Regularised synthesis of the magnetic field using the L-curve approach

Abstract

Ill-posed problems occur frequently in science and engineering. Regularisation methods are used for computing stable solutions to the ill-posed problems. The purpose of regularisation is to incorporate more information about the desired solution in order to stabilise the problem and find a useful and stable solution. The most common and well-known form of regularisation is that of Tikhonov in which the regularised solution is searched as a minimizer of the weighted combination of the residual norm and a side constraint. The weight given to the minimisation of the side constraint is controlled by the regularisation parameter. Thus, the regularisation parameter is an important factor that controls the quality of the regularised solution. A good regularisation parameter should fairly balance the perturbation error and the regularisation error in the regularised solution. There are several methods for choosing regularisation parameter. In the paper we demonstrate another method: the L-curve criterion for choosing the regularisation parameter. The L-curve is a plot of the seminorm of the regularised solution versus the corresponding residual norm thus expressing the compromise between minimisation of these two terms. The L-curve method efficiency is demonstrated on examples of synthesis of magnetic field of different kind.