Stabilization of Stochastic Iterative Methods for Singular and Nearly Singular Linear Systems

We consider linear systems of equations, Ax = b, with an emphasis on the case where A is singular. Under certain conditions, necessary as well as sufficient, linear deterministic iterative methods generate sequences {xk} that converge to a solution, as long as there exists at least one solution. This convergence property can be impaired when these methods are implemented with stochastic simulation, as is often done in important classes of large-scale problems. We introduce additional conditions and novel algorithmic stabilization schemes under which {xk} converges to a solution when A is singular, and may also be used with substantial benefit when A is nearly singular. Report LIDS-P-2878, December 2011 (Revised November 2012) Massachusetts Institute of Technology, Cambridge, MA Laboratory for Information and Decision Systems