A finite element exterior-to-interior reconstruction in the fractional conductivity inverse problem
Abstract
We develop a regularized finite element method for the fractional conductivity equation, recovering the conductivity on an exterior observation region from noisy Dirichlet-to-Neumann measurements and using this reconstruction to determine the interior conductivity. Mesh-localized inputs and mass-lumped Tikhonov regularization discretize the exterior determination principle of Covi--Railo--Zimmermann (2026, Calc. Var. PDE). We prove exterior L2- and L∞-convergence with error estimates, including the logarithmic correction at the critical Hölder exponent. The exterior reconstruction enables the Liouville reduction to the Calderón problem for the fractional Schrödinger equation. Separated measurements recover the interior potential, after which overlapping measurements determine the exterior potential tail. Together with the recovered exterior conductivity, the tail supplies the exterior Cauchy data for a Tikhonov-regularized finite element least-squares continuation. We prove Hs-convergence of the transformed conductivity and L2-convergence of the interior conductivity, together with logarithmic state-bias and conductivity error estimates for full observation spaces. The admissible reconstructions remain uniformly positive and bounded. We present one- and two-dimensional numerical experiments illustrating the accuracy of the exterior and interior reconstructions and their sensitivity to measurement noise and discretization.
Related
- Finite element exterior calculus for spectra and pseudospectra of advection-diffusion of differential forms
- HDG methods in finite element exterior calculus
- An interior penalty nonconforming finite element method for strain gradient elasticity
- The Stokes Operator on Exterior Domains in Homogeneous Weighted Function Spaces:From Weak Theory to ℋ∞-calculus to Fractional Domains
- Rotating bases for finite element exterior calculus: closed-form change of basis under vertex permutations
BibTeX
@misc{dwivedi2026finite,
title = {{A finite element exterior-to-interior reconstruction in the fractional conductivity inverse problem}},
author = {Mukul Dwivedi and Jesse Railo and Andreas Rupp},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10484},
eprint = {2610.10484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10484},
note = {CC0},
}
RIS
TY - UNPB
TI - A finite element exterior-to-interior reconstruction in the fractional conductivity inverse problem
AU - Mukul Dwivedi
AU - Jesse Railo
AU - Andreas Rupp
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10484
UR - https://arxiv.org/abs/2610.10484
AB - We develop a regularized finite element method for the fractional conductivity equation, recovering the conductivity on an exterior observation region from noisy Dirichlet-to-Neumann measurements and using this reconstruction to determine the interior conductivity. Mesh-localized inputs and mass-lumped Tikhonov regularization discretize the exterior determination principle of Covi--Railo--Zimmermann (2026, Calc. Var. PDE). We prove exterior L^2- and L^∞-convergence with error estimates, including the logarithmic correction at the critical Hölder exponent. The exterior reconstruction enables the Liouville reduction to the Calderón problem for the fractional Schrödinger equation. Separated measurements recover the interior potential, after which overlapping measurements determine the exterior potential tail. Together with the recovered exterior conductivity, the tail supplies the exterior Cauchy data for a Tikhonov-regularized finite element least-squares continuation. We prove H^s-convergence of the transformed conductivity and L^2-convergence of the interior conductivity, together with logarithmic state-bias and conductivity error estimates for full observation spaces. The admissible reconstructions remain uniformly positive and bounded. We present one- and two-dimensional numerical experiments illustrating the accuracy of the exterior and interior reconstructions and their sensitivity to measurement noise and discretization.
ER -