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8
Preface
thesolutionstability,aboutconstructionofsuchapproximatesolution
oftheinverseproblem,whichslightlydiffersfromtrueoneinasenseof
smallnessofmeasurementerrorlevel.
Theaimofthisworkisthedevelopmentandapplicationofthe
asymptotic,thenumerical-analyticalmethodstotheproblemsofde-
terminationofthegeometricalandphysical-mechanicalparametersof
thelocalinhomogeneitybymeansofelasticwavesfields.Achievementof
thisaimconsistsofsolvingofacomplexofsuchquestions:(i)elabora-
tionofnewanddevelopmentoftheexistingmethodsofresearchofthe
correspondingdirectproblemsofelasticitytheoryonthebaseofintegral
equations,definedatclosedmanifoldsoratmanifoldswithboundary;(ii)
creationoftheeffectivealgorithmsofthenumerical-analyticalresearch
oftheinverseproblems,beingconsidered,onthebaseoftheobtained
approximatesolutionsofthecorrespondingintegralequationsandthe
theoryofill-posedproblems;(iii)algebraisationoftheofferedalgorithms,
takingintoaccountthevolumeofcomputingwork,whichisnecessary
fortheirnumericalrealisation,andcarryingoutofthenumericalexper-
imentsonmodelexamples.
Theproblemdefinedanddiscussedissolvedwiththeuseofinte-
gralequationtheory,theWiener-Hopftechnique,matchedasymptotic
expansionsmethod,contourintegrals,successivediffractions,stationary
phaseandsaddle-pointmethods,theFourierandtheRadonintegral
transforms,ill-posedproblemstheory(constructionofnormalpseudoso-
lutionsandregularizationoperators,dependentonintegerparameter,
theTikhonovregularizationmethodwithstabilisingfunctionalofthe
firstorder,theprincipleofiterativeregularizationonthebaseofthe
Newton-Kantorovichmethod).Withtheaimofscatteringdatamod-
elingonehasusedtheGalerkinmethod,thecollocationsmethod,the
methodofmechanicalquadraturesforthenumericalsolvingofintegral
(integro-differential)equations,andalsothemethodofvariablessepara-
tion.
InChapter1,thecorrespondingstatementsofthedirectandinverse
elastodynamicboundaryscatteringproblemsaredescribed.Theneces-
saryinformationfromthetheoryofill-posedproblemsisgivenaswellas
somegeneralschemesofregularizedsolutionsconstruction.Besidesthat,
thesolutionsofseveralsimpleinverseproblemsarefoundandapplication
ofregularizationtechniquesareillustrated.Inparticular,theproblemof
reconstructionofthesourceradiationintensityatacoustichalf-space
issolved.RegularizationofproblemsisdonebymeansoftheGauss-
Weierstrassintegraloperator.Inthefollowingsectionofthischapter,
