On the convergence of simulation-based iterative methods for solving singular linear systems

Abstract

We consider the simulation-based solution of linear systems of equations, $Ax=b$, of various types frequently arising in large-scale applications, where $A$ is singular. We show that the convergence properties of iterative solution methods are frequently lost when they are implemented with simulation (e.g., using sample average approximation), as is often done in important classes of large-scale problems. We focus on special cases of algorithms for singular systems, including some arising in least squares problems and approximate dynamic programming, where convergence of the residual sequence $\{Ax_{k}-b\}$ may be obtained, while the sequence of iterates $\{x_{k}\}$ may diverge. For some of these special cases, under additional assumptions, we show that the iterate sequence is guaranteed to converge. For situations where the iterates diverge but the residuals converge to zero, we propose schemes for extracting from the divergent sequence another sequence that converges to a solution of $Ax=b$.