  In the context of Cosmic Microwave Background data analysis, we study the
solution to the equation that transforms scanning data into a map. As
originally suggested in "messenger" methods for solving linear systems, we
split the noise covariance into uniform and non-uniform parts and adjust their
relative weights during the iterative solution. With simulations, we study mock
instrumental data with different noise properties, and find that this "cooling"
or perturbative approach is particularly effective when there is significant
low-frequency noise in the timestream. In such cases, a conjugate gradient
algorithm applied to this modified system converges faster and to a higher
fidelity solution than the standard conjugate gradient approach. We give an
analytic estimate for the parameter that controls how gradually the linear
system should change during the course of the solution.
