A Backward/Forward Recovery Approach for the Preconditioned Conjugate Gradient Algorithm

Several recent papers have introduced a periodic verification mechanism to detect silent errors in iterative solvers. Chen [PPoPP'13, pp. 167--176] has shown how to combine such a verification mechanism (a stability test checking the orthogonality of two vectors and recomputing the residual) with checkpointing: the idea is to verify every $d$ iterations, and to checkpoint every $c \times d$ iterations. When a silent error is detected by the verification mechanism, one can rollback to and re-execute from the last checkpoint. In this paper, we also propose to combine checkpointing and verification, but we use algorithm-based fault tolerance (ABFT) rather than stability tests. ABFT can be used for error detection, but also for error detection and correction, allowing a forward recovery (and no rollback nor re-execution) when a single error is detected. We introduce an abstract performance model to compute the performance of all schemes, and we instantiate it using the preconditioned conjugate gradient algorithm. Finally, we validate our new approach through a set of simulations.