Variational regularization method of solving the Cauchy problem for Laplace's equation: Innovation of the Grad—Shafranov (GS) reconstruction

The simplified linear model of Grad—Shafranov (GS) reconstruction can be reformulated into an inverse boundary value problem of Laplace's equation. Therefore, in this paper we focus on the method of solving the inverse boundary value problem of Laplace's equation. In the first place, the variational regularization method is used to deal with the ill-posedness of the Cauchy problem for Laplace's equation. Then, the ‘L-Curve’ principle is suggested to be adopted in choosing the optimal regularization parameter. Finally, a numerical experiment is implemented with a section of Neumann and Dirichlet boundary conditions with observation errors. The results well converge to the exact solution of the problem, which proves the efficiency and robustness of the proposed method. When the order of observation error δ is 10⁻¹, the order of the approximate result error can reach 10⁻³. (geophysics, astronomy, and astrophysics)