  We propose extensions and improvements of the statistical analysis of
distributed multipoles (SADM) algorithm put forth by Chipot et al. in [6] for
the derivation of distributed atomic multipoles from the quantum-mechanical
electrostatic potential. The method is mathematically extended to general
least-squares problems and provides an alternative approximation method in
cases where the original least-squares problem is computationally not
tractable, either because of its ill-posedness or its high-dimensionality. The
solution is approximated employing a Monte Carlo method that takes the average
of a random variable defined as the solutions of random small least-squares
problems drawn as subsystems of the original problem. The conditions that
ensure convergence and consistency of the method are discussed, along with an
analysis of the computational cost in specific instances.
