Splines and PDEs: From Approximation Theory to Numerical Linear Algebra: Cetraro, Italy 2017 PDF
Preview Splines and PDEs: From Approximation Theory to Numerical Linear Algebra: Cetraro, Italy 2017
Lecture Notes in Mathematics 2219 CIME Foundation Subseries Angela Kunoth · Tom Lyche Giancarlo Sangalli Stefano Serra-Capizzano Splines and PDEs: From Approximation Theory to Numerical Linear Algebra Cetraro, Italy 2017 Tom Lyche · Carla Manni Hendrik Speleers Editors Lecture Notes in Mathematics 2219 Editors-in-Chief: Jean-MichelMorel,Cachan BernardTeissier,Paris AdvisoryBoard: MichelBrion,Grenoble CamilloDeLellis,Princeton AlessioFigalli,Zurich DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Athens GáborLugosi,Barcelona MarkPodolskij,Aarhus SylviaSerfaty,NewYork AnnaWienhard,Heidelberg Moreinformationaboutthisseriesathttp://www.springer.com/series/304 Angela Kunoth (cid:129) Tom Lyche (cid:129) Giancarlo Sangalli (cid:129) Stefano Serra-Capizzano Splines and PDEs: From Approximation Theory to Numerical Linear Algebra Cetraro, Italy 2017 Tom Lyche, Carla Manni, Hendrik Speleers Editors 123 Authors AngelaKunoth TomLyche MathematicalInstitute DepartmentofMathematics UniversityofCologne UniversityofOslo Cologne,Germany Oslo,Norway GiancarloSangalli StefanoSerra-Capizzano DepartmentofMathematics DepartmentofScienceandHigh UniversityofPavia Technology Pavia,Italy UniversityofInsubria Como,Italy Editors TomLyche CarlaManni DepartmentofMathematics DepartmentofMathematics UniversityofOslo UniversityofRomeTorVergata Oslo,Norway Rome,Italy HendrikSpeleers DepartmentofMathematics UniversityofRomeTorVergata Rome,Italy ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics C.I.M.E.FoundationSubseries ISBN978-3-319-94910-9 ISBN978-3-319-94911-6 (eBook) https://doi.org/10.1007/978-3-319-94911-6 LibraryofCongressControlNumber:2018954553 MathematicsSubjectClassification(2010):Primary:65-XX;Secondary:65Dxx,65Fxx,65Nxx ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Thefourchaptersofthisbookcollectthemaintopicsthathavebeenlecturedatthe C.I.M.E.summerschool“SplinesandPDEs:RecentAdvancesfromApproximation Theoryto StructuredNumerical Linear Algebra” held in Cetraro, July 2–7, 2017. Theaimofthesummerschoolhasbeentogiveanintroductiontothemostadvanced mathematical developments and numerical methods originating in the numerical treatmentofPDEsbasedonsplinefunctions. Arenewedinterestinsplinemethodshasbeenstimulatedinthelastdecadebythe success of isogeometric analysis. The large research activity around isogeometric methods shows that splines yield a powerful tool to PDE discretizations. In this generalperspective,the progressofisogeometricanalysiswenthandin handwith theformulationofnewproblemsrequiringnewtechniquestoaddressthemproperly. This gave rise to novel (spline) results in different areas of classical numerical analysis,rangingfromapproximationtheorytostructurednumericallinearalgebra. Thesedevelopmentsmotivatedthetopicsofthesummerschool. Thefirstchapter“FoundationsofSplineTheory:B-Splines,SplineApproxima- tion, and Hierarchical Refinement” is written by us. It provides a comprehensive and self-contained introduction to B-splines and their properties, discusses the approximation power of spline spaces, and gives a review on hierarchical spline bases. Thesecondchapter“AdaptiveMultiscaleMethodsfortheNumericalTreatment ofSystemsofPDEs”byAngelaKunothisdevotedtonumericalschemesbasedon B-splines and B-spline-typewavelets as a particularmultiresolutiondiscretization methodology,inthecontextofcontrolproblems. The third chapter “Generalized Locally Toeplitz Sequences: A Spectral Anal- ysis Tool for Discretized Differential Equations” by Carlo Garoni and Stefano Serra-Capizzano presents the theory of generalized locally Toeplitz sequences, a frameworkforcomputingandanalyzingthespectraldistributionofmatricesarising fromthenumericaldiscretizationofdifferentialequations. The last chapter “Isogeometric Analysis: Mathematical and Implementational Aspects,withApplications”byThomasJ.R.Hughes,GiancarloSangalli,andMattia Taniprovidesanoverviewofthemathematicalpropertiesofisogeometricanalysis, v vi Preface discusses computationally efficient isogeometric algorithms, and presents some isogeometricbenchmarkapplications. We express our deepestgratitude to all the people who have contributedto the success of this C.I.M.E. summer school: the invited lecturers, the seminar and contributed talk speakers, and the authors who have contributed to this C.I.M.E. Foundation Subseries book. In addition, we thank all the participants, from 11 countries,thatenthusiasticallycontributedtothesuccessoftheschool.Lastbutnot least, we also thankC.I.M.E., in particular Elvira Mascolo (the C.I.M.E. director) andPaoloSalani(theC.I.M.E.scientificsecretary)fortheir continuoussupportin theorganizationoftheschool. Oslo,Norway TomLyche Rome,Italy CarlaManni Rome,Italy HendrikSpeleers Acknowledgments ThisC.I.M.E.activitywascarriedoutwiththecollaborationandfinancialsupportof INdAM(IstitutoNazionalediAltaMatematica)andEMS(EuropeanMathematical Society). It was also partially supported by the MIUR “Futuro in Ricerca 2013” Programmethroughtheproject“DREAMS.” vii Contents 1 FoundationsofSplineTheory:B-Splines,SplineApproximation, andHierarchicalRefinement............................................... 1 TomLyche,CarlaManni,andHendrikSpeleers 2 AdaptiveMultiscaleMethodsfortheNumericalTreatmentof SystemsofPDEs.............................................................. 77 AngelaKunoth 3 GeneralizedLocallyToeplitzSequences:ASpectralAnalysis ToolforDiscretizedDifferentialEquations............................... 161 CarloGaroniandStefanoSerra-Capizzano 4 IsogeometricAnalysis:MathematicalandImplementational Aspects,withApplications.................................................. 237 ThomasJ.R.Hughes,GiancarloSangalli,andMattiaTani ix Chapter 1 Foundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement TomLyche,CarlaManni,andHendrikSpeleers Abstract Thischapterpresentsanoverviewofpolynomialsplinetheory,withspe- cialemphasisontheB-splinerepresentation,splineapproximationproperties,and hierarchicalspline refinement. We start with the definition of B-splines by means of a recurrence relation, and derive several of their most important properties. In particular,weanalyzethepiecewisepolynomialspacetheyspan.Then,wepresent the construction of a suitable spline quasi-interpolant based on local integrals, in order to show how well any function and its derivatives can be approximated in a given spline space. Finally, we provide a unified treatment of recent results on hierarchical splines. We especially focus on the so-called truncated hierarchical B-splines and their main properties. Our presentation is mainly confined to the univariate spline setting, but we also briefly address the multivariate setting via the tensor-product construction and the multivariate extension of the hierarchical approach. 1.1 Introduction Splines,inthebroadsenseoftheterm,arefunctionsconsistingofpiecesofsmooth functionsgluedtogetherinacertainsmoothway.Besidestheirtheoreticalinterest, they have application in several branches of the sciences including geometric modeling,signalprocessing,dataanalysis,visualization,numericalsimulation,and probability,just to mentiona few. There is a large variety of spline species, often referredtoasthezooofsplines.Themostpopularspeciesistheonewherethepieces arealgebraicpolynomialsandinter-smoothnessisimposedbymeansofequalityof T.Lyche DepartmentofMathematics,UniversityofOslo,Oslo,Norway e-mail:[email protected] C.Manni·H.Speleers((cid:2)) DepartmentofMathematics,UniversityofRomeTorVergata,Rome,Italy e-mail:[email protected];[email protected] ©SpringerNatureSwitzerlandAG2018 1 T.Lycheetal.(eds.),SplinesandPDEs:FromApproximationTheorytoNumerical LinearAlgebra,LectureNotesinMathematics2219, https://doi.org/10.1007/978-3-319-94911-6_1
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