Tokamak Plasma Boundary Reconstruction Using Toroidal Harmonics and an Optimal Control Method

This paper proposes a new fast and stable algorithm for the reconstruction of the plasma boundary from discrete magnetic measurements taken at several locations surrounding the vacuum vessel. The resolution of this inverse problem takes two steps. In the first one, we transform the set of measurements into Cauchy conditions on a fixed contour Γ_O close to the measurement points. This is done by least-squares fitting a truncated series of toroidal harmonics functions to the measurements. The second step consists in solving a Cauchy problem for the elliptic equation satisfied by the flux in the vacuum and for the overdetermined boundary conditions on Γ_O previously obtained with the help of toroidal harmonics. It is reformulated as an optimal control problem on a fixed annular domain of external boundary Γ_O and fictitious inner boundary Γ_I. A regularized Kohn-Vogelius cost function, which depends on the value of the flux on Γ_I, measures the discrepancy between the solution to the equation satisfied by the flux obtained using Dirichlet conditions on Γ_O and the one obtained using Neumann conditions. This function is minimized. The method presented here has led to the development of software, called VacTH-KV, which enables plasma boundary reconstruction in any tokamak.