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INVERSE PROBLEMS

Introduction

The concept of well -posed and ill posed- problems goes back to Hadamard . When Hadamard wrote his 1902 paper, defining ill posed  those problems whose solution does not exist or it is not unique or it if is not stable under perturbations on data,
Hadamard [1,2] introduced this notion of well-posed and gave an example of an
ill-posed problem via the Cauchy problem for Laplace's equation.

A- Well-posed problems
A mathematical problem is well-posed, if it verifies the following conditions:
1) The existence of the solution,
2) The unicity of the solution,
3) The stability of the solution.
    The solution depends continuously on the data, in some reasonable topology.

Precisely, let be Ω  a complete metric space, and A an operator. 
Consider the equation:           
                                 Au(x,t) = f    
This equation is properly wel-posed if:

• La solution u(x,t) exits V fε S
• La solution u(x,t) is unique
• La solution u(x,t) depends continuously on f

Examples:

• Dirichlet Problem ( Laplace’s equation with initial conditions)
• Heat conduction with initial conditions

B- Ill-posed problems (Tichonov)
If the one of the above conditions are not verified, then we say the problem is not well-posed on the sense of Hadamard: The mathematical problem is ill-posed.
For example, the Inverse Problems are often ill-posed.Problems.
If the problem is well-posed, then it is possible to find a solution on a computer with a stable algortithm.
In contrario, the problem must be re-formulated for numerical analysis. We use the process known as regularization (including additional assumptions, such as smoothness of solution,etc..)

References
1.   Jacques Hadamard (1902): Sur les problèmes aux dérivées partielles et leur  signification physique. Princeton University Bulletin, 49-52

2. J. Hadamard, Lectures on Cauchy’s problem in linear partial differential equations, Dover, New York, 1952. MR 14:474

D- The Inverse Problems

1. Definitions

2. Domain (Non-exhaustive list)

• Mechanics (Parameters Characterization,...)
• Powder and Granular Mechanics ( Dead zones detection,...)
• Dame and Fracture Mechanics (Non destructive control, ...)
• Impact Mechanics (Identifcation of transient stresses, ...)
• Particle Physics
• Medical Imaging (Tomography, Echocardiography, scanner, ...),
• Biology ( Bones and tissues identification,..)
• Geophysics (Seismic imaging, Land mines detection, Oil prospection, ..)
• Meteorology (Particles Analysis in the atmosphere, ...)
• Telecommunications and IT (Satellite, radar, Capacity planning Networks, ..)
• Econometrics (Finances )

3. Mathematical Classification

4. Regularization

5- Used numerical methods (FDE, FEM, BFEM)

E- Resources