Abstract
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.
Highlights
In many scientific, medical and industrial problems, one has to retrieve unknown coefficients of a governing differential equation (PDE) from measurements of its solution
It is well known that the Schrödinger differential equation can be reduced to the following Schrödinger integral equations at ±∞
Such Volterra-type integral equations can be derived using the variation of constants and we refer to [20] for a discussion about the existence and uniqueness of solutions of these integral equations
Summary
Medical and industrial problems, one has to retrieve unknown coefficients of a governing differential equation (PDE) from (partial) measurements of its solution. This way, properties of materials can be studied in a medium that we do not have direct physical access to. For example, a well-known problem is estimating the elastic parameters of the subsurface from surface measurements. The governing PDE is a wave equation, and the measurements consist of a trace of its solution on the boundary of the domain. We focus on the inverse problem for the 1D static wave/Helmholtz equation o d n 2 d
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