## Section: New Results

### High-performance computing on next generation architectures

#### Numerical recovery strategies for parallel resilient Krylov linear solvers

As the computational power of high performance computing (HPC) systems continues to increase by using a huge number of cores or specialized processing units, HPC applications are increasingly prone to faults. In this paper, we present a new class of numerical fault tolerance algorithms to cope with node crashes in parallel distributed environments. This new resilient scheme is designed at application level and does not require extra resources, i.e., computational unit or computing time, when no fault occurs. In the framework of iterative methods for the solution of sparse linear systems, we present numerical algorithms to extract relevant information from available data after a fault, assuming a separate mechanism ensures the fault detection. After data extraction, a well chosen part of missing data is regenerated through interpolation strategies to constitute meaningful inputs to restart the iterative scheme. We have developed these methods, referred to as Interpolation-Restart techniques, for Krylov subspace linear solvers. After a fault, lost entries of the current iterate computed by the solver are interpolated to define a new initial guess to restart the Krylov method. A well suited initial guess is computed by using the entries of the faulty iterate available on surviving nodes. We present two interpolation policies that preserve key numerical properties of well-known linear solvers, namely the monotonic decrease of the A-norm of the error of the conjugate gradient or the residual norm decrease of GMRES. The qualitative numerical behavior of the resulting scheme have been validated with sequential simulations, when the number of faults and the amount of data losses are varied. Finally, the computational costs associated with the recovery mechanism have been evaluated through parallel experiments.

More details on this work can be found in [7].

#### Interpolation-restart strategies for resilient eigensolvers

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.

More details on this work can be found in [8].