Mat-1.3656 Seminar on
numerical analysis and computational science
Monday, Feb 2, 2011, room
U322 at 14.15, Eirola & Stenberg
Jérémi Darde
A new framework to solve the inverse obstacle problem: the "exterior approach"
Inverse obstacle problems arise in many situations in engineering and
physics: medical imaging, non destructive testing of structures,
control of the plasma's boundary in a Tokamak... They consist in
finding the geometry of an heterogeneity in a domain from the knowledge
of a Cauchy data (both Dirichlet data and Neumann data) on a subpart of
the boundary of this domain. Mathematically, we search a function u and
an obstacle O such that u verifies an elliptic PDE outside O, some
known condition on O, and fits the Cauchy data. Many methods exist to
solve this problem, most of them relies on optimization techniques.
We propose a new framework to solve the inverse obstacle problem. It is
based on the construction of a decreasing sequence of open domains that
contain the searched obstacle. At each step of this method, we first
obtain an approximation of u outside the current open domain by solving
a Cauchy problem using the quasi-reversibility method. Then, we update
the domain using a level-set technique to obtain a better "exterior
approximation" of the obstacle.
We present some theoretical justifications of this method and some
numerical illustrations of its efficiency for two different 2-D
problems:
- the inverse obstacle problem with Dirichlet condition
- the non-destructive testing of elastoplastic media in the antiplane case.