Randomly sparsified Richardson iteration: A dimension‐independent sparse linear solver

Abstract

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 . So far, a complete mathematical explanation for this success has proven elusive.The family of methods has not yet been extended to the important case of linear system solves. In this paper, we propose a new scheme based on repeated random sparsification that is capable of solving sparse linear systems in arbitrarily 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 is too large to store as a dense vector.