Fault Tolerant Computation of Hyperbolic Partial Differential Equations with the Sparse Grid Combination Technique

As the computing power of supercomputers continues to increase exponentially the mean time between failures (MTBF) is decreasing. Checkpoint-restart has historically been the method of choice for recovering from failures. However, such methods become increasingly inefficient as the time required to complete a checkpoint-restart cycle approaches the MTBF. There is therefore a need to explore different ways of making computations fault tolerant. This thesis studies generalisations of the sparse grid combination technique with the goal of developing and analysing a holistic approach to the fault tolerant computation of partial differential equations (PDEs). Sparse grids allow one to reduce the computational complexity of high dimensional problems with only small loss of accuracy. A drawback is the need to perform computations with a hierarchical basis rather than a traditional nodal basis. We survey classical error estimates for sparse grid interpolation and extend results to functions which are non-zero on the boundary. The combination technique approximates sparse grid solutions via a sum of many coarse approximations which need not be computed with a hierarchical basis. Study of the combination technique often assumes that approximations satisfy an error splitting formula. We adapt classical error splitting results to our slightly different convention of combination level. Literature on the application of the combination technique to hyperbolic PDEs is scarce, particularly when solved with explicit finite difference methods. We show a particular family of finite difference discretisations for the advection equation solved via the method of lines has solutions which satisfy an error splitting formula. As a consequence, classical error splitting based estimates are readily applied to finite difference solutions of many hyperbolic PDEs. Our analysis also reveals how repeated combinations throughout the computation leads to a reduction in approximation error. Generalisations of the combination technique are studied and developed at depth. The truncated combination technique is a modification of the classical method used in practical applications and we provide analogues of classical error estimates. Adaptive sparse grids are then studied via a lattice framework. A detailed examination reveals many results regarding combination coefficients and extensions of classical error estimates. The framework is also applied to the study of extrapolation formula. These extensions of the combination technique provide the foundations for the development of the general coefficient problem. Solutions to this problem allow one to combine any collection of coarse approximations on nested grids. Lastly, we show how the combination technique is made fault tolerant via application of the general coefficient problem. Rather than recompute coarse solutions which fail we instead find new coefficients to combine remaining solutions. This significantly reduces computational overheads in the presence of faults with only small loss of accuracy. The latter is established with a careful study of the expected error for some select cases. We perform numerical experiments by computing combination solutions of the scalar advection equation in a parallel environment with simulated faults. The results support the preceding analysis and show that the overheads are indeed small and a significant improvement over traditional checkpoint-restart methods

Resiliency in numerical algorithm design for extreme scale simulations

This work is based on the seminar titled ‘Resiliency in Numerical Algorithm Design for Extreme Scale Simulations’ held March 1–6, 2020, at Schloss Dagstuhl, that was attended by all the authors. Advanced supercomputing is characterized by very high computation speeds at the cost of involving an enormous amount of resources and costs. A typical large-scale computation running for 48 h on a system consuming 20 MW, as predicted for exascale systems, would consume a million kWh, corresponding to about 100k Euro in energy cost for executing 1023 floating-point operations. It is clearly unacceptable to lose the whole computation if any of the several million parallel processes fails during the execution. Moreover, if a single operation suffers from a bit-flip error, should the whole computation be declared invalid? What about the notion of reproducibility itself: should this core paradigm of science be revised and refined for results that are obtained by large-scale simulation? Naive versions of conventional resilience techniques will not scale to the exascale regime: with a main memory footprint of tens of Petabytes, synchronously writing checkpoint data all the way to background storage at frequent intervals will create intolerable overheads in runtime and energy consumption. Forecasts show that the mean time between failures could be lower than the time to recover from such a checkpoint, so that large calculations at scale might not make any progress if robust alternatives are not investigated. More advanced resilience techniques must be devised. The key may lie in exploiting both advanced system features as well as specific application knowledge. Research will face two essential questions: (1) what are the reliability requirements for a particular computation and (2) how do we best design the algorithms and software to meet these requirements? While the analysis of use cases can help understand the particular reliability requirements, the construction of remedies is currently wide open. One avenue would be to refine and improve on system- or application-level checkpointing and rollback strategies in the case an error is detected. Developers might use fault notification interfaces and flexible runtime systems to respond to node failures in an application-dependent fashion. Novel numerical algorithms or more stochastic computational approaches may be required to meet accuracy requirements in the face of undetectable soft errors. These ideas constituted an essential topic of the seminar. The goal of this Dagstuhl Seminar was to bring together a diverse group of scientists with expertise in exascale computing to discuss novel ways to make applications resilient against detected and undetected faults. In particular, participants explored the role that algorithms and applications play in the holistic approach needed to tackle this challenge. This article gathers a broad range of perspectives on the role of algorithms, applications and systems in achieving resilience for extreme scale simulations. The ultimate goal is to spark novel ideas and encourage the development of concrete solutions for achieving such resilience holistically.Peer Reviewed"Article signat per 36 autors/es: Emmanuel Agullo, Mirco Altenbernd, Hartwig Anzt, Leonardo Bautista-Gomez, Tommaso Benacchio, Luca Bonaventura, Hans-Joachim Bungartz, Sanjay Chatterjee, Florina M. Ciorba, Nathan DeBardeleben, Daniel Drzisga, Sebastian Eibl, Christian Engelmann, Wilfried N. Gansterer, Luc Giraud, Dominik G ̈oddeke, Marco Heisig, Fabienne Jezequel, Nils Kohl, Xiaoye Sherry Li, Romain Lion, Miriam Mehl, Paul Mycek, Michael Obersteiner, Enrique S. Quintana-Ortiz, Francesco Rizzi, Ulrich Rude, Martin Schulz, Fred Fung, Robert Speck, Linda Stals, Keita Teranishi, Samuel Thibault, Dominik Thonnes, Andreas Wagner and Barbara Wohlmuth"Postprint (author's final draft

Magic-State Functional Units: Mapping and Scheduling Multi-Level Distillation Circuits for Fault-Tolerant Quantum Architectures

Quantum computers have recently made great strides and are on a long-term path towards useful fault-tolerant computation. A dominant overhead in fault-tolerant quantum computation is the production of high-fidelity encoded qubits, called magic states, which enable reliable error-corrected computation. We present the first detailed designs of hardware functional units that implement space-time optimized magic-state factories for surface code error-corrected machines. Interactions among distant qubits require surface code braids (physical pathways on chip) which must be routed. Magic-state factories are circuits comprised of a complex set of braids that is more difficult to route than quantum circuits considered in previous work [1]. This paper explores the impact of scheduling techniques, such as gate reordering and qubit renaming, and we propose two novel mapping techniques: braid repulsion and dipole moment braid rotation. We combine these techniques with graph partitioning and community detection algorithms, and further introduce a stitching algorithm for mapping subgraphs onto a physical machine. Our results show a factor of 5.64 reduction in space-time volume compared to the best-known previous designs for magic-state factories.Comment: 13 pages, 10 figure