Seminar in Numerical Analysis: Maya de Buhan (Université Paris Descartes)

A new approach to solve the inverse scattering problem for the wave equation

In this talk, we propose a new method to solve the following inverse problem: we aim at reconstructing, from boundary measurements, the location, the shape and the wave propagation speed of an unknown obstacle surrounded by a medium whose properties are known. Our strategy combines two methods recently developed by the authors:

1 - the Time-Reversed Absorbing Condition method: It combines time reversal techniques and absorbing boundary conditions to reconstruct and regularize the signal in a truncated domain that encloses the obstacle. This enables us to reduce the size of the computational domain where we solve the inverse problem, now from virtual internal measurements.

2 - the Adaptive Inversion method: It is an inversion method which looks for the value of the unknown wave propagation speed in a basis composed by eigenvectors of an elliptic operator. Then, it uses an iterative process to adapt the mesh and the basis and improve the reconstruction.

We present several numerical examples in two dimensions to illustrate the efficiency of the combination of both methods. In particular, our strategy allows (a) to reduce the computational cost, (b) to stabilize the inverse problem and (c) to improve the precision of the results.

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