Characterizing the impact of soft errors on iterative methods in scientific computing

  • Authors:

  • Manu Shantharam;Sowmyalatha Srinivasmurthy;Padma Raghavan

  • Affiliations:

  • The Pennsylvania State University, University Park, PA, USA;The Pennsylvania State University, University Park, PA, USA;The Pennsylvania State University, University Park, PA, USA

  • Venue:
  • Proceedings of the international conference on Supercomputing
  • Year:
  • 2011

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Abstract

The increase in on-chip transistor count facilitates achieving higher performance, but at the expense of higher susceptibility to soft errors. In this paper, we characterize the challenges posed by soft errors for large-scale applications representative of workloads on supercomputing systems. Such applications are typically based on the computational solution of partial differential equation models using either explicit or implicit methods. In both cases, the execution time of such applications is typically dominated by the time spent in their underlying sparse matrix vector multiplication kernel (SpMV, t ← A • y). We provide a theoretical analysis of the impact of a single soft error through its propagation by a sequence of sparse matrix vector multiplication operations. Our analysis indicates that a single soft error in some ith component of the vector y can corrupt the entire resultant vector in a relatively short sequence of SpMV operations. Additionally, the propagation pattern corresponds to the sparsity structure of the coefficient matrix A and the magnitude of the error grows non-linearly as(||Ai||2∗)k, after k SpMV operations, where, ||Ai∗||2 is the 2-norm of the ith row of A. We corroborate this analysis with empirical observations on a model heat equation using explicit method and well known sparse matrix systems (matrices from a test suite) for the implicit method using iterative solvers such as CG, PCG and SOR. Our results indicate that explicit schemes will suffer from soft error induced numerical instabilities, thus exacerbating intrinsic stability issues for such methods, that impose constraints on relative time and space step sizes. For implicit schemes, linear solver performance through widely used CG and PCG schemes, degrades by a factor as high as 200x, whereas, a stationary scheme such as SOR is inherently soft error resilient. Our results thus indicate the need for new approaches to achieve soft error resiliency in such methods and a critical evaluation of the tradeoffs among multiple metrics, including, performance, reliability and energy.