Positive-energy D-bar method for acoustic tomography: a computational study

TL;DR

This paper introduces a computational D-bar method for acoustic tomography that effectively reconstructs potentials at positive energies, demonstrating robustness and accuracy through numerical simulations.

Contribution

It develops a new D-bar based reconstruction technique that operates without exceptional points for positive energy acoustic tomography.

Findings

Method works well for energies between 10^{-5} and 5.

Reconstruction accuracy improves at smaller energy values.

Numerical tests confirm robustness against noise.

Abstract

A new computational method for reconstructing a potential from the Dirichlet-to-Neumann map at positive energy is developed. The method is based on D-bar techniques and it works in absence of exceptional points -- in particular, if the potential is small enough compared to the energy. Numerical tests reveal exceptional points for perturbed, radial potentials. Reconstructions for several potentials are computed using simulated Dirichlet-to-Neumann maps with and without added noise. The new reconstruction method is shown to work well for energy values between $10^{-5}$ and $5$, smaller values giving better results.