Significance of solutions of the inverse Biot-Savart problem in thick superconductors

TL;DR

This paper investigates the theoretical challenges of reconstructing current distributions in thick superconductors from surface magnetic field measurements, highlighting the ill-posed nature of the inverse Biot-Savart problem and the impact of assumptions on solutions.

Contribution

It demonstrates that the inverse problem is inherently ill-posed for thick superconductors and shows how assumptions influence the solutions, emphasizing fundamental limitations.

Findings

The inverse Biot-Savart problem in thick superconductors is inherently ill-posed.

Additional assumptions significantly affect the reconstructed current distributions.

No models or algorithms can fully overcome the fundamental difficulties of the inverse problem.

Abstract

The evaluation of current distributions in thick superconductors from field profiles near the sample surface is investigated theoretically. A simple model of a cylindrical sample, in which only circular currents are flowing, reduces the inversion to a linear least squares problem, which is analyzed by singular value decomposition. Without additional assumptions about the current distribution (e.g. constant current over the sample thickness), the condition of the problem is very bad, leading to unrealistic results. However, any additional assumption strongly influences the solution and thus renders the solutions again questionable. These difficulties are unfortunately inherent to the inverse Biot-Savart problem in thick superconductors and cannot be avoided by any models or algorithms.