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2.Numericalproblemanalysis
Introducingintothesystemofequations(2.28)nodalvariablesdeterminedby
boundaryconditions,theexpressionsonthediagonalofmatrixHshouldbemod-
ified.ElementslyingonthediagonalofthematrixHassociatedwithagivenedge
nodearemultipliedbyalargenumber,whiletheappropriateelementofvectorb
isreplacedbyavariableintheedgenodemultipliedbythesamelargefactorand
diagonalelement.Thisisrepeatedforalledgenodes.Theeffectofthisisthatthe
unmodifiedelementsofthematrixHareverysmallcomparedtothemodified
elements.ThismodificationofmatrixHretainsthepropertiesoftheoutputmatrix,
i.e.itisbanded,sparseandsymmetrical,butmuchworseconditioned[153-155].
2.3.Formulatingtheinverseproblem
Theoppositeproblemisthesearchforthedistributionofconductivityfor
knownpotentialvalues.Thetaskofthereconstructionalgorithmconsistsofdeter-
miningsuchavectorγ,forwhichtheobjectivefunctionreachesatleastthelocal
minimum.Thevalueoftheobjectivefunctionatthedeterminedmeasurement
datadependsonthematrixЈ.Onthebasisofapreciselysolvedsimpleproblem,
thevaluesofpotentialsaredeterminedinallnodesofthegridassumingthatthe
distributionofconductivityintheareaisknown.Inthereconstructionprocess,
sensitivityanalysisisanimportantissue.Itdeterminesthederivativeoftheob-
jectivefunctionrelativetothedecisionvariables(relativetothevectorγofthe
electricalconductivitydistribution).Sensitivityanalysisinconnectionwithany
optimizationmethodgivestheanswertothequestion,whatcorrectionstothe
decisionvariablesshouldbemadeinordertoreducethevalueoftheobjective
function.Inversetransformationcanbeapproachedinavarietyofways.Using
thesensitivityanalysis,thegradientoftheobjectivefunctionisdetermined.There
aretwowaystosolvetheproblem:directdifferentiationoftheobjectivefunction
againstdecisionvariablesortheuseofaconjugatevariable.Inthedirectdiffer-
entiationmethod,thebasicproblemistodeterminetheconductivitydistribution
thatprovidesthelocalminimumoftheobjectivefunctionF.Aprerequisitefor
determiningtheminimumoftheobjectivefunctionduetothedesignvariablesγ
istoequateitsfirstderivativetozero
(2.29)
where:ЈJacobimatrixwithdimensionsn×m.
Theelementofthismatrixisexpressedbythefollowingformula:
(2.30)
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