Carleman-based reconstruction algorithm for the waves
Résumé
We present a globally convergent numerical algorithm based on global Carleman estimates to reconstruct the speed of propagation of waves in a bounded domain with Dirichlet boundary conditions from a single measurement of the boundary flux of the solutions in a finite time interval. The global convergence of the proposed algorithm naturally arises from the proof of the Lipschitz stability of the corresponding inverse problem for both sufficiently large observation time and boundary using global Carleman inequalities. The speed of propagation is supposed to be independent of time but varying in space with a trace and normal derivative known at the boundary and belonging to a certain admissible set that limits the speed fluctuations with respect to a given exterior point x0. In order to recover the speed, we also require a single experiment with null initial velocity and initial deformation having some monotonicity properties in the direction of x − x0. We perform numerical simulations in the discrete setting in order to illustrate and to validate the feasibility of the algorithm in both one and two dimensions in space. As proved theoretically, we verify that the numerical reconstruction is achieved for any admissible initial guess, even in the presence of small random disturbances on the measurements.
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