  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.
