CMB data analysis commonly involves solution of large systems of equations. Finding appropriate solvers for these systems is a subject of ongoing research and one of the aims of B3DCMB project. The fact that, in general, the system matrix cannot be assembled, suggests using the Krylov subspace iterative methods as they only require the action of the matrix on a given vector (and no other manipulations with the matrix). The conjugate gradient method with a block-diagonal preconditioner is among the most popular and succesful solvers in CMB data analysis. Recently, the PCG solver mentioned above was further improved by considering so-called two level preconditioner, which enriches the block-diagonal preconditioner by the approximation of the slowest eigenvectors, which often harm the convergence of the PCG solver; see also the post on this blog from March 19, 2018.
In a recent paper, we investigate another new approach, messenger-field, proposed by Elsner & Wandelt in the paper entitled Efficient Wiener filtering without preconditioning. We showed that the technique proposed therein correspond to fixed point iterations of an appropriately preconditioned original system and we proved (and illustrated numerically) that significant gains are possible if the conjugate gradient method is used instead of the fixed point iterations. This motivates us to look for better preconditioning techniques as the most promising way to address the problem of solving large systems in the CMB data analysis. Additionally, the work on understanding the messenger-field approach motivated us to pay attention to the choice of the initial vector. This is often ommited from considerations, however, as our numerical experiments suggest, a proper (yet very cheap) choice can bring significant savings in the solution time.