A Regularised Total Least Squares Approach for 1D Inverse Scattering

We study the inverse scattering problem for a Schrodinger 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.

Automated summary

In this paper, we study the inverse scattering problem for a Schrodinger operator related to a static wave operator with variable velocity using the GLM integral equation. We assume the presence of noisy scattering data and 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 regularize the problem, we formulate a variational total least squares problem and prove the existence of minimizers under certain regularity assumptions. Finally, we compute the regularized solution of the GLM equation numerically using the total least squares method in a discrete sense.