  We propose a new approach to the numerical solution of radiative transfer
equations with certified a posteriori error bounds. A key role is played by
stable Petrov--Galerkin type variational formulations of parametric transport
equations and corresponding radiative transfer equations. This allows us to
formulate an iteration in a suitable, infinite dimensional function space that
is guaranteed to converge with a fixed error reduction per step. The numerical
scheme is then based on approximately realizing this iteration within
dynamically updated accuracy tolerances that still ensure convergence to the
exact solution. To advance this iteration two operations need to be performed
within suitably tightened accuracy tolerances. First, the global scattering
operator needs to be approximately applied to the current iterate within a
tolerance comparable to the current accuracy level. Second, parameter dependent
linear transport equations need to be solved, again at the required accuracy of
the iteration. To ensure that the stage dependent error tolerances are met, one
has to employ rigorous a posteriori error bounds which, in our case, rest on a
Discontinuous Petrov--Galerkin (DPG) scheme. These a posteriori bounds are not
only crucial for guaranteeing the convergence of the perturbed iteration but
are also used to generate adapted parameter dependent spatial meshes. This
turns out to significantly reduce overall computational complexity. Since the
global operator is only applied, we avoid the need to solve linear systems with
densely populated matrices. Moreover, the approximate application of the global
scatterer accelerated through low-rank approximation and matrix compression
techniques. The theoretical findings are illustrated and complemented by
numerical experiments with non-trivial scattering kernels.
