Solving linear equations with messenger-field and conjugate gradients techniques - an application to CMB data analysis

TL;DR

This paper compares messenger-field and conjugate gradient methods for solving linear systems in CMB data analysis, showing conjugate gradients often outperform messenger-field techniques, especially with good initial guesses.

Contribution

It demonstrates that preconditioned conjugate gradient methods generally outperform messenger-field techniques and highlights the importance of initial vectors in CMB map-making.

Findings

Conjugate gradient methods typically require fewer iterations.

Performance depends significantly on the initial vector.

Conjugate gradients can be more efficient for high signal-to-noise data.

Abstract

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.