The Detection and Correction of Silent Errors in Pipelined Krylov Subspace Methods

As computational machines become larger and more complex, the probability of hardware failure rises. ``Silent errors'', or bit flips, may not be immediately apparent but can cause detrimental effects to algorithm behavior. In this work, we examine an algorithm-based approach to silent error detection in the context of pipelined Krylov subspace methods, in particular, Pipe-PR-CG, for the solution of linear systems. Our approach is based on using finite precision error analysis to bound the differences between quantities which should be equal in exact arithmetic. By monitoring select quantities during the iteration, we can detect when these bounds are violated, which indicates that a silent error has occurred. We use this approach to develop a fault-tolerant variant and also suggest a strategy for dynamically adapting the detection criteria. Our numerical experiments demonstrate the effectiveness of our approach.