Section: New Results
Inverse acoustic problems
The determination of the shape of an obstacle from its effects on known acoustic or electromagnetic waves is an important problem in many technologies such as sonar, radar, geophysical exploration, medical imaging and nondestructive testing. This inverse obstacle problem (IOP) is difficult to solve, especially from a numerical viewpoint, because it is ill-posed and nonlinear  . Moreover, the precision in the reconstruction of the shape on an obstacle strongly depends on the quality of the given far-field pattern (FFP) measurements: the range of the measurements set and the level of noise in the data. Indeed, the numerical experiments performed in the resonance region (for example  ,  ,  ,  ), that is for a wavelength that is approximately equal to the diameter of the obstacle, tend to indicate that in practise, and at least for simple shapes, a unique and reasonably good solution of the IOP can be often computed using only one incident wave and full aperture far-field data (FFP measured only at a limited range of angles), as long as the aperture is larger than . For smaller apertures the reconstruction of the shape of an obstacle becomes more difficult and nearly impossible for apertures smaller than /4 .
Given that, and the fact that from a mathematical viewpoint, the FFP can be determined on the entire S1 from its knowledge on a subset of S1 because it is an analytic function, we propose a solution methodology to extend the range of FFP data when measured in a limited aperture and not on the entire sphere S1 . Therefore it would be possible to solve numerically the IOP when only limited aperture measurements are available. However, due to the analyticity nature of the FFP, the reconstruction or the extension of the far-field pattern from limited measurements is an inverse problem that is severely ill-posed and therefore very challenging from a numerical viewpoint. Indeed preliminary numerical results  indicate that the reconstruction of the FFP using the discrete L2 minimization with the standard Tikhonov regularization is very sensitive to the noise level in the data. The procedure is successful only when the range of measurements is very large which is not realistic for most applications.
We propose a multi-step procedure for extending/reconstructing the FFP from the knowledge of limited measurements. The proposed solution methodology addresses the ill-posedness nature of this inverse problem using a total variation of the FFP coefficients as a penalty term. Consequently the new cost function is no longer differentiable. We restore the differentiability to the cost function using a perturbation technique  which allows us to apply the Newton algorithm for computing the minimum. The multi-step feature of the proposed method consists in extending the FFP at each step by an n degrees increment.
We investigate the effect of the frequency regime and the noise level of the performance of the proposed solution methodology. Preliminary results obtained in the case of two-dimensional sound-soft disk-shaped scatterer have been performed. They illustrate the potential of the solution methodology for enriching the FFP measurements for various frequencies and levels of noise.