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Many environmental problems contain incomplete data in the initial or boundary conditions. How do we solve problems for which some of the initial and/or boundary conditions are unknown? Using a new technique, the sentinel method, this book answers these questions and others as they pertain to inverse problems in environmental pollution, such as pollution of underground and surface waters, thermal pollution, and air pollution.
Table of Contents
Introduction Physical Motivation Historical Background Description of the Method Identification of Pollution in an Aquifer Modelling of Pollution Transport in an Aquifer A Sentinel Attached to Each Parameter Some Examples of Similar Problems Flow Rate Numerical Experiments Sentinels and Pseudo-Inverse Identification of Pollution in a Lake Pollution of a Lake Sentinels Numerical Experiment Adjoint State Numerical Details HUM Method Time and Space Discretization Optimal Emplacement of Sensors Rectangular Domain Physical Motivations Modelling the Physical System Exact Solution of the Direct Problem Sentinels for Inversion Direct Method Numerical Experiments Sensitivity to the Size of the Observatory Sentinels in a River Oxygen Kinetics and Polluted Water Convection-Dispersion-Reaction Equation Exact Solution of the State Equation Sentinels for a River (Evolution Regime) Numerical Experiments Our First Nonlinear Problem Linear Case Non-Linear Case Non-Linear Problems Position of the Problem Sentinels of the Linearized Problem Building the Generalized Inverse Example Dispersion Coefficients Motivation Linearized System Linearized System Sentinels Non-Linear Problem Numerical Results Position of a Source Position of the Problem Linearization Sentinels of the Linearized Problem Non-Linear Problem Numerical Experiments Unknown Position and Flow Rate Sentinels of the Linearized Problem (1) Sentinels of the Linearized Problem (2) Moving Source Recapitulation Definitions Inverse Problems A Convergence Result Gauss-Newton Method Shallow Waters The Movement of Tides: Saint-Venant Shallow Water Equations Numerical Solution of Shallow Water Equations Weak Formulation of the Problem Reaction-Convection-Dispersion Equations Sentinels Appendix Sentinels with a Given Sensibility and Duality Duality Existence of a Solution and Functional Spaces References
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CHOICE – Outstanding Academic Title – Award Winner
CHOICE – 2018 Outstanding Academic Title – Award Winner
Shingo Research and Professional Publication Award Winner
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