Solving linear equations with messenger-field and conjugate gradients techniques - an application to CMB data analysis
This work clarifies the relationship between two solver families for CMB practitioners, but the findings are incremental and confirm known numerical linear algebra principles.
The authors compare messenger-field and conjugate gradient solvers for linear systems in CMB data analysis, showing that conjugate gradients generally outperform messenger-field in iteration count and time-to-solution, and that preconditioned conjugate gradient performance depends on the starting vector, especially for high signal-to-noise maps.
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.