Is the Multigrid Method Fault Tolerant? The Multilevel Case

Computing at the exascale level is expected to be affected by a significantly higher rate of faults, due to increased component counts as well as power considerations. Therefore, current day numerical algorithms need to be reexamined as to determine if they are fault resilient, and which critical operations need to be safeguarded in order to obtain performance that is close to the ideal fault-free method. In a previous paper, a framework for the analysis of random stationary linear iterations was presented and applied to the two grid method. The present work is concerned with the multigrid algorithm for the solution of linear systems of equations, which is widely used on high performance computing systems. It is shown that the Fault-Prone Multigrid Method is not resilient, unless the prolongation operation is protected. Strategies for fault detection and mitigation as well as protection of the prolongation operation are presented and tested, and a guideline for an optimal choice of parameters is devised.Comment: 25 pages, 10 figure

Is the Multigrid Method Fault Tolerant? The Two-Grid Case

The predicted reduced resiliency of next-generation high performance computers means that it will become necessary to take into account the effects of randomly occurring faults on numerical methods. Further, in the event of a hard fault occurring, a decision has to be made as to what remedial action should be taken in order to resume the execution of the algorithm. The action that is chosen can have a dramatic effect on the performance and characteristics of the scheme. Ideally, the resulting algorithm should be subjected to the same kind of mathematical analysis that was applied to the original, deterministic variant. The purpose of this work is to provide an analysis of the behaviour of the multigrid algorithm in the presence of faults. Multigrid is arguably the method of choice for the solution of large-scale linear algebra problems arising from discretization of partial differential equations and it is of considerable importance to anticipate its behaviour on an exascale machine. The analysis of resilience of algorithms is in its infancy and the current work is perhaps the first to provide a mathematical model for faults and analyse the behaviour of a state-of-the-art algorithm under the model. It is shown that the Two Grid Method fails to be resilient to faults. Attention is then turned to identifying the minimal necessary remedial action required to restore the rate of convergence to that enjoyed by the ideal fault-free method.Comment: 27 pages and 6 figure