Solving linear equations with messenger-field and conjugate gradients
  techniques - an application to CMB data analysis
  We discuss linear system solvers invoking a messenger-field and compare them
with (preconditioned) conjugate gradients approaches. We show that the
messenger-field techniques correspond to fixed point iterations of an
appropriately preconditioned initial system of linear equations. We then argue
that a conjugate gradient solver applied to the same preconditioned system, or
equivalently a preconditioned conjugate gradient solver using the same
preconditioner and applied to the original system, will in general ensure at
least a comparable and typically better performance in terms of the number of
iterations to convergence and time-to-solution. We illustrate our conclusions
on two common examples drawn from the Cosmic Microwave Background data
analysis: Wiener filtering and map-making. In addition, and contrary to the
standard lore in the CMB field, we show that the performance of the
preconditioned conjugate gradient solver can depend importantly on the starting
vector. This observation seems of particular importance in the cases of
map-making of high signal-to-noise sky maps and therefore should be of
relevance for the next generation of CMB experiments.
