A numerical method for reconstructing the potential in fractional Calder\'{o}n problem with a single measurement

TL;DR

This paper introduces a numerical approach for reconstructing potentials in fractional Calderón problems using finite differences and Tikhonov regularization, demonstrating stability and accuracy through numerical experiments.

Contribution

The paper presents a novel numerical method combining finite difference discretization and Tikhonov regularization for the fractional Calderón problem with a single measurement.

Findings

Method effectively reconstructs potentials with high accuracy.

Numerical experiments confirm stability and efficiency.

Guidelines for choosing regularization parameters are provided.

Abstract

In this paper, we develop a numerical method for determining the potential in one and two dimensional fractional Calder\'{o}n problems with a single measurement. Finite difference scheme is employed to discretize the fractional Laplacian, and the parameter reconstruction is formulated into a variational problem based on Tikhonov regularization to obtain a stable and accurate solution. Conjugate gradient method is utilized to solve the variational problem. Moreover, we also provide a suggestion to choose the regularization parameter. Numerical experiments are performed to illustrate the efficiency and effectiveness of the developed method and verify the theoretical results.