Monochromatic reconstruction algorithms for two-dimensional multi-channel inverse problems

Abstract

We consider two inverse problems for the multi-channel two-dimensional Schrödinger equation at fixed positive energy, i.e. the equation $-\Delta \psi +V(x)\psi =E\psi$ at fixed positive $E$, where $V$ is a matrix-valued potential. The first is the Gel'fand inverse problem on a bounded domain $D$ at fixed energy and the second is the inverse fixed-energy scattering problem on the whole plane ${\mathbb{R}}^2$. We present in this paper two algorithms which give efficient approximate solutions to these problems: in particular, in both cases we show that the potential $V$ is reconstructed with Lipschitz stability by these algorithms up to $O(E^{-(m-2)/2})$ in the uniform norm as $E\to +\infty$, under the assumptions that $V$ is $m$-times differentiable in $L^1$, for $m\ge 3$, and has sufficient boundary decay.

References in corpus (3)

• The d-bar approach to approximate inverse scattering at fixed energy in three dimensions
• A global stability estimate for the Gel'fand-Calderon inverse problem in two dimensions
• Global uniqueness and reconstruction for the multi-channel Gel'fand-Calderón inverse problem in two dimensions

Cited by in corpus (5)

• Convexification of Restricted Dirichlet-to-Neumann Map
• Numerical approximation of the potential in the two-dimensional inverse scattering problem
• Recovery of $L^p$-potential in the plane
• Positive-energy D-bar method for acoustic tomography: a computational study
• Energy and regularity dependent stability estimates for near-field inverse scattering in multidimensions