A multi-level solver for Gaussian constrained CMB realizations
  We present a multi-level solver for drawing constrained Gaussian realizations
or finding the maximum likelihood estimate of the CMB sky, given noisy sky maps
with partial sky coverage. The method converges substantially faster than
existing Conjugate Gradient (CG) methods for the same problem. For instance,
for the 143 GHz Planck frequency channel, only 3 multi-level W-cycles result in
an absolute error smaller than 1 microKelvin in any pixel. Using 16 CPU cores,
this translates to a computational expense of 6 minutes wall time per
realization, plus 8 minutes wall time for a power spectrum-dependent
precomputation. Each additional W-cycle reduces the error by more than an order
of magnitude, at an additional computational cost of 2 minutes. For comparison,
we have never been able to achieve similar absolute convergence with
conventional CG methods for this high signal-to-noise data set, even after
thousands of CG iterations and employing expensive preconditioners. The solver
is part of the Commander 2 code, which is available with an open source license
at http://commander.bitbucket.org/.
