Inverse Problems and Carleman Estimates: Global Uniqueness, Global Convergence and Experimental Data

344 Pages·2021·6.296 MB·English

by  Michael V. Klibanov, Jingzhi Li

MichaelV.Klibanov,JingzhiLi
InverseProblemsandCarlemanEstimates
Inverse and Ill-Posed Problems
Series
|
Edited by
Sergey I. Kabanikhin, Novosibirsk, Russia;
Almaty, Kazakhstan
Volume 63
Michael V. Klibanov, Jingzhi Li
Inverse Problems
and Carleman
Estimates
|
Global Uniqueness, Global Convergence and
Experimental Data
MathematicsSubjectClassification2010
35R30,65N21
Authors
Prof.Dr.MichaelV.Klibanov Prof.JingzhiLi
UniversityofNorthCarolinaatCharlotte SouthernUniversityofScienceandTechnology
DepartmentofMathematicsandStatistics DepartmentofMathematics
9201UniversityCityBlvd. 1088XueyuanAve
Charlotte,NC28223,USA Shenzhen518005,P.R.China
[email protected] [email protected]
ISBN978-3-11-074541-2
e-ISBN(PDF)978-3-11-074548-1
e-ISBN(EPUB)978-3-11-074555-9
ISSN1381-4524
LibraryofCongressControlNumber:2021940405
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2021WalterdeGruyterGmbH,Berlin/Boston
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
|
MichaelVictorKlibanovdedicatesthisbooktohisfamily:Ada(mother),Victor(fa-
ther), Vera (wife), Olga and Gregory (children), Luba (sister), and Leah and Victor
(grandchildren).Theyhavebeenalwaysverysupportiveofhimandhismathematical
effort.JingzhiLidedicatesthisbooktohisfamily:Fengying(mother),Shide(father),
Ning(wife),MengyaandMengqi(children).Theyhavebeenalwaysverysupportiveof
himandhismathematicalinterest.

Preface
In1981,almost40yearsago,A.L.BukhgeimandM.V.Klibanovintroduced,forthefirst
time,thepowerfultoolofCarlemanestimatesinthefieldofInverseProblems[51].The
techniqueof[51]isnowcalledthe“Bukhgeim–Klibanovmethod”(BK);see[48,49,
120,122]forthefirstdetailedproofsoftheoremsof[51].AccordingtoGoogleScholar,
thepaper[51]currently(2021)hasmorethan570citations;seehttps://scholar.google.
com/citations?user=pFmp7LMAAAAJ&hl=en.
While initially the BK method was thought only as a tool for proofs of global
uniqueness and stability theorems for coefficient inverse problems, it was discov-
ered recently that the ideas of BK generate a powerful numerical method for these
problems,theso-calledconvexificationmethod.
ThisbooksummarizesthemainanalyticalandnumericalresultsofM.V.Klibanov
andJ.LiaboutthetechniqueofCarlemanestimates,whichtheyhaveobtainedsince
thepublication[51].
GivenaPartialDifferentialEquation(PDE),aCoefficientInverseProblem(CIP)for
itistheproblemoffindingeitheroneorseveralcoefficientsofthatequationfromaddi-
tionalboundarymeasurements.CIPshavearapidlygrowingnumberofapplications
inmanyfields;seeChapters7–12.
Thefollowingfourtopicsarediscussedinthisbook:
1. Topic1:DerivationofCarlemanestimatesandconditionalstabilityestimatesfor
someill-posedCauchyproblems,Chapter2.
2. Topic 2: Global uniqueness for multidimensional CIPs on the basis of the BK
method,Chapter3.
3. Topic3:Carlemanestimatesfornumericalmethodsforill-posedCauchyproblems
forPDEs,Chapters4and5.
4. Topic4:TheconvexificationgloballyconvergentnumericalconceptforCIPs:afar
reachingconsequenceoftheBKmethod,Chapters6–12.
https://doi.org/10.1515/9783110745481-201

Acknowledgments
MichaelVictorKlibanovwassupportedbytwoUSArmyResearchLaboratoryandUS
ArmyResearchOfficegrantsW911NF-15-1-0233andW911NF-19-1-0044.M.V.Klibanov
heartilyappreciatesthesupportandunderstandingofDr.JosephD.Myers,aProgram
ManagerofTheUSArmyResearchOffice.
Jingzhi Li was partially supported by the NSF of China No.11971221, Shenzhen
Sci-TechFundNos.JCYJ20190809150413261andJCYJ20170818153840322,andGuang-
dongProvincialKeyLaboratoryofComputationalScienceandMaterialDesignNo.
2019B030301001.
https://doi.org/10.1515/9783110745481-202