Linearization versus preconditioning: Which approach is best for solving multilevel transfer problems?

We present a critical analysis of linearization and preconditioning, the two most used approaches proposed for achieving the required linearity in the iterative solution of the multilevel transfer problem. By distinguishing from the outset between the response of the radiation held to the source function and opacity perturbations, we are able to demonstrate that if the linearization strategy, on which the local approximate Lambda-operator option of the multilevel transfer code MULTI is based, is applied neglecting the terms coming from the response of the radiation held to the opacity perturbations, one then recovers the same equations obtained using the preconditioning technique of Rybicki & Hummer. It is also shown that if this preconditioning technique is applied taking into account the response of the radiation field to both the source function and opacity variations, one then ends up with the same equations found via the linearization method. Thus these two approaches to the numerical solution of the multilevel transfer problem turn out to be essentially the same, because similar equations are obtained if the same information is taken into account. Finally, it is pointed out that, if one wishes to guarantee positivity for the atomic level populations, it is necessary to neglect the terms associated with the response of the radiation field to the opacity perturbations. Neglecting such terms does not deteriorate the convergence rate of multilevel transfer methods that make use of a local approximate operator.