Recently, a class of algorithms combining classical fixed point iterations
with repeated random sparsification of approximate solution vectors has been
successfully applied to eigenproblems with matrices as large as 10108×10108. So far, a complete mathematical explanation for their success
has proven elusive. Additionally, the methods have not been extended to linear
system solves.
In this paper we propose a new scheme based on repeated random sparsification
that is capable of solving linear systems in extremely high dimensions. We
provide a complete mathematical analysis of this new algorithm. Our analysis
establishes a faster-than-Monte Carlo convergence rate and justifies use of the
scheme even when the solution vector itself is too large to store.Comment: 27 pages, 2 figure