Many environmental problems contain incomplete data in the initial or boundary conditions. How is this data found? How do we solve problems for which some of the initial and/or boundary conditions are unknown? Using the sentinel method, a new mathematical technique developed for environmental problems, the author answers these questions and others as they pertain to inverse problems in environmental systems.
IntroductionPhysical MotivationHistorical BackgroundDescription of the MethodIdentification of Pollution in an AquiferModelling of Pollution Transport in an AquiferA Sentinel Attached to Each ParameterSome Examples of Similar ProblemsFlow RateNumerical ExperimentsSentinels and Pseudo-InverseIdentification of Pollution in a LakePollution of a LakeSentinelsNumerical ExperimentAdjoint StateNumerical DetailsHUM MethodTime and Space DiscretizationOptimal Emplacement of SensorsRectangular DomainPhysical MotivationsModelling the Physical SystemExact Solution of the Direct ProblemSentinels for InversionDirect MethodNumerical ExperimentsSensitivity to the Size of the ObservatorySentinels in a RiverOxygen Kinetics and Polluted WaterConvection-Dispersion-Reaction EquationExact Solution of the State EquationSentinels for a River (Evolution Regime)Numerical ExperimentsOur First Nonlinear ProblemLinear CaseNon-Linear CaseNon-Linear ProblemsPosition of the ProblemSentinels of the Linearized ProblemBuilding the Generalized InverseExampleDispersion CoefficientsMotivationLinearized SystemLinearized System SentinelsNon-Linear ProblemNumerical ResultsPosition of a SourcePosition of the ProblemLinearizationSentinels of the Linearized ProblemNon-Linear ProblemNumerical ExperimentsUnknown Position and Flow RateSentinels of the Linearized Problem (1)Sentinels of the Linearized Problem (2)Moving SourceRecapitulationDefinitionsInverse ProblemsA Convergence ResultGauss-Newton MethodShallow WatersThe Movement of Tides: Saint-Venant Shallow Water EquationsNumerical Solution of Shallow Water EquationsWeak Formulation of the ProblemReaction-Convection-Dispersion EquationsSentinelsAppendixSentinels with a Given Sensibility and DualityDualityExistence of a Solution and Functional SpacesReferences
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