Gelfand-Levitan-Marchenko and model order reduction methods in inverse scattering

Abstract

Inverse scattering problems arise in many applications, especially in imaging. In this thesis we studied frequency domain inverse scattering problems for the Helmholtz and the Schrödinger operators using both classical inverse scattering and modern reduced order model techniques. We started by revisiting the classical Gelfand-Levitan-Marchenko (GLM) integral equation method for solving the inverse Schrödinger scattering problem in 1D. The inverse Schrödinger scattering problem is interesting for imaging purposes, since it is possible to transform the Helmholtz and the (frequency domain) acoustic wave equation to the Schrödinger equation using a coordinate transform. In particular, we considered the GLM method with noise in the data, where we contributed an error bound for the solution of the unregularised GLM equation. We also proposed a regularised total least squares formulation in the infinite dimensional setting and we showed well posedness. Moreover, we studied the 1D scattering problem for the Helmholtz operator and we developed a GLM theory exclusively for the Helmholtz problem. In particular, we derived a generalised GLM equation in the space of tempered distributions for reconstructing the Jost solutions of the Helmholtz operator. To do so, we had to examine the asymptotic behaviour of the Jost solutions of the Helmholtz operator in terms of the wavenumber. After studying classical inverse scattering methods based on the GLM approach, we continued by studying inversion methods based on reduced order models (ROMs). We started with the inverse Schrödinger scattering problem of retrieving the scattering potential in 1D Schrödinger equation using boundary data. For that reason, we proposed a two-step approach inspired by a previously-published ROM-based method. We presented explicit expressions allowing the exact reconstruction of the ROM-matrices from boundary data and proposed a new data-assimilation approach for approximating the state from these matrices. Given the estimates of the states, the scattering potential is obtained by solving a Lippmann-Schwinger type integral equation. Finally, we combined the traditional FWI method with reduced order models and we proposed a new nonlinear inversion method for the inverse Helmholtz scattering problem. In particular, the input of our misfit functional consisted of the stiffness matrix of the ROM projection. In this case, we studied the well posedness of the nonlinear optimization problem and we derived the optimality condition. We finally compared numerically the ROM based FWI method with the conventional FWI method.