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

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.

References in corpus (8)

• Detection of Gravitational Lensing in the Cosmic Microwave Background
• Efficient Wiener filtering without preconditioning
• Wiener filter reloaded: fast signal reconstruction without preconditioning
• Making maps of Cosmic Microwave Background polarization for B-mode studies: the POLARBEAR example
• Accelerating Cosmic Microwave Background map-making procedure through preconditioning
• Iterative map-making with two-level preconditioning for polarized Cosmic Microwave Background data sets
• Cosmic Microwave Background Mapmaking with a Messenger Field
• Preconditioner-free Wiener filtering with a dense noise matrix

Cited by in corpus (5)

• Efficient Optimal Reconstruction of Linear Fields and Band-powers from Cosmological Data
• Wiener filtering and pure E/B decomposition of CMB maps with anisotropic correlated noise
• Improved Gibbs samplers for Cosmic Microwave Background power spectrum estimation
• Accelerating linear system solvers for time domain component separation of cosmic microwave background data
• Cooling Improves Cosmic Microwave Background Map-Making When Low-Frequency Noise is Large