Preconditioner-free Wiener filtering with a dense noise matrix

This work extends the Elsner & Wandelt (2013) iterative method for efficient, preconditioner-free Wiener filtering to cases in which the noise covariance matrix is dense, but can be decomposed into a sum whose parts are sparse in convenient bases. The new method, which uses multiple messenger fields, reproduces Wiener filter solutions for test problems, and we apply it to a case beyond the reach of the Elsner & Wandelt (2013) method. We compute the Wiener filter solution for a simulated Cosmic Microwave Background map that contains spatially-varying, uncorrelated noise, isotropic $1/f$ noise, and large-scale horizontal stripes (like those caused by the atmospheric noise). We discuss simple extensions that can filter contaminated modes or inverse-noise filter the data. These techniques help to address complications in the noise properties of maps from current and future generations of ground-based Microwave Background experiments, like Advanced ACTPol, Simons Observatory, and CMB-S4.