Reconstruction and stability for piecewise smooth potentials in the plane

TL;DR

This paper demonstrates the recovery of piecewise smooth potentials with jump discontinuities in the plane from boundary measurements and scattering data, providing stability estimates for the reconstruction process.

Contribution

It introduces a method to recover discontinuous potentials using Bukhgeim's approach and combines it with existing formulas for fixed energy scattering data.

Findings

Potentials with jump discontinuities can be reconstructed from the Dirichlet-to-Neumann map.

The method enables recovery from scattering amplitude at fixed energy.

Provides stability estimates for approximate knowledge of discontinuity locations.

Abstract

We show that potentials with jump discontinuities can be recovered from the Dirichlet-to-Neumann map using Bukhgeim's method. Combining with known formulas, this enables the recovery from the scattering amplitude at a fixed energy. We also provide a priori stability estimates for reconstruction from the Dirichlet-to-Neumann map as well as from the scattering amplitudegiven an approximate knowledge of the location of the discontinuities.