Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering

344 Pages·2017·9.09 MB·English

by  Christian Soize (auth.)

Interdisciplinary Applied Mathematics 47
Christian Soize
Uncertainty
Quantifi cation
An Accelerated Course with Advanced
Applications in Computational
Engineering
Uncertainty Quantification
Interdisciplinary Applied Mathematics
Volume 47
Editors
S.S.Antman,UniversityofMaryland,CollegePark,MD,USA
[email protected]
L.Greengard,NewYorkUniversity,NewYork,NY,USA
[email protected]
P.J.Holmes,PrincetonUniversity,Princeton,NJ,USA
[email protected]
Advisors
L.Glass,McGillUniversity,Montreal,QB,Canada
A.Goriely,UniversityofOxford,Oxford,UK
R.Kohn,NewYorkUniversity,NewYork,USA
P.S.Krishnaprasad,UniversityofMaryland,CollegePark,MD,USA
J.D.Murray,UniversityofOxford,Oxford,UK
C.Peskin,NewYorkUniversity,NewYork,USA
S.S.Sastry,UniversityofCalifornia,Berkeley,CA,USA
J.Sneyd,UniversityofAuckland,Auckland,NewZealand
Moreinformationaboutthisseriesathttp://www.springer.com/series/1390
Christian Soize
Uncertainty Quantification
An Accelerated Course with Advanced
Applications in Computational Engineering
123
ChristianSoize
LaboratoireMode´lisationetSimulation
Multi-Echelle(MSME)
Universite´Paris-EstMarne-la-Valle´e
(UPEM)
Marne-la-Valle´e,France
ISSN0939-6047 ISSN2196-9973 (electronic)
InterdisciplinaryAppliedMathematics
ISBN978-3-319-54338-3 ISBN978-3-319-54339-0 (eBook)
DOI10.1007/978-3-319-54339-0
LibraryofCongressControlNumber:2017937717
Mathematics Subject Classification (2010): 15B52, 35Q62, 35Q74, 35R30, 35R60, 37N15, 47H40,
60G60,60H25,60J22,62B10,62C10,62F15
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Foreword
Innumerical modeling and simulation,some degree of uncertainty isinevitable in
the ability of the model to truly describe the physics of interest and/or in the data
this model uses to assist in describing these physics. For this reason, and because
numericalpredictionsareoftenthebasisofengineeringdecisions,uncertaintyquan-
tification has been a subject of concern for many years. With the advent of multi-
scaleandmultiphysicsmodeling,findingpracticalandyetrigorouswaystointerpret
uncertaintyandcharacterizeitsimpactontheassessmentofprobableoutcomeshas
becomeevermorechallenging.Thisbookcontributestopavingthewayformeeting
thisobjectiveoneday.
Thebookfocusesprimarilyonfundamentalnotionsinthestochasticmodelingof
uncertaintiesandtheirquantificationincomputationalmodels.Mechanicalsystems
are the privileged applications because of the additional expertise of the author in
thisfield.Allofaleatory,epistemic,parametric,andnonparametricuncertaintiesare
addressed.
The book begins with a description of those fundamental mathematical tools
ofprobabilityandstatisticsthataredirectlyusefulforuncertaintyquantification.It
proceedswithawell-carriedoutdescriptionofsomebasicandafewmoreadvanced
methodsforconstructingstochasticmodelsofuncertainties.Itpaysparticularatten-
tiontotheproblemofcalibratingandidentifyingastochasticmodelofuncertainties
when experimental data is available. For completeness, italsooverviews the main
approachesthathavebeendevelopedforanalyzingthepropagationofuncertainties
incomputationalmodels.Coveringallofthesetopicsinlessthan330pages,mak-
ingtheminterestingtofaculty,students,researchers,andpractitioners,isadaunting
challenge.Theauthorhasovercomethisdifficultybyskillfullyorganizingthecon-
tentofthisbookandreferringtomathematicalproofsratherthanderivingthemin
themiddleofhigh-leveldescriptions.
I have been privileged to read and study the lecture notes on which this book
is based. I have found them to be among the mostuseful ones for my understand-
ing of the topic. Consequently, I invited the author to deliver in February 2015 at
StanfordUniversityanacceleratedcourseonUncertaintyQuantificationinCompu-
tational Mechanics based on these lecture notes. Given the success of this course,
v
vi Foreword
I invited the author again in June 2016 to give a shorter version at the Aberdeen
ProvingGroundsiteoftheArmyResearchLaboratory(ARL).Thecoursewaswell
attendedbyscientistsandengineersfromARLrepresentingfivedifferentlaboratory
divisionsandseveralgraduatestudents.Forallthesereasons,Iexpectthisbookto
beausefulreferenceandasourceofinspiration.
StanfordUniversity CharbelFarhat
Stanford,CA,USA
November23,2016
Preface
Thisbookresultsfromacoursedevelopedbytheauthorandreflectsbothhisown
andcollaborativeresearchregardingthedevelopmentandimplementationofuncer-
taintyquantification(UQ)techniquesforlarge-scaleapplicationsoverthelasttwo
decades.
Theobjectivesofthisbookaretopresentfundamentalnotionsforthestochastic
modelingofuncertaintiesandtheirquantificationincomputationalmodelsencoun-
tered in computational sciences and engineering. The text covers basic methods
andnoveladvancedtechniquestoquantifyuncertaintiesinlarge-scaleengineering
and science models. It focuses on aleatory and epistemic uncertainties, parametric
uncertainties (associated with the parameters of computational models), and non-
parametric uncertainties (induced by modeling errors). The text covers mainly the
basic methods and novel advanced methodologies for constructing the stochastic
modeling of uncertainties. To this effect, it presents only the fundamental math-
ematical tools of probability and statistics that are directly useful for uncertainty
quantification and overviews the main approaches for studying the propagation of
uncertainties in computational models. Important methods are also presented for
performing robust analysis of computational models with respect to uncertainties,
robustupdatingofsuchcomputationalmodels,robustoptimization,anddesignun-
deruncertainties.Italsocarriesoutthecalibrationandidentificationofthestochas-
tic model of uncertainties when experimental data is available. The methods are
illustratedonadvancedapplicationsincomputationalengineering,suchasincom-
putational structural dynamics and vibroacoustics of complex mechanical systems
andinmicromechanicsandmultiscalemechanicsofheterogeneousmaterials.
This book is intended to be a graduate-type textbook for graduate students, en-
gineers, doctoral students, postdocs, researchers, assistant professors, professors,
etc. For learning a difficult interdisciplinary domain, such as the UQ domain, the
fundamentaldifficultyisnottheaccesstotheknowledgebutisrelatedtotheunder-
standingoftheknowledgeorganization,tothelevelofexpertisethatisrequiredfor
solving a complex problem, and to the use of methods that are scientifically vali-
dated.Suchanexpertiseisobtainedbythinkingandnotbycarryingoutexercises.
vii
viii Preface
Theaimoftheauthoristoproposeagraduate-typetextbookthatleadsthereaderto
thinkandnottotrainhim/herwithexercises.
Thewritingstylehasdeliberatelybeenchosenasshortanddirect,avoidingun-
necessary mathematical details that may prevent access to a quick understanding
of the concepts, ideas, and methods. Nevertheless, all the mathematics tools that
arepresentedareabsolutelycorrectandscientificallyrigorous.Alltherequiredhy-
potheses for using them are given and explained. Any approximation that is intro-
duced is commented and the limitations are specified. The useful references are
givenforareaderwhoisinterestedinfindingthemathematicaldetailsforproving
some mathematical results. A short paragraph is written at the beginning of each
chapter,summarizingtheobjectivesofthechapterandexplainingtheinterconnec-
tionsbetweenthechapters.Thechaptershavealargenumberofsubsectionswhich
will allow readers to clearly find and learn about specific topics that are useful to
fulfillthebroaderchapterobjectives.
Anobjectiveofthisbookistopresentconstructivemethodsforsolvingadvanced
applications and not to re-explain in detail the classical statistical tools, which are
alreadydevelopedinexcellenttextbooksinwhichthereareacademicexamplesthat
allowfortrainingtheundergraduatestudents.
Themainobjectiveiseffectivelytonotrecopyonemoretimetheclassicalstatis-
ticalmethodsforsolvingacademicproblemsinsmalldimensionwithaverysmall
numberofscalarrandomvariables.Thebookpresentsclassicalandadvancedmath-
ematical tools from the probability theory and novel methodologies in computa-
tional statistics that are necessary for solving the large-scale computational mod-
els with uncertainty quantification in high dimension. The book presents a set of
mathematical tools,their organization, and theirinterconnections, which allow for
constructingefficientmethodologiesthatarerequiredforsolvingchallenginglarge-
scale problems that are encountered in computational engineering. The book pro-
posestothereadersaclearandidentifiedstrategyforsolvingcomplexproblems.
The book includes several topics that are not covered in published research
monographs or texts on UQ. This includes random matrix theory for uncertainty
quantification,asignificantpartofthetheoryonstochasticdifferentialequations,the
useofmaximumentropyandpolynomialchaostechniquestoconstructpriorproba-
bilitydistributions,theidentificationofnon-Gaussiantensor-valuedrandomfieldsin
highstochasticdimensionbysolvingstatisticalinverseproblemrelatedtostochastic
boundaryvalueproblem,andtherobustanalysisfordesignandoptimization.Many
of these topics appear in research papers, but this is the first time that all of these
topicsarepresentedinaunifiedmanner.Anothernovelaspectofthisbookisthefact
thatitaddressesuncertaintyquantificationforlarge-scaleengineeringapplications,
which differentiates the methods proposed in this book from classical approaches.
Thetheory,themethodologies,theirimplementation,andtheirexperimentalvalida-
tions are illustrated by large-scale applications, including computational structural
dynamicsandvibroacousticsandsolidmechanicsofcontinuummedia.
Universite´ Paris-EstMarne-la-Valle´e ChristianSoize
Marne-la-Valle´e,France
November24,2016
Acknowledgments
I thank Professor Charbel Farhat from Stanford University for inviting me to give
thisacceleratedcoursethatIhaveespeciallywrittenforthisoccasionandgivingme
theopportunitytowritesuchanacceleratedcourseonthebasisofresearchyearsthat
Ihavecarriedoutintheareaofuncertaintyquantification.Thankstohisinvitation,
IspentaveryexcitingperiodforteachingandresearchatStanfordUniversity.
Ithankallthecolleagues,researchers,doctoralstudents,andpostdocswhohave
workedwithmesinceyear2000onimportantsubjectsrelatedtouncertaintyquan-
tificationandwhoaremycoauthorsinthepapersreferencedinthisbook:
AllainJM,ArnouxA,ArnstM,AvalosJ,BatouA,Capiez-LernoutE,CapillonR,
CataldoE,ChebliH,ChenC,Cle´mentA,ClouteauD,CottereauR,DesceliersC,
Duchereau J, Duhamel D, Durand JF, Ezvan O, Farhat C, Fernandez C,
Funfschilling C, Gagliardini L, Ghanem R, Gharbi H, Grimal Q, Guilleminot J,
Heck JV, Kassem M, Le TT, Leissing T, Lestoille N, Macocco K, Mbaye L,
Mignolet MP, Naili S, Nguyen MT, Nouy A, Ohayon R, Pellissetti M, Perrin G,
Poloskov IE, Pradlwarter H, Rochinha FA, Ritto TG, Sakji S, Sampaio R,
SchuellerGI,andTalmantM.
IwouldliketoparticularlythankProfessorRogerGhanemfromtheUniversity
ofSouthernCaliforniaandProfessorMarcMignoletfromArizonaStateUniversity,
withwhichwehavealwayscarriedoutexcitingscientificcollaborations.
Ithankmygreatfriend,ProfessorRogerOhayon,fromCNAMinFrancewho,in
additionofourfructuousandintensivescientificcollaborations,hascarefullyreread
themanuscriptofthisbook.
ix