# The detection and correction of silent errors in pipelined Krylov subspace methods

## TL;DR

This paper introduces a method to detect and correct silent errors in large-scale computations using pipelined Krylov subspace methods.

## Contribution

A novel algorithm-based approach for silent error detection in pipelined Krylov subspace methods is proposed.

## Key findings

- Silent errors can be detected by monitoring deviations in quantities that should be equal in exact arithmetic.
- A fault-tolerant variant of Pipe-PR-CG was developed and tested successfully.
- The proposed method allows dynamic adaptation of detection criteria for improved reliability.

## Abstract

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.