Applied Mathematics Research for Exascale Computing

Dongarra, Jack; Hittinger, Jeffrey; Bell, Jennifer; Chacòn, Luis; Falgout, Robert D.; Heroux, Michael A.; Hovland, Paul; Ng, Esmond G.; Webster, Clayton G.; Wild, Stefan M.

doi:10.2172/1149042

Cited by 78 publications

(97 citation statements)

References 37 publications

(32 reference statements)

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“…Fourth, we report on reasonable MFlop rates and vectorisation efficiency as we fuse the solution of multiple preconditioning problems into one adaptive grid; despite the fact that we employ a low order scheme which is notoriously bandwidth bound and do not rely on sophisticated smoother optimisation techniques [Kowarschik et al 2000]. This renders the algorithmic mindset well-suited for upcoming machine generations that are expected to obtain a significant part of their capability from vectorisation's extreme concurrency in combination with constrained memory [Dongarra et al 2014]. Fifth, we demonstrate how complex scaling and various choices of relaxation and very few operators allow a user to obtain a set of solvers that can be tailored to many problems.…”

Section: Introductionmentioning

confidence: 91%

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Reps

Weinzierl

2017

ACM Trans. Math. Softw.

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We introduce a family of implementations of low order, additive, geometric multilevel solvers for systems of Helmholtz equations arising from Schrödinger equations. Both grid spacing and arithmetics may comprise complex numbers and we thus can apply complex scaling to the indefinite Helmholtz operator. Our implementations are based upon the notion of a spacetree and work exclusively with a finite number of precomputed local element matrices. They are globally matrix-free.Combining various relaxation factors with two grid transfer operators allows us to switch from additive multigrid over a hierarchical basis method into a Bramble-Pasciak-Xu (BPX)-type solver, with several multiscale smoothing variants within one code base. Pipelining allows us to realise full approximation storage (FAS) within the additive environment where, amortised, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realise a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as enabler for full multigrid (FMG) cycling-the grid unfolds throughout the computation-allows us to reduce the cost per unknown per order of accuracy.The present paper primary contributes towards software realisation and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR and vectorisation-friendly, complex scaled, matrix-free FMG cycles delivers a mature implementation blueprint for solvers of Helmholtz equations in general. For this blueprint, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs.

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Section: Introductionmentioning

confidence: 91%

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Reps

Weinzierl

2017

ACM Trans. Math. Softw.

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“…This more than 3 times the number of nodes the parallel multi-grid solver alone could scale. In a recent report released by the Exascale Mathematics Working Group at Lawrence Livermore National Laboratory, time-parallel integration techniques are highlighted as a potential path in overcoming the limitations of strong scaling in evolution problems and calls for more research in this direction [5]. In this paper we demonstrate how a popular method for time-parallel integration, Parareal, with slight modifications, may be made resilient towards hardware faults.…”

Section: Introductionmentioning

confidence: 94%

“…In [5], resilience to faults is identified as being critical for future exascale HPC systems. The techniques needed to achieve a thousand fold increase in computational capacity expected over the next decade, are predicted to also increase the rate of faults on large systems.…”

Section: Resiliencementioning

confidence: 99%

Fault Tolerance in the Parareal Method

Nielsen

Hesthaven

2016

Proceedings of the ACM Workshop on Fault-Tolerance for HPC at Extreme Scale

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Parallel-in-time integration is an often advocated approach for extracting parallelism in the solution of PDEs beyond what is possible using spacial domain decomposition techniques. Due to the comparatively low parallel efficiency of parallel-in-time integration techniques, they are primarily of interest as an extension for classical approaches at parallelism. As such, potential applications are expected to scale across several hundreds, or possibly thousands of nodes, making algorithmic resilience towards hardware induced errors highly relevant. In this work we develop a scheduling scheme for the parareal algorithm that is resilient to node-loss. The fault-tolerant scheme is based on a popular approach introduced by E. Aubanel in [1], modified with a set of MPI interface extensions for implementing recovery strategies available in the ULFM framework. In addition, we demonstrate how the parareal algorithm may be made resilient towards Silent-Data-Corruption (SDC) errors by viewing it as a point-iterative method, locally monitoring the residual between consecutive iterations so to discard potentially corrupt iterations.

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“…(17) has already been shown in (14). To obtain the identity (18) we multiply out the tensor product, complete the square to recover the first term, and then use that…”

Section: Definition 9 (Energy Norms)mentioning

confidence: 99%

“…All of these of factors contribute to a lower overall reliability of the machine in terms of random bit-flipping and corruption of logic states from extraneous sources such as cosmic rays and, failures of individual components. The issue is further exacerbated by the additional numbers of hardware components required for an exascale system [8][9][10]14].…”

Section: Introductionmentioning

confidence: 99%

Is the Multigrid Method Fault Tolerant? The Two-Grid Case

Ainsworth

Glusa

2017

SIAM J. Sci. Comput.

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Abstract. The predicted reduced resiliency of next-generation high performance computers means that it will become necessary to take into account the effects of randomly occurring faults on numerical methods. Further, in the event of a hard fault occurring, a decision has to be made as to what remedial action should be taken in order to resume the execution of the algorithm. The action that is chosen can have a dramatic effect on the performance and characteristics of the scheme. Ideally, the resulting algorithm should be subjected to the same kind of mathematical analysis that was applied to the original, deterministic variant.The purpose of this work is to provide an analysis of the behaviour of the multigrid algorithm in the presence of faults. Multigrid is arguably the method of choice for the solution of large-scale linear algebra problems arising from discretization of partial differential equations and it is of considerable importance to anticipate its behaviour on an exascale machine. The analysis of resilience of algorithms is in its infancy and the current work is perhaps the first to provide a mathematical model for faults and analyse the behaviour of a state-of-the-art algorithm under the model. It is shown that the Two Grid Method fails to be resilient to faults. Attention is then turned to identifying the minimal necessary remedial action required to restore the rate of convergence to that enjoyed by the ideal fault-free method. IntroductionPresident Obama's executive order in the summer of 2015 establishing the National Strategic Computing Initiative 1 committed the US to the development of a capable exascale computing system. Given that the performance of the current number one machine Tianhe-2 is roughly one thirtieth of that of an exascale system, it is easy to underestimate the challenge posed by this task. One way to envisage the scale of the undertaking is that the combined processing power of the entire TOP 500 list is less than half of one exaflop (10 18 floating point operations per second).It is widely accepted that an exascale machine should respect a 20MW power envelope. Tianhe-2 already consumes 18MW of power, and if it were possible to simply upscale to exascale using the current technology, would require around 2010 Mathematics Subject Classification. 65F10, 65N22, 65N55, 68M15.

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Applied Mathematics Research for Exascale Computing

Cited by 78 publications

References 37 publications

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Fault Tolerance in the Parareal Method

Is the Multigrid Method Fault Tolerant? The Two-Grid Case

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