Self-stabilizing iterative solvers

TL;DR: It is shown how to use the idea of self-stabilization, which originates in the context of distributed control, to make fault-tolerant iterative solvers, and has promise to become a useful tool for constructing resilient solvers more generally.

Abstract: We show how to use the idea of self-stabilization, which originates in the context of distributed control, to make fault-tolerant iterative solvers. Generally, a self-stabilizing system is one that, starting from an arbitrary state (valid or invalid), reaches a valid state within a finite number of steps. This property imbues the system with a natural means of tolerating transient faults. We give two proof-of-concept examples of self-stabilizing iterative linear solvers: one for steepest descent (SD) and one for conjugate gradients (CG). Our self-stabilized versions of SD and CG require small amounts of fault-detection, e.g., we may check only for NaNs and infinities. We test our approach experimentally by analyzing its convergence and overhead for different types and rates of faults. Beyond the specific findings of this paper, we believe self-stabilization has promise to become a useful tool for constructing resilient solvers more generally.