We introduce a novel algorithm-based fault-tolerance scheme to detect and repair soft transient faults (silent data corruption, bitflips) in multigrid solvers: by applying the full approximation scheme (FAS) variant of multigrid to linear systems, we prove invariants that enable fault detection and correction, and ultimately lead to a black-box protection of the smoothing stage. A statistical analysis for a wide range of prototypical problems demonstrates the efficiency of our approach, especially compared with full checksum protection. In particular, the overhead of our new method is negligible in the fault-free case, since we only employ readily available quantities.
AinsworthMGlusaC (2016) Multigrid at scale? In: KarasözenBManguoğluMTezer-SezginMGöktepeSUğurÖ (eds.) Numerical Mathematics and Advanced Applications ENUMATH 2015. New York: Springer International Publishing, pp. 237–253.
2.
AltenberndM (2015) Konzepte für fehlertolerante Mehrgitterverfahren. Master’s Thesis, TU Dortmund, Fakultät für Mathematik, Lehrstuhl 3 für Angewandte Mathematik und Numerik.
3.
BastianPEngwerCGöddekeD. (2014) EXA-DUNE: Flexible PDE solvers, numerical methods and applications. In: LopesLŽilinskasJCostanA. (eds.) Euro-Par 2014: Parallel Processing Workshops (Lecture Notes in Computer Science, vol. 8806). New York: Springer, pp. 530–541.
4.
BosilcaGDelmasRDongarraJLangouJ (2009) Algorithm-based fault tolerance applied to high performance computing. Journal of Parallel and Distributed Computing69(4): 410–416.
5.
BraessD (2001) Finite Elements – Theory, Fast Solvers and Applications in Solid Mechanics (2nd edn). Cambridge: Cambridge University Press.
6.
BrandtA (1977) Multi-level adaptive solutions to boundary-value problems. Mathematics of Computation31: 333–390.
7.
BriggsWLHensonVEMcCormickSF (2000) A Multigrid Tutorial: Second Edition. Philadelphia, PA: Society for Industrial and Applied Mathematics.
8.
CasasMde SupinskiBRBronevetskyGSchulzM (2012) Fault resilience of the algebraic multi-grid solver. In: Proceedings of the 26th ACM International Conference on Supercomputing (ICS ’12). New York: ACM Press, pp. 91–100.
9.
ChenZ (2011) Algorithm-based recovery for iterative methods without checkpointing. In: Proceedings of the 20th International Symposium on High Performance Distributed Computing (HPDC ’11), pp. 73–84.
10.
DongarraJHittingerJBellJ. (2014) Applied mathematics research for exascale computing. Technical report, U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research Program.
11.
ElliottJHoemmenMMuellerF (2014) Evaluating the impact of SDC on the GMRES iterative solver. In: Proceedings of the 2014 IEEE 28th International Parallel and Distributed Processing Symposium (IPDPS’14), pp. 1193–1202. doi:10.1109/IPDPS.2014.123.
12.
ElliottJKharbasKFialaDMuellerFFerreiraKEngelmannC (2012) Combining partial redundancy and checkpointing for HPC. In: Proceedings of the 2012 IEEE International Conference on Distributed Computing Systems (ICDCS’12), pp. 615–626.
13.
FerreiraKStearleyJLarosJHIII. (2011) Evaluating the viability of process replication reliability for exascale systems. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis (SC’11), pp. 44:1–44:12.
14.
FialaDMuellerFEngelmannCRiesenRFerreiraKBrightwellR (2012) Detection and correction of silent data corruption for large-scale high-performance computing. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis (SC’12), pp. 78:1–78:12.
LammersD (2010) The era of error-tolerant computing. IEEE Spectrum47(11): 15–15.
22.
LiDChenZWuPVetterJS (2013) Rethinking algorithm-based fault tolerance with a cooperative software-hardware approach. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis (SC’13), pp. 44:1–44:12.
23.
NiXMenesesEJainNKaléLV (2013) ACR: Automatic checkpoint/restart for soft and hard error protection. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis (SC’13), pp. 7:1–7:12.
24.
RoparsTMartsinkevichTVGuermoucheASchiperACappelloF (2013) SPBC: Leveraging the characteristics of MPI HPC applications for scalable checkpointing. In: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis (SC’13), pp. 8:1–8:12.
25.
SaoPVuducR (2013) Self-stabilizing iterative solvers. In: Proceedings of the Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems (ScalA ’13), pp. 4:1–4:8.
26.
SchöllABraunCKochteMAWunderlichHJ (2015) Low-overhead fault-tolerance for the preconditioned conjugate gradient solver. In: Proceedings of the International Symposium on Defect and Fault Tolerance in VLSI and Nanotechnology Systems (DFT’15), pp. 60–65.
27.
SloanJKumarRBronevetskyG (2012) Algorithmic approaches to low overhead fault detection for sparse linear algebra. In: Proceedings of the 42nd Annual IEEE/IFIP International Conference on Dependable Systems and Networks (DSN ’12), pp. 1–12.
28.
SloanJKumarRBronevetskyG (2013) An algorithmic approach to error localization and partial recomputation for low-overhead fault tolerance. In: Proceedings of the 43rd Annual IEEE/IFIP International Conference on Dependable Systems and Networks (DSN ’13), pp. 1–12.
29.
SnirMWisniewskiRWAbrahamJA. (2014) Addressing failures in exascale computing. International Journal of High Performance Computing Applications28(2): 129–173.
30.
StearleyJRRiesenRLarosJHIII. (2010) Redundant computing for exascale systems. Technical Report SAND2010-8709, Sandia National Laboratories.
31.
TeranishiKHerouxMA (2014) Toward local failure local recovery resilience model using MPI-ULFM. In: EuroMPI/ASIA ’14, pp. 51:51–51:56.
32.
TrottenbergUOosterleeCWSchullerA (2001) Multigrid. New York: Academic Press.
33.
TurekSGöddekeDBeckerCBuijssenSHWobkerH (2010) FEAST – Realisation of hardware-oriented numerics for HPC simulations with finite elements. Concurrency and Computation: Practice and Experience22(6): 2247–2265.
