Identification of singular parameters in partial differential equations
- Slim Chaabane, Faculté des sciences de Sfax, Tunisia
- Houssem Haddar, research team Inria IDEFIX / Centre Inria Saclay – Île-de-France
The goal of this associate team is to contribute to the analysis of inverse problems where the sought parameters lack regularity. A typical example is the inverse geometrical problem where the geometry to recover from given data represents the discontinuity set for some physical coefficients in a PDE model. This problem arises in a variety of applications like geophysics (e.g. the parameter being the sound velocity), non destructive testing (e.g. the parameter being the crack’s impedance, dielectric properties of deposits), medical imaging (e.g. the parameter being the conductivity), etc… For this type of problems, a classical formulation of the inverse problem as an optimisation problem would be faced in general with the lack of differentiability of the state variable with respect to the discontinuity location. We explore two main strategies to address this issue. The first one is based on the design of a suitable misfit functional that would be differentiable although the state variable is not. This is the case for example of the Kohn-Vogelius cost function for selfadjoint operators as it has been previously established by the team members. The second strategy would be to develop optimization free inversion procedures that avoid the derivative of the state variable. This is the case for instance of sampling methods that have been developed for cracks by the team members.
Key words: A6. – Modeling, simulation and control, A6.2. – Scientific computing, Numerical Analysis \& Optimization, A6.2.1. – Numerical analysis of PDE and ODE, A6.2.6. – Optimization, A6.3.1. – Inverse problems
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