................ 119
Index ................................................................ 123
Chapter 1
Introduction
1.1 IntroductoryExamples
1.1.1 OptimalHeating
Let Ω be a bounded three-dimensional domain with boundary Γ, which represents a
body that has to be heated. We may act along the boundary by setting a temperature
u=u(x)and,inthatmanner,change thetemperaturedistributioninsidethebody. The
goaloftheproblemconsistsingettingascloseaspossibletoagivendesiredtemperature
distributionz (x)inΩ .
d
Mathematically,theproblemmaybewrittenasfollows:
(cid:2) (cid:2)
1 α
min J(y,u)= (y(x)−z (x))2dx+ u(x)2ds,
d
2 Ω 2 Γ
subjectto:
⎫
−Δ y=0 inΩ ,⎬
∂y =ρ(u−y) inΓ,⎭Stateequation
∂n
u ≤u(x)≤u , Controlconstraints
a b
where u ,u ∈ R such that u ≤ u . The control constraints are imposed if there is
a b a b
a technological limitation on the maximum or minimum value of the temperature to
be controlled. The scalar α>0 can be interpreted as a control cost, which, as a by-
product,leadstomoreregularsolutionsoftheoptimizationproblem.Thefunctionρ(x)
©TheAuthor(s)2015 1
J.C.DelosReyes,NumericalPDE-ConstrainedOptimization,
SpringerBriefsinOptimization, DOI10.1007/978-3-319-13395-9 1