The solution of large eigenproblems is involved in many scientific and engineering applications when for instance, stability analysis is a concern. For large simulation in material physics or thermo-acoustics, the calculation can last for many hours on large parallel platforms. 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 eigenproblems. Consequently, it becomes critical to design parallel eigensolvers 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. Because we focus on numerical remedies we do not consider parallel implementations nor parallel experiments but only numerical experiments. We assume that a separate mechanism ensures the fault detection and that a system layer provides support for setting back the environment (processes,. . .) in a running state. Once the system is in a running state, after a fault, our main objective is to provide robust resilient schemes so that the eigensolver may keep converging in the presence of the fault without restarting the calculation from scratch. For this purpose, we extend the interpolation-restart (IR) strategies initially introduced for the solution of linear systems in a previous work to the solution of eigenproblems in this paper. For a given numerical scheme, the IR strategies consist of extracting relevant spectral 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. One of the main features of this numerical remedy is that it does not require extra resources, i.e., computational unit or computing time, when no fault occurs. In this paper, we revisit a few state-of-the-art methods for solving large sparse eigenvalue problems namely the Arnoldi methods, subspace iteration methods and the Jacobi-Davidson method, in the light of our IR strategies. For each considered eigensolver, we adapt the IR strategies to regenerate as much spectral information as possible. Through extensive numerical experiments, we study the respective robustness of the resulting resilient schemes with respect to the MTBF and to the amount of data loss via qualitative and quantitative illustrations. 1. Introduction. The computation of eigenpairs (eigenvalues and eigenvectors) of large sparse matrices is involved in many scientific and engineering applications such as when stability analysis is a concern. To name a few, it appears in structural dynamics, thermodynamics, thermo-acoustics, quantum chemistry. With the permanent increase of the computational power of high performance computing (HPC) systems by using a larger and larger number of CPU cores or specialized processing units, HPC applications are increasingly prone to faults. To guarantee fault tolerance, two classes of strategies are required. One for the fault detection and the other for fault correction. Faults such as computational node crashes are obvious to detect while silent faults may be challenging to detect. To cope with silent faults, a duplication strategy is commonly used for fault detection [18, 39] by comparing the outputs, while triple modular redundancy (TMR) is used for fault detection and correction [34, 37]. However, the additional computational resources required by such replication strategies may represent a severe penalty. Instead of replicating computational resources, studies [7, 36] propose a time redundancy model for fault detection. It consists in repeating computation twice on the same resource. The advantage of time redundancy models is the flexibility at application level; software developers can indeed select only a set of critical instructions to protect. Recomputing only some instructions instead of the whole application lowers the time redundancy overhead [25]. In some numerical simulations, data naturally satisfy well defined mathematical properties. These properties can be efficiently exploited for fault detection through a periodical check of the numerical properties during computation [10]. Checkpoint/restart is the most studied fault recovery strategy in the context of HPC systems. The common checkpoint/restart scheme consists in periodically saving data onto a reliable storage device such as a remote disk. When a fault occurs, a rollback is performed to the point of the most recent and consistent checkpoint. According to the implemented checkpoint strategy, all processes
Read more
9
