Section: New Results

A Nash game for the coupled problem of conductivity identification and data completion

Participants : Abderrahmane Habbal, Rabeb Chamekh [PhD, LAMSIN, Univ. Tunis Al Manar] , Moez Kallel [LAMSIN, Univ. Tunis Al Manar] , Nejib Zemzemi [Inria Bordeaux, EPI CARMEN] .

In this work, we are interested in solving the electrocardiography inverse problem which could be reduced to the data completion problem for the Poisson equation. The difficulty comes from the fact that the conductivity values of the torso organs like lungs, bones, liver,...etc, are not known and could be patient dependent. Our goal is to construct a methodology allowing to solve both data completion and conductivity optimization problems at the same time.

In   [92], [101] a Nash game approach was developed to tackle the data completion problem. Our algorithm turned out to be efficient and robust with respect to noisy data. In a first attempt, presented in [17], we formulated the identification-completion problem as a Stackelberg game. Some numerical experiments were successful in this joint identification, but some were not. Which led us to develop new formulations, with direct impact on the technological modus operandi in the theoretical tomography process. In few words, the new formulations are based on the remark that not all the over-specified data are necessary to ensure the existence and uniqueness for the Cauchy problem, since by Holmgren theorem, only a piece of these data (over a subset of the boundary with non zero superficial measure) is necessary and sufficient. Presently, we are investigating the ability of these new formulations to ensure identifiability of the conductivity coefficients, for Poisson and linear elasticity equations.