The paper presents a numerical method for determining the potential in one and two-dimensional fractional Calderón problems with a single measurement. The key highlights are:
A 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. A suggestion is provided to choose the regularization parameter.
Numerical experiments are performed to illustrate the efficiency and effectiveness of the developed method and verify the theoretical results.
For the one-dimensional case, a logarithmic stability estimate is derived for the potential reconstruction. A similar result is obtained for the two-dimensional case.
The numerical schemes leverage the Toeplitz matrix structure for efficient computations using fast Fourier transform techniques.
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