Inverse problems for elliptic operators with discrete data and sources: an electromagnetic example

This paper presents the formalism required to describe mathematically the inversion problem for the Grad Shafranov´s equation in toroidal geometry. An algorithm based on the Singular Value Decomposition (SVD) is described capable of finding the unique solution of minimal norm of a least squares problem and to define the optimum number of detectors and sources points in order to maximize the accuracy of the solution and to minimize the cost of the measurement system (number of detectors). An example is presented of an application to Tokamak nuclear fusion experiments. The inversion procedure is applied to the reconstruction of the equilibrium plasma current distribution. The detectors consist of a discrete set of pick-up coils placed all around the vessel measuring the magnetic induction (flux density) and a discrete set of loops measuring the magnetic flux