Topics in applied analysis and optimisation, CIM-WIAS Workshop, TAAO 2017

406 Pages·2019·4.598 MB·English

by  Hintermuller M., Rodrigues J.F (ed.)

CIM Series in Mathematical Sciences
Michael Hintermüller
José Francisco Rodrigues  Editors
Topics in
Applied
Analysis and
Optimisation
Partial Differential Equations, Stochastic
and Numerical Analysis
CIM Series in Mathematical Sciences
SeriesEditors:
IreneFonseca
DepartmentofMathematicalSciences
CenterforNonlinearAnalysis
CarnegieMellonUniversity
Pittsburgh,PA, USA
José Francisco Rodrigues
CMAF&IO, Faculdade de Ciências
Universidade de Lisboa
Lisboa, Portugal
The CIM Series in Mathematical Sciences is published on behalf of and in
collaboration with the Centro Internacional de Matemática (CIM) in Portugal.
Proceedings,lecturecoursematerial fromsummerschools andresearch monographs
willbeincludedinthenewseries.
More information about this series at http://www.springer.com/series/11745
Joint CIM-WIAS Workshop, TAAO 2017,
Lisbon, Portugal, December 6-8, 2017
123
Editors
Michael Hintermüller José Francisco Rodrigues
Weierstrass Institute for Applied Analysis CMAF&IO, Faculdade de Ciências
and Stochastics Universidade de Lisboa
Berlin, Germany Lisboa, Portugal

ISSN 2364-950X ISSN 2364-9518 (electronic)
CIM Series in Mathematical Sciences
ISBN 978-3-030-33115-3 ISBN 978-3-030-33116-0 (eBook)
https://doi.org/10.1007/978-3-030-33116-0

Mathematics Subject Classification (2010): 35-06, 35Qxx, 35Rxx, 49-06, 49Nxx, 60-06, 65-06, 65Kxx,
6 5Mxx, 65Zxx

© Springer Nature Switzerland AG 2019

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Preface
ThisvolumeoftheSpringer-CIMSeriesisrelatedtoaworkshopon”TopicsinAp-
pliedAnalysisandOptimisation(Stochastic,PartialDifferentialEquationsandNu-
mericalAnalysis)”whichwasheldinLisbonfromDecember6–8,2017.Itbrought
together twenty four speakers in applied mathematics invited from both, the Por-
tuguese International Center for Mathematics (CIM) and the Weierstrass-Institute
for Applied Analysis and Stochastics (WIAS) Berlin, Germany. Both institutions
are members of ERCOM (European Research Centres on Mathematics). CIM is
a not-for-profit, privately-run association that aims at developing and promoting
research in Mathematics. At present CIM has 20 associates, including three Por-
tuguese Universities, thirteen Research Centres in the Mathematical Sciences, the
InstituteofTelecommunications,andthreenationalscientificsocieties,respectively
ofMathematics,ofStatisticsandofMechanics.WIASisamemberoftheLeibniz
Associationandconductsprojectorientedresearchinappliedmathematicswiththe
aimofsolvingcomplexproblemsintechnology,scienceandtheeconomyaswellas
biomedicine.
We also feel that a brief word on the birth of the idea of having the aforemen-
tioned workshop in Lisbon and on editing the present special volume within the
Springer-CIM Seriesis in order:In fact,during a Workshopon “Emerging Devel-
opmentsinInterfacesandFreeBoundaries”attheMathematicalResearchInstitute
in Oberwolfach, yet another member of the Leibniz Association in Germany and
ERCOM,thetwoeditorsofthisvolumecametogetherinordertodevelopaformat
to foster joint research between ERCOM members. As a result, a scientific event
was created with the aim to present and discuss current scientific interests among
the research groups of the Weierstrass Institute in Berlin and mathematics centres
in Portugal, in particular the Centro de Matemática, Aplicações Fundamentais e
InvestigaçãoOperacional(CMAFcIO),oftheUniversityofLisbon,andtheCentro
deMatemáticaoftheUniversityofCoimbra(CMUC).
This CIM -WIAS workshop finally brought together a selection of experts in
Europe, and launched and strengthened further scientific collaborations in applied
mathematics.Topicsofparticularinterestincludedpartialdifferentialequationswith
applications to material sciences, thermodynamics and laser dynamics, scientific
v
vi Preface
computing, nonlinear optimisation and stochastic analysis. In the outcome of the
workshop, this collective book gathers fourteen contributions from four specific
topicalareasthatwereaddressedinthemeeting.
ThethreesurveysonnonsmoothoptimisationstartwithachapterbyA.Alphonse,
M. Hintermüller and C. N. Rautenberg on elliptic stationary quasi-variational in-
equalities(QVIs),whichisaclassofquasi-equilibriumnon-convexandnon-smooth
problemsthathasrecentlyreceivedincreasinginterest.Latestprogressinthatfield
includes the development of numerical solutions algorithms. In particular for the
so-called gradient-constrained case, a Moreau-Yosida technique combined with a
variablesplittingapproachcanbeaddressedefficientlybyalternatingminimization
schemes. The latter are the subject of the contribution by M. A. T. Figueiredo on
the Alternating Direction Method of Multipliers (ADMM). The scope of that sec-
tion,however,ismuchwiderthanjustaddressing(discretized)QVIs.Rathergeneral
classes of minimization problems are considered. The third survey, which is by J.
F. Rodrigues and L. Santos, presents a general framework for a class of problems
with gradient type constraints that can be formulated as stationary and evolution
variationalandquasi-variationalinequalitiesandisillustratedwithseveralphysical
applications.
The four contributions that related to stochastic methods involve a statistical
approachtoconstructingnon-asymptoticconfidencesetsin2-Wassersteinspaceby
J.Ebert,V.Spokoiny,andA.Suvorikova;recentmodelinganddevelopmentsfrom
stochasticgeometrytoanalysespatialmultiopcommunicationsystemsbyB.Jahnel
and W. König; a model to deal with uncertain friction for the transport of natural
gas through passive networks of pipelines, involving a simplification of the Euler
equationsandtheuseofaMarkovchainMonteCarlomethod,isconsideredbyH.
Heitsch and N. Strogies; and the use of Malliavin calculus to study invariant and
quasi-invariantmeasuresforthetwodimensionalEulerequationissurveyedbyA.
B.CruzeiroandA.Symeonides.
Partial Differential Equations for dissipative and conservative models are ubiq-
uitous in mathematical-physics problems, like in the contribution by M. Kantner,
A. Mielke, M. Mittnenzweig and N. Rotundo on the modeling of semiconductors,
from quantum mechanics to devices, respecting fundamental principles of non-
equilibrium thermodynamics. These models are also of importance in describing
gradientstructuresfortwo-phaseflowsofconcentrationsuspensions,asinthechap-
terbyD.Peschka,M.Thomas,T.Ahnert,A.MünchandB.Wagner.Thenumerical
solutionsforelectrolyteflowsmodeledbycouplingtheNernst-Planck-Poissondrift
diffusionandNavier-StokesequationsisconsideredbyJ.Fuhrmann,C.Guhlke,A.
Linke,C.MerdonandR.Müller,whilemodelsofdynamicdamageandphase-field
fracture,andtheirvarioustimediscretisationsarepresentedbyT.Roubíček.
Analytic and geometrical insights into modeling phase change problems are
presented in the remaining three papers. In this respect, S. Amstutz and N. van
Goethem present a survey and motivation of the incompatibility operator, a recent
geometrical object introduced for a novel approach to elasto-plasticity problems,
includingamodelforcontinuawithdislocations;P.Colli,G.GilardiandJ.Sprekels
showthewell-posednessandstabilityforaclassicalnonlocalphase-fieldsystemof
Preface vii
viscousCahn-Hilliardtype;andS.DipierroandE.Valdinocipresentrecentprogress
onthefractionalAllen-Cahnequationforlong-rangephasecoexistencemodels.
Finally, we would like to acknowledge financial support from CMAFcIO and
CMUCfortheeventheldattheFaculdadedeCiênciasdaUniversidadedeLisboa
andtheWeierstrassInstituteBerlin.FortechnicalhelpinputtingtogethertheLaTeX
collectionofarticles,oursincerethanksgotoAnjaSchröter(WIAS)andtoAssis
Azevedo(UMinho).
February7,2019.
MichaelHintermüller(Berlin)
JoséFranciscoRodrigues(Lisboa)
Contents
Preface............................................................ v
RecentTrendsandViewsonEllipticQuasi-VariationalInequalities ::::: 1
AmalAlphonse,MichaelHintermüller,andCarlosN.Rautenberg
1 Introduction.............................................. 1
1.1 Thebasicsetting andproblemformulation ............ 3
2 Someexistencetheory ..................................... 4
2.1 CompactnessandMoscoconvergence ................ 4
2.2 Orderapproaches.................................. 7
3 Solutionmethodsandalgorithms ............................ 9
3.1 ContractionresultsforT............................ 9
3.2 ThemapK7!PKandextensionstoLions–Stampacchia. 11
3.3 Orderapproaches:solutionmethodsform„f”andM„f”.. 14
3.4 Regularizationmethods ............................ 16
3.5 Gerhardt-typeregularizationforthegradientcase....... 19
3.6 Drawbacksoftheiteration yn+1 =T„yn” .............. 20
4 Optimalcontrolproblems .................................. 21
5 Differentiability........................................... 22
5.1 DirectionaldifferentiabilityforQVIs ................. 24
6 Conclusion............................................... 28
References ..................................................... 28
The Incompatibility Operator: from Riemann’s Intrinsic View of
GeometrytoaNewModelofElasto-Plasticity ::::::::::::::::::::::: 33
SamuelAmstutz,NicolasVanGoethem
1 Ontheoriginofcurvatureinscienceandthebirthofintrinsicviews 34
2 Curvatureinnonlinearelasticity ............................. 35
3 Incompatibilityinlinearizedelasticityandpathintegralformulae . 37
4 ThelegacyofEkkehartKröner:thegeometryofacrystalwith
dislocations .............................................. 39
4.1 Thegeometricapproachatthemacroscale............. 39
ix
x Contents
4.2 Paralleldisplacementandcurvature .................. 41
4.3 Thenon-Riemanniancrystalmanifold ................ 43
4.4 Internalandexternalobservers ...................... 44
4.5 Inelasticeffectsandnotionofeigenstrain.............. 45
5 A geometric conception of linearized elasticity: the intrinsic
approach ................................................ 45
5.1 Gaussvs.Riemanninlinearizedelasticity ............. 45
5.2 Ciarlet’sintrinsicapproachtolinearizedelasticity ...... 46
6 Theclassicalroutetoplasticity .............................. 48
6.1 Themathematicalapproaches:twoperspectives ........ 48
6.2 Conventional(0th-order)elasto-plasticitymodels....... 50
7 Gradientelasto-plasticityforcontinuawithdislocations:
towardsanincompatibility-drivenmodel...................... 51
7.1 Thesizeeffect .................................... 51
7.2 Gradientmodels .................................. 52
7.3 Ourapproach:agradientmodelbasedonthestrain
incompatibility.................................... 52
7.4 Linkwithclassicalelasto-plasticitymodels ............ 54
8 Theincompatibilityoperator:functionalframework ............ 55
8.1 Divergence-freelifting,Greenformulaandapplications . 55
8.2 Saint-Venant compatibility conditions andBeltrami
decomposition .................................... 58
8.3 Orthogonaldecompositions ......................... 58
8.4 Boundaryvalueproblemsfortheincompatibility ....... 59
9 Towardsanintrinsicapproachtolinearizedelasto-plasticity...... 60
9.1 Objectivityandprincipleofvirtualpowers ............ 60
9.2 Constitutivelaw................................... 62
9.3 Equilibriumequations.............................. 62
9.4 Interpretationoftheexternalpowerandkinematical
framework ....................................... 63
9.5 Existenceresultsandelasticlimit .................... 63
9.6 Example:barintraction ............................ 64
9.7 Incrementalformulationofhardeningproblems ........ 66
Acknowledgements.............................................. 67
References ..................................................... 67
NonlocalPhaseFieldModelsofViscousCahn–HilliardType::::::::::: 71
PierluigiColli,GianniGilardi,JürgenSprekels
1 Introduction.............................................. 71
1.1 Aboutthemodelandrelatedproblems ................ 72
1.2 Nonlocaloperators ................................ 74
1.3 Overviewofsomerelatedcontribution................ 76
1.4 Aboutwell-posednessandregularityresults ........... 77
1.5 Theoptimalcontrolproblemforalogarithmicpotential . 77
1.6 Commentingontheoptimalcontrolproblem........... 78