On Soft Errors in the Conjugate Gradient Method: Sensitivity and Robust Numerical Detection

Abstract

The conjugate gradient (CG) method is the most widely used iterative scheme for the solution of large sparse systems of linear equations when the matrix is symmetric positive definite. Although more than 60 years old, it is still a serious candidate for extreme-scale computations on large computing platforms. On the technological side, the continuous shrinking of transistor geometry and the increasing complexity of these devices affect dramatically their sensitivity to natural radiation and thus diminish their reliability. One of the most common effects produced by natural radiation is the single event upset which consists in a bit-flip in a memory cell producing unexpected results at the application level. Consequently, future extreme-scale computing facilities will be more prone to errors of any kind, including bit-flips, during their calculations. These numerical and technological observations are the main motivations for this work, where we first investigate through extensive numerical experiments the sensitivity of CG to bit-flips in its main computationally intensive kernels, namely the matrix-vector product and the preconditioner application. We further propose numerical criteria to detect the occurrence of such soft errors and assess their robustness through extensive numerical experiments.