Optimal map-making with singularities

In this work, we investigate the optimal map-making technique for the linear system $\bm{d}=\bm{A}\bm{x}+\bm{n}$ while carefully taking into account singularities that may come from either the covariance matrix $\bm{C} = \<\bm{n}\bm{n}^t\>$ or the main matrix $\bm{A}$. We first describe the general optimal solution, which is quite complex, and then use the modified pseudo inverse to create a near-optimal solution, which is simple, robust, and can significantly alleviate the unwanted noise amplification during map-making. The effectiveness of the nearly optimal solution is then compared to that of the naive co-adding solution and the standard pseudo inverse solution, showing noticeable improvements. Interestingly, all one needs to get the near-optimal solution with singularity is just a tiny change to the traditional optimal solution that is designed for the case without singularity.