Modern Solvers for Helmholtz Problems

247 Pages·2017·5.622 MB·English

by  Domenico Lahaye, Jok Tang, Kees Vuik (eds.)

Geosystems Mathematics
Domenico Lahaye
Jok Tang
Kees Vuik
Editors
Modern Solvers
for Hemholtz
Problems
Geosystems Mathematics
Serieseditors
W.Freeden
Kaiserslautern,Germany
M.Z.Nashed
Orlando,Florida,USA
Thisseriesprovidesanidealframeandforumforthepublicationofmathematical
key technologiesand their applicationsto geoscientific and geo-related problems.
Current understanding of the highly complex system Earth with its interwoven
subsystems and interacting physical, chemical, and biological processes is not
only driven by scientific interest but also by the growing public concern about
the future of our planet, its climate, its environment and its resources. In this
situation mathematics provides concepts, tools, methodology and structures to
characterize, model and analyze this complexity at various scales. Modern high
speed computersare increasinglyenteringallgeo-disciplines.Terrestrial,airborne
as well as spaceborne data of higher and higher quality become available. This
fact has not only influenced the research in geosciences and geophysics, but also
increased relevant mathematical approaches decisively as the quality of available
datawasimproved.
Geosystems Mathematics showcases important contributions and helps to
promote the collaboration between mathematics and geo-disciplines. The closely
connectedseriesLectureNotesinGeosystemsMathemacticsandComputingoffers
theopportunitytopublishsmallbooksfeaturingconcisesummariesofcutting-edge
research,newdevelopments,emergingtopicsandpracticalapplications.AlsoPhD
theses may be evaluated, provided that they represent a significant and original
scientificadvance.
Editedby
• WilliFreeden(UniversityofKaiserslautern,Germany)
• M.ZuhairNashed(UniversityofCentralFlorida,Orlando,USA)
Inassociationwith
• Hans-PeterBunge(MunichUniversity,Germany)
• RoussosG.Dimitrakopoulos(McGillUniversity,Montreal,Canada)
• YalchinEfendiev(TexasA&MUniversity,CollegeStation,TX,USA)
• AndrewFowler(UniversityofLimerick,Ireland&UniversityofOxford,UK)
• BulentKarasozen(MiddleEastTechnicalUniversity,Ankara,Turkey)
• JürgenKusche(UniversityofBonn,Germany)
• LiqiuMeng(TechnicalUniversityMunich,Germany)
• VolkerMichel(UniversityofSiegen,Germany)
• NilsOlsen(TechnicalUniversityofDenmark,KongensLyngby,Denmark)
• HelmutSchaeben(TechnicalUniversityBergakademieFreiberg,Germany)
• OtmarScherzer(UniversityofVienna,Austria)
• FrederikJ.Simons(PrincetonUniversity,NJ,USA)
• ThomasSonar(TechnicalUniversityofBraunschweig,Germany)
• Peter J.G. Teunissen, Delft University of Technology, The Netherlands and
CurtinUniversityofTechnology,Perth,Australia)
• Johannes Wicht (Max Planck Institute for Solar System Research, Göttingen,
Germany).
Formoreinformationaboutthisseriesathttp://www.springer.com/series/13389
Domenico Lahaye • Jok Tang • Kees Vuik
Editors
Modern Solvers for
Helmholtz Problems
Editors
DomenicoLahaye JokTang
DelftInstituteofAppliedMathematics DelftInstituteofAppliedMathematics
DelftUniversityofTechnology DelftUniversityofTechnology
Delft,TheNetherlands VORtechB.V.
Delft,TheNetherlands
KeesVuik
DelftInstituteofAppliedMathematics
DelftUniversityofTechnology
Delft,TheNetherlands
ISSN2510-1544 ISSN2510-1552 (electronic)
GeosystemsMathematics
ISBN978-3-319-28831-4 ISBN978-3-319-28832-1 (eBook)
DOI10.1007/978-3-319-28832-1
LibraryofCongressControlNumber:2016962706
MathematicsSubjectClassification(2010):35-XX,65FXX,35J05
©SpringerInternationalPublishingAG2017
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Foreword
TheHelmholtzequationrepresentsthetime-independentpartofthewaveequation
for electromagnetic, seismic, and acoustic waves. It was named after the scientist
Hermann von Helmholtz (1821–1894). Today, it is one of the most used partial
differentialequationsinnumericalsimulation.
In the multibillion seismic imaging industry, being my field of expertise,
longitudinal wave fields are generated in complex geological media. These wave
fieldsshowabroadrangeofspace-variantwavenumbers.Thisbroadrangeiscaused
byseismicsourceswithabandwidthuptofiveoctaves(startingatafewHz)andby
rock velocitiesthat rangefrom a few hundredm/s to many thousandm/s. In such
achallengingnaturalenvironment,accuratesolutionsneedtobecomputedthatare
used for the design of effective data collection geometries, for understanding the
very complex seismic responses, and for making imaging algorithms that utilize
solversinreversetime.
Notorious problems are the accuracy of high-wavenumber solutions, where
avoiding numerical dispersion requires very fine spatial sampling. For the very
sizeable geological models, this makes these traditional solvers economically not
feasible.
I complementthe authorsforgivingan elegantoverviewofHelmholtzsolvers,
with emphasis on the latest developments. The book is particularly valuable by
showing the reader how to derive and use solvers that are independent of the
wavenumber.Thiscouldfind wideapplicationin allwave field simulationswhere
sizeablemodelsandhighwavenumbersareoflargeinterest.
I hopethat the new insightsin this bookwill be widely used in academicsand
industrytobettersolvethemultipleforwardandinverseproblemsthatplayacritical
roleintheincreasingamountofwavefieldapplicationsworldwide.
ProfessorofGeosciencesEmeritus,TUDelft Dr.A.J.(Guus)Berkhout
DirectoroftheCentreforGlobal
Socio-EconomicChange
v
Contents
PartI Algorithms:NewDevelopmentsandAnalysis
RecentResultsonDomainDecompositionPreconditioning
fortheHigh-FrequencyHelmholtzEquationUsingAbsorption ........... 3
IvanG.Graham,EuanA.Spence,andEeroVainikko
High Order Transparent Boundary Conditions
fortheHelmholtzEquation..................................................... 27
LotharNannen
On the Optimality of Shifted Laplacian in a Class
ofPolynomialPreconditionersfortheHelmholtzEquation................. 53
SiegfriedCoolsandWimVanroose
PartII Algorithms:PracticalMethodsandImplementations
HowtoChoosetheShiftintheShiftedLaplacePreconditioner
fortheHelmholtzEquationCombinedwithDeflation....................... 85
D.LahayeandC.Vuik
TheMultilevelKrylov-MultigridMethodfortheHelmholtz
EquationPreconditionedbytheShiftedLaplacian .......................... 113
YogiA.Erlangga,LuisGarcíaRamos,andReinhardNabben
A Geometric Multigrid Preconditioner for the Solution
of the Helmholtz Equation in Three-Dimensional
HeterogeneousMediaonMassivelyParallelComputers.................... 141
H.Calandra,S.Gratton,andX.Vasseur
PartIII ImplementationsandIndustrialApplications
Some ComputationalAspectsof the Time andFrequency
DomainFormulationsofSeismicWaveformInversion...................... 159
René-ÉdouardPlessix
vii
viii Contents
Optimized Schwarz Domain Decomposition Methods
forScalarandVectorHelmholtzEquations................................... 189
X.AntoineandC.Geuzaine
Computationally Efficient Boundary Element Methods
forHigh-FrequencyHelmholtzProblemsinUnboundedDomains ........ 215
TimoBetcke,Elwinvan’tWout,andPierreGélat
Contributors
X.Antoine InstitutElieCartan deLorraine,UniversitédeLorraine,InriaNancy-
GrandEstEPISPHINX,Vandoeuvre-lès-Nancy,France
Timo Betcke Department of Mathematics, University College London, London,
UK
H.Calandra TOTALE&PResearchandTechnologyUSA,Houston,TX,USA
SiegfriedCools DepartmentofMathematicsandComputerScience,Universityof
Antwerp,Antwerp,Belgium
Yogi A. Erlangga Mathematics Department, Nazarbayev University, Astana,
Kazakhstan
LuisGarcíaRamos InstitutfürMathematik,TUBerlin,Berlin,Germany
PierreGélat DepartmentofMechanicalEngineering,UniversityCollegeLondon,
London,UK
C.Geuzaine InstitutMontefioreB28,UniversitédeLiège,Liège,Belgium
IvanG.Graham DepartmentofMathematicalSciences,UniversityofBath,Bath,
UK
S.Gratton INPT-IRIT,UniversityofToulouseandENSEEIHT,Toulouse,France
D.Lahaye DIAM,TUDelft,Delft,TheNetherlands
ReinhardNabben InstitutfürMathematik,TUBerlin,Berlin,Germany
LotharNannen TechnischeUniversitätWien,Wien,Austria
René-ÉdouardPlessix ShellGlobalSolutionsInternational,Rijswijk,TheNether-
lands
EuanA.Spence DepartmentofMathematicalSciences,UniversityofBath,Bath,
UK
ix
x Contributors
EeroVainikko InstituteofComputerScience,UniversityofTartu,Tartu,Estonia
WimVanroose DepartmentofMathematicsandComputerScience,Universityof
Antwerp,Antwerp,Belgium
X.Vasseur ISAE-SUPAERO,Toulouse,France
C.Vuik DIAM,TUDelft,Delft,TheNetherlands
Elwinvan’tWout SchoolofEngineeringandFacultyofMathematics,Pontificia
UniversidadCatólicadeChile,Santiago,Chile