Cosmic microwave background (CMB) experiments chase faint signals from the primordial universe by continuously scanning the sky for years. Their cameras are typically read out hundreds of times per second, delivering the so-called time ordered data, which are nothing but the image of the sky unfolded in a much larger domain. *Map-making* aims at folding back these raw data in order to get a representation of the sky signal. This process takes the form of a linear inverse problem that pose serious computational challenges because of the size of the domains involved. In most application the explicit solution is not an option and we have to resort to the preconditioned conjugate gradient method (PCG), which starts from an initial guess for the sky signal and iteratively refine the solution. This process can take hundreds of iterations to converge – if ever – and is typically the most demanding part of CMB data analysis pipelines.

In a recent paper to appear on Astronomy&Astrophysics we tackle this computational challenge by exploiting a novel technique: the two-level preconditioners built using the Arnoldi algorithm. This method is capable of isolating and curing the small part of the problem that slows down the PCG convergence. As a result, we get better convergence in fewer PCG steps. Most notably, the (fairly expensive) precomputation that our method requires can be recycled for many similar map-making problems yielding equally good performances in all of them. This result is extremely valuable for modern CMB data analyses workflows because they typically require large sets of Monte Carlo simulations in which hundreds of similar map-making problems have to be solved.