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Recently, a class of algorithms combining classical fixed point iterationswith repeated random sparsification of approximate solution vectors has beensuccessfully applied to eigenproblems with matrices as large as $10^{108}\times 10^{108}$. So far, a complete mathematical explanation for their successhas proven elusive. Additionally, the methods have not been extended to linearsystem solves.  In this paper we propose a new scheme based on repeated random sparsificationthat is capable of solving linear systems in extremely high dimensions. Weprovide a complete mathematical analysis of this new algorithm. Our analysisestablishes a faster-than-Monte Carlo convergence rate and justifies use of thescheme even when the solution vector itself is too large to store.

Randomly sparsified Richardson iteration is really fast