Optimized Schwarz Method for Poisson’s Equation in Rectangular Domains

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

doi:10.1007/978-3-319-93873-8_51

Cited by 5 publications

(6 citation statements)

References 4 publications

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“…We are able to prove convergence for any number of subdomains. In a sense, we are able to offer a more practical convergence result than that of the whole plane in the work of Magoulès et al, and that of Garay et al, but for a less general partition of the subdomains. We use techniques taken from both of these papers, but tailored to our particular setup.…”

Section: Introductionmentioning

confidence: 95%

“…While this was the first proof of the convergence of asynchronous optimized Schwarz, and this gave the numerical experiments a sound theoretical basis, the fact that it was for the whole plane, left open the problem of analyzing convergence for more practical bounded domains. More recently, in the work of Garay et al, a convergence analysis of the solution of Poisson's equation using optimized Schwarz on a rectangular domain has been presented, in which the domain is divided into p × q (overlapping) rectangles in a 2D configuration. That is, the artificial interfaces have cross‐points.…”

Section: Introductionmentioning

confidence: 99%

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

Haddad

Garay

Magoulès

et al. 2019

Numerical Linear Algebra App

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Summary Convergence of both synchronous and asynchronous optimized Schwarz algorithms for the shifted Laplacian operator on a bounded rectangular domain, in a one‐way subdivision of the computational domain, with overlap, is shown. Convergence results are obtained under very mild conditions on the size of the subdomains and on the amount of overlap. A couple of results are also given, relating the convergence rate of the asynchronous method to changes in the size of the domain. Numerical experiments illustrate the theoretical results.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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

Haddad

Garay

Magoulès

et al. 2019

Numerical Linear Algebra App

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“…In this paper, we would like to consider only the two-subdomain decomposition described above, which allows us to obtain asymptotically accurate formulas of the best transmission parameters. When the physical domain is decomposed into arbitrary number of subdomains (cross points may be present), we refer the reader to [25] for a convergence analysis of the synchronous optimized Schwarz method with Robin transmission condition and to [26] for the asynchronous case. Beyond the continuous level analysis, see [18] for an algebraic analysis for problems defined on irregular physical domains.…”

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confidence: 99%

The influence of domain truncation on the performance of optimized Schwarz methods

Xu¹

2018

etna

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Optimized Schwarz methods enhance convergence using optimized transmission conditions between subdomains. The optimization is usually performed for a model problem on an unbounded domain and two subdomains represented by half spaces. The influence of the domain decomposition geometry on the convergence and the optimized parameters is thus lost in the process, and it is not even theoretically clear if the results published for the unbounded domain still hold in concrete applications where the domains are bounded. We prove here rigorously for a two-subdomain decomposition that the asymptotic performance of optimized Schwarz methods derived from an unbounded domain analysis still holds in the case of a bounded domain, but the constants in the best choice of parameters and convergence rate estimates are influenced by the domain truncation. We obtain accurate estimates for this influence and show theoretically that the domain truncation has more remarkable influence for the slowly converging optimized Schwarz methods than for those converging fast. When the subdomain size is very small, our new optimized parameters lead to much faster algorithms than those obtained from an unbounded domain analysis. We illustrate our theoretical results with numerical experiments.

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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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Optimized Schwarz Method for Poisson’s Equation in Rectangular Domains

Cited by 5 publications

References 4 publications

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

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

The influence of domain truncation on the performance of optimized Schwarz methods

Resiliency in numerical algorithm design for extreme scale simulations

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