Emrullah Fatih Yetkin

On rounding error resilience, maximal attainable accuracy and parallel performance of the pipelined Conjugate Gradients method for large-scale linear systems in PETSc

Pipelined Krylov solvers typically display better strong scaling compared to standard Krylov methods for large linear systems. The synchronization bottleneck is mitigated by overlapping time-consuming global communications with computations. To achieve this hiding of communication, pipelined methods feature additional recurrence relations on auxiliary variables. This paper analyzes why rounding error effects have a significantly larger impact on the accuracy of pipelined algorithms. An algebraic model for the accumulation of rounding errors in the (pipelined) CG algorithm is derived. Furthermore, an automated residual replacement strategy is proposed to reduce the effect of rounding errors on the final solution. MPI parallel performance tests implemented in PETSc on an Intel Xeon X5660 cluster show that the pipelined CG method with automated residual replacement is more resilient to rounding errors while maintaining the efficient parallel performance obtained by pipelining.

Hard Faults and Soft-Errors: Possible Numerical Remedies in Linear Algebra Solvers

On future large-scale systems, the mean time between failures (MTBF) of the system is expected to decrease so that many faults could occur during the solution of large problems. Consequently, it becomes critical to design parallel numerical linear algebra kernels that can survive faults. In that framework, we investigate the relevance of approaches relying on numerical techniques, which might be combined with more classical techniques for real large-scale parallel implementations. Our main objective is to provide robust resilient schemes so that the solver may keep converging in the presence of the hard fault without restarting the calculation from scratch. For this purpose, we study interpolation-restart (IR) strategies. For a given numerical scheme, the IR strategies consist of extracting relevant information from available data after a fault. After data extraction, a well-selected part of the missing data is regenerated through interpolation strategies to constitute a meaningful input to restart the numerical algorithm. In this paper, we revisit a few state-of-the-art methods in numerical linear algebra in the light of our IR strategies. Through a few numerical experiments, we illustrate the respective robustness of the resulting resilient schemes with respect to the MTBF via qualitative illustrations.

Analyzing the Effect of Local Rounding Error Propagation on the Maximal Attainable Accuracy of the Pipelined Conjugate Gradient Method

Pipelined Krylov subspace methods typically offer improved strong scaling on parallel HPC hardware compared to standard Krylov subspace methods for large and sparse linear systems. In pipelined methods the traditional synchronization bottleneck is mitigated by overlapping time-consuming global communications with useful computations. However, to achieve this communication hiding strategy, pipelined methods introduce additional recurrence relations for a number of auxiliary variables that are required to update the approximate solution. This paper aims at studying the influence of local rounding errors that are introduced by the additional recurrences in the pipelined Conjugate Gradient method. Specifically, we analyze the impact of local round-off effects on the attainable accuracy of the pipelined CG algorithm and compare to the traditional CG method. Furthermore, we estimate the gap between the true residual and the recursively computed residual used in the algorithm. Based on this estimate we suggest an automated residual replacement strategy to reduce the loss of attainable accuracy on the final iterative solution. The resulting pipelined CG method with residual replacement improves the maximal attainable accuracy of pipelined CG, while maintaining the efficient parallel performance of the pipelined method. This conclusion is substantiated by numerical results for a variety of benchmark problems.