A divergence-free global least-squares approximation method for magnetic field interpolation

Magnetic field interpolation is a major bottleneck in transport computations. To ease this problem, we present an accurate and efficient divergence-free global least-squares (DivFree-GLS) interpolation method for a given set of N_{train} magnetic field point values. The method is mesh-free, the distribution of the training set can be arbitrary, and based on a data-driven reconstruction of the magnetic field using a global expansion of basis functions. The M_{terms} << N_{train} expansion coefficients are obtained by minimizing the cost function of the L2 norm of the difference between the ground truth and the approximate magnetic field and the divergence free condition. An exponential decay of the approximation error as function of M_{terms} is observed and compared with the less favorable algebraic decay of the local spline method with N_{train}. In addition, the DivFree-GLS method exhibits a significant reduction of the computational complexity, compared to local splines, while maintaining a small error in the divergence, even in the presence of magnetic islands and stochasticity. Applications of the DivFree-GLS method to the computation of Poincaré sections using magnetic field data obtained with the M3D-C1 and NIMROD MHD codes are presented and compared with local methods currently in use.