多重网格方法是否具有容错性？双网格情况
Is the Multigrid Method Fault Tolerant? The Two-Grid Case
AI总结：
本文分析了多重网格算法在故障下的行为，发现双网格方法对故障不具容错性，并探讨了恢复收敛速度的必要补救措施。
摘要：
预测的下一代高性能计算机的降级韧性意味着必须考虑随机故障对数值方法的影响。在发生硬故障时，必须决定采取何种补救措施以恢复算法执行。所选措施对方案的性能和特性有重大影响。理想情况下，所得算法应接受与原始确定性变体相同的数学分析。本文旨在分析多重网格算法在故障存在下的行为。多重网格可能是求解大规模线性代数问题的首选方法，这些问题是来自偏微分方程离散化的问题，其在exascale机器上的行为至关重要。算法韧性分析仍处于初级阶段，本文可能是首次提供故障数学模型并分析当前最先进算法在该模型下的行为的尝试。证明了双网格方法对故障不具韧性。然后转向识别恢复到理想无故障方法收敛速度的最小必要补救措施。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
