Hard Faults and Soft-Errors: Possible Numerical Remedies in Linear Algebra Solvers

Agullo, Emmanuel; Cools, Siegfried; Giraud, Luc; Moreau, Alexandre; Salas, Pablo; Vanroose, Wim; Yetkin, Emrullah Fatih; Zounon, Mawussi

doi:10.1007/978-3-319-61982-8_3

Cited by 5 publications

(6 citation statements)

References 21 publications

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“…Preconditioning is omitted in this section for simplicity of notation but without loss of generality. 1 In this section we use a notation with bars to indicate variables that are computed in a finite precision setting. Furthermore, for variables that are defined recursively in the algorithm, we differentiate between the recursively computed variable and the 'actual' variable, i.e.…”

Section: Analysis Of the Attainable Accuracy Of P(l)-cg In Finite Pre...mentioning

confidence: 99%

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The Communication-Hiding Conjugate Gradient Method with Deep Pipelines

Cornelis,

Cools,

Vanroose

2018

Preprint

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Krylov subspace methods are among the most efficient solvers for large scale linear algebra problems. Nevertheless, classic Krylov subspace algorithms do not scale well on massively parallel hardware due to synchronization bottlenecks. Communication-hiding pipelined Krylov subspace methods offer increased parallel scalability by overlapping the time-consuming global communication phase with computations such as spmvs, hence reducing the impact of the global synchronization and avoiding processor idling. One of the first published methods in this class is the pipelined Conjugate Gradient method (p-CG). However, on large numbers of processors the communication phase may take much longer than the computation of a single spmv. This work extends the pipelined CG method to deeper pipelines, denoted as p(l)-CG, which allows further scaling when the global communication phase is the dominant time-consuming factor. By overlapping the global all-to-all reduction phase in each CG iteration with the next l spmvs (deep pipelining), the method hides communication latency behind additional computational work. The p(l)-CG algorithm is derived from similar principles as the existing p(l)-GMRES method and by exploiting operator symmetry. The p(l)-CG method is also compared to other Krylov subspace methods, including the closely related classic CG and D-Lanczos methods and the pipelined CG method by Ghysels et al.. By analyzing the maximal accuracy attainable by the p(l)-CG method it is shown that the pipelining technique induces a trade-off between performance and numerical stability. A preconditioned version of the algorithm is also proposed and storage requirements and performance estimates are discussed. Experimental results demonstrate the possible performance gains and the attainable accuracy of deeper pipelined CG for solving large scale symmetric linear systems.

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Section: Analysis Of the Attainable Accuracy Of P(l)-cg In Finite Pre...mentioning

confidence: 99%

“…Additionally, other error effects (e.g. delayed convergence due to loss of basis orthogonality in finite precision [29,32,46,23,6], hard faults and soft errors [1], etc.) may affect the convergence of pipelined CG.…”

mentioning

confidence: 99%

The Communication-Hiding Conjugate Gradient Method with Deep Pipelines

Cornelis,

Cools,

Vanroose

2018

Preprint

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“…The analysis in this work focuses on understanding the impact of local rounding errors in the multi-term recurrence relation pipelined BiCGStab algorithm on the attainable precision of the iterative solution. Other sources of errors, such as rounding errors due to loss of basis orthogonality [45], system noise related errors [49] or hard faults/soft errors [1] are not considered in this study. These topics will be considered in the context of pipelined Krylov methods as part of future work.…”

Section: Discussionmentioning

confidence: 99%

Analyzing and improving maximal attainable accuracy in the communication hiding pipelined BiCGStab method

Cools

2019

Parallel Computing

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Pipelined Krylov subspace methods avoid communication latency by reducing the number of global synchronization bottlenecks and by hiding global communication behind useful computational work. In exact arithmetic pipelined Krylov subspace algorithms are equivalent to classic Krylov subspace methods and generate identical series of iterates. However, as a consequence of the reformulation of the algorithm to improve parallelism, pipelined methods may suffer from severely reduced attainable accuracy in a practical finite precision setting. This work presents a numerical stability analysis that describes and quantifies the impact of local rounding error propagation on the maximal attainable accuracy of the multi-term recurrences in the preconditioned pipelined BiCGStab method. Theoretical expressions for the gaps between the true and computed residual as well as other auxiliary variables used in the algorithm are derived, and the elementary dependencies between the gaps on the various recursively computed vector variables are analyzed. The norms of the corresponding propagation matrices and vectors provide insights in the possible amplification of local rounding errors throughout the algorithm. Stability of the pipelined BiCGStab method is compared numerically to that of pipelined CG on a symmetric benchmark problem. Furthermore, numerical evidence supporting the effectiveness of employing a residual replacement type strategy to improve the maximal attainable accuracy for the pipelined BiCGStab method is provided.

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“…Buna bağlı olarak literatürde çip üzerindeki devre elemanlarının arttırılmış ışınıma karşı dayanımlarını test eden çok sayıda deneysel çalışma bulunmaktadır [15]. Yine benzer bir şekilde, büyük ölçekli entegre devrelerin farklı tasarım seviyelerinde geçici hataların etkilerinin incelenmesine ilişkin birçok çalışma da literatürde mevcuttur [16]. Öte yandan eğer geçici hatanın daha yüksek seviyelerdeki etkileri (örneğin sayısal simülasyonlar üzerindeki etkisi) ölçülmek isteniyorsa bu tip donanım seviyesinde yapılacak modellemeler yetersiz kalacaktır.…”

Section: Yük Akışı Probleminde Geçici Hata Modellemesi (Modelling The...unclassified

Güç akışı analizinin geçici hata duyarlılığının değerlendirilmesi

Yetkin

2022

Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi

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Günümüzün güç sistemleri detaylı modelleme ihtiyaçları nedeniyle çok büyük boyutlara ulaşabilmektedir ve belirli koşullar için sistemin tek bir anlık görüntüsünün çözümü bile büyük boyutlu denklem sistemlerinin çözümünü gerektirir. Bu nedenle de makul bir sürede sonuçları elde etmek için modern yüksek başarımlı hesaplama ortamları kullanılmalıdır. Bununla birlikte, yüksek başarımlı hesaplama ortamlarında artan bileşen sayısı nedeniyle, sessiz hata olasılığı da artar. Sessiz hatalar, x-ışınları, kozmik parçacık etkileri gibi nedenlerle cihaz bileşenlerinde oluşabilen çeşitli dalgalanmalardan kaynaklı arızalar olarak tanımlanabilir. Bu tür hatalar genellikle herhangi bir hesaplama anında herhangi bir kayan nokta işleminde yaşanan bir bit-kayması ile modellenebilir. Bu makalede, büyük ölçekli güç akışı simülasyonları üzerindeki sessiz hata etkileri incelenmektedir. Genel olarak yük akışı hesaplamaları, sistem doğrusal olmayan denklemlerle modellendiği için, Newton-Raphson yöntemi kullanılarak yapılır ve çözüm süreci, her yinelemede Jakobiyen matrisinin tersini almak için doğrusal bir çözücünün kullanılmasını gerektirir. Bu çalışmada, özellikle yenilenebilir enerji kaynaklarının sistemlere eklenmesi ile çok büyük boyutlara ulaşılabilen elektrik yük akış problemlerinde kullanılan matematiksel yöntemlerin sessiz-hatalara karşı hassasiyetleri incelenerek, karşılaşılabilecek sorunlar ortaya konulmuştur.

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Hard Faults and Soft-Errors: Possible Numerical Remedies in Linear Algebra Solvers

Cited by 5 publications

References 21 publications

The Communication-Hiding Conjugate Gradient Method with Deep Pipelines

The Communication-Hiding Conjugate Gradient Method with Deep Pipelines

Analyzing and improving maximal attainable accuracy in the communication hiding pipelined BiCGStab method

Güç akışı analizinin geçici hata duyarlılığının değerlendirilmesi

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