A data-driven approach to solving a 1D inverse scattering problem

In this paper, we extend a recently proposed approach for inverse scattering with Neumann boundary conditions [Druskin et al., Inverse Probl. 37, 075003 (2021)] to the 1D Schrödinger equation with impedance (Robin) boundary conditions. This method approaches inverse scattering in two steps: first, to extract a reduced order model (ROM) directly from the data and, subsequently, to extract the scattering potential from the ROM. We also propose a novel data-assimilation (DA) inversion method based on the ROM approach, thereby avoiding the need for a Lanczos-orthogonalization (LO) step. Furthermore, we present a detailed numerical study and A comparison of the accuracy and stability of the DA and LO methods

A data-driven approach to solving a 1D inverse scattering problem

In this paper, we extend a recently proposed approach for inverse scattering with Neumann boundary conditions [Druskin et al., Inverse Probl. 37, 075003 (2021)] to the 1D Schrödinger equation with impedance (Robin) boundary conditions. This method approaches inverse scattering in two steps: first, to extract a reduced order model (ROM) directly from the data and, subsequently, to extract the scattering potential from the ROM. We also propose a novel data-assimilation (DA) inversion method based on the ROM approach, thereby avoiding the need for a Lanczos-orthogonalization (LO) step. Furthermore, we present a detailed numerical study and A comparison of the accuracy and stability of the DA and LO methods

A regularised total least squares approach for 1D inverse scattering

We study the inverse scattering problem for a Schrödinger operator related to a static wave operator with variable velocity, using the GLM (Gelfand–Levitan–Marchenko) integral equation. We assume to have noisy scattering data, and we derive a stability estimate for the error of the solution of the GLM integral equation by showing the invertibility of the GLM operator between suitable function spaces. To regularise the problem, we formulate a variational total least squares problem, and we show that, under certain regularity assumptions, the optimisation problem admits minimisers. Finally, we compute numerically the regularised solution of the GLM equation using the total least squares method in a discrete sense

A distributional Gelfand–Levitan–Marchenko equation for the Helmholtz scattering problem on the line

We study an inverse scattering problem for the Helmholtz equation on the whole line. The goal of this paper is to obtain a Gelfand–Levitan–Marchenko (GLM)-type equation for the Jost solution that corresponds to the 1D Helmholtz differential operator. We assume for simplicity that the refraction index is of compact support. Using the asymptotic behavior of the Jost solutions with respect to the wave-number, we derive a generalized Povzner–Levitan representation in the space of tempered distributions. Then, we apply the Fourier transform on the scattering relation that describes the solutions of the Helmholtz scattering problem and we derive a generalized GLM equation. Finally, we discuss the possible application of this new generalized GLM equation to the inverse medium problem

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

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

A Regularised Total Least Squares Approach for 1D Inverse Scattering

We study the inverse scattering problem for a Schrödinger operator related to a static wave operator with variable velocity, using the GLM (Gelfand–Levitan–Marchenko) integral equation. We assume to have noisy scattering data, and we derive a stability estimate for the error of the solution of the GLM integral equation by showing the invertibility of the GLM operator between suitable function spaces. To regularise the problem, we formulate a variational total least squares problem, and we show that, under certain regularity assumptions, the optimisation problem admits minimisers. Finally, we compute numerically the regularised solution of the GLM equation using the total least squares method in a discrete sense

A Regularised Total Least Squares Approach for 1D Inverse Scattering

We study the inverse scattering problem for a Schrödinger operator related to a static wave operator with variable velocity, using the GLM (Gelfand–Levitan–Marchenko) integral equation. We assume to have noisy scattering data, and we derive a stability estimate for the error of the solution of the GLM integral equation by showing the invertibility of the GLM operator between suitable function spaces. To regularise the problem, we formulate a variational total least squares problem, and we show that, under certain regularity assumptions, the optimisation problem admits minimisers. Finally, we compute numerically the regularised solution of the GLM equation using the total least squares method in a discrete sense