Synchronous and asynchronous optimized Schwarz methods for one‐way subdivision of bounded domains

Haddad, Mireille; Garay, José C.; Magoulès, Frédéric; Szyld, Daniel B.

doi:10.1002/nla.2279

Cited by 5 publications

(3 citation statements)

References 26 publications

(58 reference statements)

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“…Numerical experiments in Section 9 illustrate our theoretical results and complement the numerical results in [14,33], showing that indeed asynchronous optimized Schwarz methods can be effective; see also [23]. Several appendices present detailed proofs of some of the results mentioned in the paper.…”

supporting

confidence: 54%

“…In [10], a convergence analysis of the classical Schwarz method is presented for a bounded domain with multiple subdomains, for the case in which the subdomains form a one-way partition of the domain and in which each overlapped region is shared by two subdomains. See also [14] for an analysis of optimized Schwarz methods for Poisson's equation in a rectangular domain using a one-way subdivision of the domain. In [22] we presented a preliminary analysis of the convergence of the optimized Schwarz method in the synchronous case for a problem defined in a bounded domain and for an arbitrary number of subdomains, when the subdomains form a two-dimensional subdivision containing cross points.…”

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confidence: 99%

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Synchronous and asynchronous optimized Schwarz methods for Poisson's equation in rectangular domains

Garay¹,

Magoulès²,

Szyld³

2022

etna

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Convergence results for optimized Schwarz methods (OSM) applied as solvers for Poisson's equation in a bounded rectangular domain with Dirichlet (physical) boundary conditions and zeroth-order (Robin) artificial transmission conditions between subdomains are presented. The analysis presented applies to a continuous formulation on an arbitrary number of subdomains with cross points. Both synchronous and asynchronous versions of OSM are discussed. Convergence theorems are presented, and it is shown numerically that the hypotheses of these theorems are satisfied for certain configurations of the subdomains. Additional numerical experiments illustrate the practical behavior of the methods discussed.

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confidence: 54%

mentioning

confidence: 99%

Synchronous and asynchronous optimized Schwarz methods for Poisson's equation in rectangular domains

Garay¹,

Magoulès²,

Szyld³

2022

etna

Self Cite

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“…More general is the idea of asynchronously updating subdomains in Schwarz decompositions. In particular, asynchronous restricted additive Schwarz methods and asynchronous optimized Schwarz methods have been identified to combine algorithm-inherent resilience with scalability on pre-exascale hardware architectures (El Haddad et al, 2020; Garay et al, 2017; Glusa et al, 2019; Magoulès et al, 2017; Yamazaki et al, 2019).…”

Section: Numerical Algorithms For Resiliencementioning

confidence: 99%

Resiliency in numerical algorithm design for extreme scale simulations

Agullo

Altenbernd

Anzt

et al. 2021

The International Journal of High Performance Computing Applica

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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.

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Asynchronous global–local non-invasive coupling for nonlinear monotone patches: Application to plasticity problems

El Kerim,

Gosselet,

Magoulès

2024

Computer Methods in Applied Mechanics and Engineering

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Synchronous and asynchronous optimized Schwarz methods for one‐way subdivision of bounded domains

Cited by 5 publications

References 26 publications

Synchronous and asynchronous optimized Schwarz methods for Poisson's equation in rectangular domains

Synchronous and asynchronous optimized Schwarz methods for Poisson's equation in rectangular domains

Resiliency in numerical algorithm design for extreme scale simulations

Asynchronous global–local non-invasive coupling for nonlinear monotone patches: Application to plasticity problems

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