We present two extensions of the method, originally developed by Kress and Rundell in 2005 for a 2D inverse boundary value problem for the Laplace equation. In particular, we consider the reconstruction of a 3D perfectly electric conductor obstacle, and the reconstruction of generalized surface impedance functions for acoustic scattering from the knowledge of far-field measurements of a scattered wave associated with a few incident plane waves. Inverse scattering problems are solved numerically by the approach based on the reformulation of a problem as a system of nonlinear and ill-posed integral equations for the unknown boundary (or boundary condition) and the measurements. The iteratively regularized Gauss-Newton method is applied to the resulting system.

[1] R. Kress, W. Rundell. Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Probl. 21(4): 1207--1223, 2005.

[2] O. Ivanyshyn Yaman, F. Le Lou\"{e}r. Material derivatives of boundary integral operators in electromagnetism and application to inverse scattering problems. Inverse Probl. 32(9): 095003, 2016.

[3] O. Ivanyshyn Yaman. Reconstruction of generalized impedance functions for 3D acoustic scattering. J. Comput. Phys, 392(1): 444--455, 2019.

This talk concerns the identification of defects in a closed waveguide which is partially embedded in a surrounding medium, from scattering measurements on the free part of the waveguide. We wish to model for example a NDT experiment on a steel cable embedded in concrete. There are two main issues: the back-scattering situation and the leakage of waves from the closed waveguide to the surrounding medium. We will first introduce Perfectly Matched Layers in the transverse direction in order to transform the structure into a junction of two closed-half waveguides, one of them being a complex stratified medium. Then, after discussing the well-posedness of the forward problem and its numerical resolution, we will show how we can solve the inverse problem with the help of a modal formulation of the Linear Sampling Method. Some 2D numerical experiments will be shown.

In this talk we will address the numerical solution of the time-harmonic inverse scattering problem for an obstacle with transmission conditions and with given far-field data. To this end we will revisit the ideas of the hybrid method [1,2,3,4,5] that combines the framework of the Kirsch-Kress decomposition method and the iterative Newton-type method.

Instead of linearizing all the equations at once as in [6,7], we will explore the possibility of in a first ill-posed step reconstructing the scattered exterior field and the interior field by imposing the far-field condition and one of the boundary conditions and then in a second step linearizing on the second boundary condition in order to update the approximation of the boundary of the obstacle. The first and second steps are then iterated until some stopping criteria is achieved.

[1] R. Kress, P. Serranho. A hybrid method for two-dimensional crack reconstruction, Inverse Probl. 21 (2): 773--784, 2005.

[2] P. Serranho. A hybrid method for inverse scattering for shape and impedance, Inverse Probl. 22 (2): 663--680, 2006.

[3] R. Kress, P. Serranho. A hybrid method for sound-hard obstacle reconstruction, J. Comput. Appl. Math. 204 (2): 418--427, 2007.

[4] P. Serranho. A hybrid method for inverse scattering for Sound-soft obstacles in $\mathbb R^{3}$. Inverse Problems and Imaging. 1(4): 691--712, 2007.

[5] O. Ivanyshyn, R. Kress, P. Serranho. Huygens’ principle and iterative methods in inverse obstacle scattering. Adv. Comput. Math. 33 (4): 413--429, 2010.

[6] A. Altundag, R. Kress. An iterative method for a two-dimensional inverse scattering problem for a dielectric. J. Inverse Ill-Posed Probl. 20 (4): 575--590, 2012.

[7] A. Altundag. Inverse obstacle scattering with conductive boundary condition for a coated dielectric cylinder. J. Concr. Appl. Math. 13 ,(1--2): 11--22, 2015.

This talk discusses the inverse source problem with a single propagating mode at multiple frequencies in an acoustic waveguide. The goal is to provide both theoretical justifications and efficient algorithms for imaging extended sources using the sampling methods. In contrast to the existing far/near field operator based on the integral over the space variable in the sampling methods, a multi-frequency far-field operator is introduced based on the integral over the frequency variable. This far-field operator is defined in a way to incorporate the possibly non-linear dispersion relation, a unique feature in waveguides. The factorization method is deployed to establish a rigorous characterization of the range support which is the support of source in the direction of wave propagation. A related factorization-based sampling method is also discussed. These sampling methods are shown to be capable of imaging the range support of the source. Numerical examples are provided to illustrate the performance of the sampling methods, including an example to image a complete sound-soft block.