The determination of stellar atmospheres is a fundamental problem of astrophysics. In specific, the determination of the magnetic field vector field of the sun's photosphere requires the non-linear fitting of theoretical profiles to observed signals acquired with spectro-polarimeters. In this presentation we briefly explain the physics behind the problem along with a discussion of the Milne-Eddington solution to the radiative transfer equation. Then we cover more in detail the mathematics of Non Linear Least-Square minimization used to obtain the solar atmosphere from a polarized signal, along with the computational approaches to compute the solutions in parallel.