Inverse Problems with Internal Functionals 英文文献资料

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1、CalderonprizelectureMay25,2011InverseProblemswithInternalFunctionalsFromCalderon'sproblemtoHybridInverseProblemsGuillaumeBalDepartmentofAppliedPhysics&AppliedMathematicsColumbiaUniversityCalderonprizelectureMay25,2011TheCalderonproblemConsidertheellipticmodel:r
(x)ru=0inXandu=gon@X:TheCalder

2、onproblemconsistsofreconstructingtheunknown(x)fromknowledgeofallpossibleCauchydata(u; 
ru)on@Xwithusolutionoftheaboveequation.Calderon(1980)showedinjectivityofthelinearizedCalderonproblemus-ingcomplexgeometricoptics(CGO)solutions.SylvesterandUhlmann(1987)showedinjectivityoftheCalderonproblem

3、forC2functionsandAstalaandP•aiv•arinta(2006)forL1functionsindimensiontwo.Alessandrini(1988)showedthatthemodulusofcontinuityoftheinverseproblemwas(essentially)logarithmic:severelossofresolution.GuillaumeBalHybridInverseProblems&InternalFunctionalsCalderonprizelectureMay25,2011CGOsolutionsandQuali

4、tativeStatementsInjectivityoftheCalderonproblemisprovedbyshowingthatq1=q2Zwhen(
qi)ui=0and(q1q2)u1u2dx=0:XStatementonthedensityofproductsof(almost-)harmonicsolutions.CGOsolutionsareoftheformu
x?n??=e1+(x)=k+ik2C;jkj=jkj;k
k=0:Property:jjjjisbounded(issmallasjj!1).Choosing1and2sucht

5、hat1+2=i2Rnandj1j;j2j!1:ZZ(qi
xlim1q2)uudx=(q1q2)edx=0:12j1j;j2j!1XXHowever,ju1j;ju2jejjtodetermineq^():theCalderonproblemisaseverelyill-posedinverseproblemwithlowresolutioncapabilities.GuillaumeBalHybridInverseProblems&InternalFunctionalsCalderonprizelectureMay25,2011HighContras

6、tandHighResolutionOpticalTomographyandElectricalImpedanceTomography,modeledbytheCalderonproblem,arelowresolutionbutHighContrastmodalities.HighresolutionmodalitiesincludeUltrasound,M.R.I.,X-rayCT.Thesemodalitiesaresometimeslowcontrast.HybridInverseProblemsareproblemsresultingfromthephysicalcoupli

7、ngbetweenaHighContrastmodalityandaHighResolutionmodality.Inthislecture,thehighresolutionmodalityisUltrasound.Thehighcontrastcomesfromelastic,electrical,oropticalpropertiesoftissues.GuillaumeBalHybridInverseProblems&Int