A parallel two-level preconditioner for Cosmic Microwave Background map-making

Generalized least square problems with nondiagonal weights arise frequently in an estimation of two dimensional images from data of cosmological as well as astro- or geo- physical observations. As the observational data sets keep growing at Moore's rate, with their volumes exceeding tens and hundreds billions of samples, the need for fast and efficiently parallelizable iterative solvers is generally recognized. In this work we propose a new iterative algorithm for solving generalized least square systems with weights given by a blockdiagonal matrix with Toeplitz blocks. Such cases are physically well motivated and correspond to measurement noise being piece-wise stationary - a common occurrence in many actual observations. Our iterative algorithm is based on the conjugate gradient method and includes a parallel two-level preconditioner (2lvl-PCG) constructed from a limited number of sparse vectors estimated from the coefficients of the initial linear system. Our prototypical application is the map-making problem in the Cosmic Microwave Background data analysis. We show experimentally that our parallel implementation of 2lvl-PCG outperforms by a factor of up to 6 the standard one-level PCG in terms of both the convergence rate and the time to solution on up to 12, 228 cores of NERSC's Cray XE6 (Hopper) system displaying nearly perfect strong and weak scaling behavior in this regime.