A Galerkin Finite Element Method for the Fractional Calder\'on Problem

TL;DR

This paper develops a Galerkin finite element approach for numerically reconstructing potentials in the fractional Calderón problem, demonstrating stability and convergence through theoretical analysis and numerical experiments.

Contribution

It introduces a stabilized Galerkin--Tikhonov method for potential reconstruction from a single measurement, with proven existence, uniqueness, and error estimates.

Findings

The method achieves stable reconstructions in 1D and 2D.

Theoretical error bounds are established for the inverse problem.

Numerical experiments confirm robustness against noise and ability to recover smooth and discontinuous potentials.

Abstract

We study a numerical reconstruction strategy for the potential in the fractional Calder\'on problem from a single partial exterior measurement. The forward model is the fractional Schr\"odinger equation in a bounded domain, with prescribed exterior Dirichlet datum and corresponding measurement of the exterior flux in an open observation set. Motivated by single-measurement uniqueness results based on unique continuation \cite{ghosh2020uniqueness}, we propose a decomposition strategy and a Galerkin--Tikhonov method to recover the potential by a stabilized least-squares quotient in a dedicated coefficient space. We prove the existence and uniqueness of the discrete reconstructor and establish conditional convergence under natural consistency and parameter choice assumptions. We further derive {\it a priori} error estimates for the reconstructed state and for the coefficient reconstruction, and combine the latter with logarithmic stability for the continuous inverse problem to obtain a total coefficient error bound. The framework cleanly separates the forward solver from the inverse reconstruction step and is compatible with practical truncation and quadrature schemes for the integral fractional Laplacian. Numerical experiments in one and two space dimensions illustrate stability with respect to noise and demonstrate reconstructions of both smooth and discontinuous potentials.