# Variational source conditions for the reconstruction of distributed fluxes

## TL;DR

This paper develops a variational source condition framework for the inverse problem of reconstructing distributed fluxes on inaccessible boundaries, achieving logarithmic convergence rates under low regularity assumptions.

## Contribution

It introduces a novel variational source condition approach for this inverse problem, linking stability estimates with regularization convergence analysis.

## Key findings

- Established a variational source condition for the inverse boundary flux problem.
- Proved logarithmic convergence rates for Tikhonov regularization under low Sobolev regularity.
- Utilized Carleman estimates and complex interpolation in the analysis.

## Abstract

This paper is devoted to the inverse problem of recovering the unknown distributed flux on an inaccessible part of boundary using measurement data on the accessible part. We establish and verify a variational source condition for this inverse problem, leading to a logarithmic-type convergence rate for the corresponding Tikhonov regularization method under a low Sobolev regularity assumption on the distributed flux. Our proof is based on the conditional stability and Carleman estimates together with the complex interpolation theory on a proper Gelfand triple.