A Galerkin Finite Element Method for the Fractional Calderón Problem

We study a numerical reconstruction strategy for the potential in the fractional Calderón problem from a single partial exterior measurement. The forward model is the fractional Schrödinger 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.

💡 Research Summary

This paper presents a novel numerical reconstruction strategy for the fractional Calderón problem, which aims to recover an unknown potential q within a bounded domain from a single partial exterior measurement. The forward model is governed by the fractional Schrödinger equation, (-∆)^s u + q u = 0 in Ω, with a prescribed exterior Dirichlet datum f and a corresponding measurement of the exterior nonlocal flux g on an open observation set W ⊂ Ω^e.

Motivated by theoretical single-measurement uniqueness results based on unique continuation properties of the fractional Laplacian, the authors develop a two-step Galerkin finite element method combined with Tikhonov regularization. The core idea is to decompose the total solution as u = (extension of f) + u0, where u0 is supported inside Ω. The inverse problem then reduces to first recovering u0 from the preprocessed measurement µ = g - (-∆)^s(extension of f) on W, which constitutes an exterior unique continuation problem involving the compact operator L: L^2(Ω) → L^2(W). This ill-posed step is regularized using a discrete least-squares formulation. Once u0_h is reconstructed, the potential q is obtained via a stabilized least-squares quotient to avoid instability where u is small.

The proposed computational framework cleanly separates the forward solver (a finite element method for the fractional Laplacian on a truncated domain) from the inverse reconstruction step. This separation is advantageous as the discrete observation operator depends only on geometry and can be precomputed. The analysis proves the existence and uniqueness of the discrete reconstructor and establishes conditional convergence under natural consistency assumptions and parameter choice rules. Key theoretical contributions include a priori error estimates for both the reconstructed state and the coefficient, and a total coefficient error bound that combines the discrete reconstruction error with the logarithmic stability inherent to the continuous inverse problem.

Numerical experiments in one and two spatial dimensions demonstrate the method’s practical feasibility. Results show stability with respect to data noise, convergence under mesh refinement, and successful reconstructions of both smooth and discontinuous (e.g., piecewise constant) potentials. The method is compatible with practical quadrature schemes for the integral fractional Laplacian, making it a robust and implementable approach for this challenging nonlocal inverse problem.