The Sentinel Method and Its Application to Environmental Pollution Problems

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