Numerical solutions to linear transfer problems of polarized radiation

Abstract

Context. A relevant class of radiative transfer problems for polarized radiation is linear, or can be linearized, and can thus be reframed as linear systems once discretized. In this context, depending on the considered physical models, there are both highly coupled and computationally expensive problems, for which state-of-the-art iterative methods struggle to converge, and lightweight ones, for which solutions can be obtained efficiently. Aims. This work aims to exploit lightweight physical models as preconditioners for iterative solution strategies to obtain accurate and fast solutions for more complex problems. Methods. We considered a highly coupled linear transfer problem for polarized radiation, which we solved iteratively using a matrix-free generalized minimal residual (GMRES) method. Different preconditioners and initial guesses, designed in a physics-based framework, are proposed and analyzed. The action of preconditioners was also computed by applying GMRES. The overall approach thus consists of two nested GMRES iterations, one for the original problem and one for its lightweight version. As a benchmark, we considered the modeling of the intensity and polarization of the solar Ca I 4227 Å line, the Sr II 4077 Å line, and the Mg II h&k lines in a semi-empirical 1D atmospheric model, accounting for partial frequency redistribution effects in scattering processes and considering a general angle-dependent treatment. Results. Numerical experiments show that using tailored preconditioners based on simplified models of the considered problem has a noticeable impact, reducing the number of iterations to convergence by a factor of 10–20. Conclusions. By designing efficient preconditioners in a physics-based context, it is possible to significantly improve the convergence of iterative processes, obtaining fast and accurate numerical solutions to the considered problems. The presented approach is general, requiring only the selection of an appropriate lightweight model, and can be applied to a larger class of radiative transfer problems in combination with arbitrary iterative procedures.