Multi-resolution Bayesian CMB component separation through Wiener-filtering with a pseudo-inverse preconditioner

We present a Bayesian model for multi-resolution CMB component separation based on Wiener filtering and/or computation of constrained realizations, extending a previously developed framework. We also develop an efficient solver for the corresponding linear system for the associated signal amplitudes. The core of this new solver is an efficient preconditioner based on the pseudo-inverse of the coefficient matrix of the linear system. In the full sky coverage case, the method gives a speed-up of 2--3x in compute time compared to a simple diagonal preconditioner, and it is easier to implement in terms of practical computer code. In the case where a mask is applied and prior-driven constrained realization is sought within the mask, this is the first time full convergence has been achieved at the full resolution of the Planck dataset. Prototype benchmark code is available at https://github.com/dagss/cmbcr .