Rechercher

sur ce site


Accueil du site > Résumés des séminaires > Labo > Experimental Data for Imaging of Refractive Indices and Shapes of Dielectrics via a Hybrid Globally Convergent/Adaptive Inverse Algorithm

Experimental Data for Imaging of Refractive Indices and Shapes of Dielectrics via a Hybrid Globally Convergent/Adaptive Inverse Algorithm

The development of numerical methods for Coefficient Inverse Problems (CIPs) faces two challenges combined : nonlinearity and ill-posedness of these problems. In the past few years a globally convergent numerical method for a CIP with single measurement data for a hyperbolic PDE was developed by the authors. However, since this method has a certain approximation, a two-stage numerical procedure was also developed. On the first stage the globally convergent method provides a good approximation for the solution. On the second stage the locally convergent so-called Finite Element Adaptive algorithm takes that approximation as the starting point for a subsequent refinement. The resulting two-stage procedure converges globally.

In this talk the above procedure will be described. Next, results for experimental time dependent data will be presented. It will be shown that the first stage provides excellent accuracy reconstruction of refractive indices and locations of dielectrics even in blind tests. Next, the adaptive algorithm complements this by an excellent reconstruction accuracy of shapes of dielectric abnormalities. Only a narrow measurement angle is used. In addition, results for backscattering data will be presented.

Finally the third result is about the regularization theory : it provides an analytical explanation on why this two-stage procedure works.

Papers [1,2] are about experimental data. The third result can be found in [3,4]. Further references to our works can be found in [1-4].

References

1.M.V. Klibanov, M.A. Fiddy, L. Beilina, N. Pantong and J. Schenk, Picosecond scale experimental verification of a globally convergent numerical method for a coefficient inverse problem, Inverse Problems, 26, 045003, 2010.

2.L. Beilina and M.V.Klibanov, Reconstruction of dielectrics from experimental data via a hybrid globally convergent/adaptive inverse algorithm, Inverse Problems, 26, 2010.

3.L. Beilina, M.V. Klibanov and M.Yu. Kokurin, Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem, Journal of Mathematical Sciences, 167, 279-325, 2010.

4.M.V. Klibanov, A.B. Bakushinskii and L. Beilina, Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess, submitted for publication, a preprint is available on-line at http://www.ma.utexas.edu/mp_arc/


CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France, Tél: +33 1 69 33 46 23 Fax: +33 1 69 33 46 46