Delayed Difference Scheme for Large Scale Scientific Simulations

Mudigere, Dheevatsa; Sherlekar, S.D.; Ansumali, Santosh

doi:10.1103/physrevlett.113.218701

Cited by 13 publications

(12 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It clearly shows the delay error from the proxy-equation approach is much smother and also lower in magnitude. Similar observations can be made when comparing the delayed time difference scheme [9] with the proxy-equation approach , FIG. 5(b).…”

supporting

confidence: 73%

See 1 more Smart Citation

Proxy-equation paradigm: A strategy for massively parallel asynchronous computations

Mittal

Girimaji

2017

Phys. Rev. E

View full text Add to dashboard Cite

Massively parallel simulations of transport equation systems call for a paradigm change in algorithm development to achieve efficient scalability. Traditional approaches require time synchronization of processing elements (PEs), which severely restricts scalability. Relaxing synchronization requirement introduces error and slows down convergence. In this paper, we propose and develop a novel "proxy equation" concept for a general transport equation that (i) tolerates asynchrony with minimal added error, (ii) preserves convergence order and thus, (iii) expected to scale efficiently on massively parallel machines. The central idea is to modify a priori the transport equation at the PE boundaries to offset asynchrony errors. Proof-of-concept computations are performed using a one-dimensional advection (convection) diffusion equation. The results demonstrate the promise and advantages of the present strategy.

show abstract

supporting

confidence: 73%

“…Currently a few numerical schemes have been developed to improve the accuracy of asynchronous computations. Recently developed asynchrony-tolerant numerical scheme [8] and the delayed difference scheme [9] attempt to counteract the delay error due to asynchrony by modifying the discretization scheme. However, the delay error still continues to be significant.…”

mentioning

confidence: 99%

Proxy-equation paradigm: A strategy for massively parallel asynchronous computations

Mittal

Girimaji

2017

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…Thus, schemes with delay on both sides will need a smaller stencil to compute derivatives. In [12], this approach was used to recover the drop in accuracy due to delay in communication in a particular application using central difference schemes. The authors further suggested that imposing delay on both sides of the stencil in central differences would suffice to maintain the accuracy under asynchronous conditions.…”

Section: Alternative Approachmentioning

confidence: 99%

“…They also proposed the possibility of deriving schemes that are tolerant to communication data asynchrony. A follow up of this work to a simple specific equation and numerical scheme has be presented in [12]. Although the authors were able to maintain second order accuracy for their chosen scheme when asynchrony is present, one can show using Taylor series that they are severely limited to low order of accuracy.…”

Section: Introductionmentioning

confidence: 99%

High-order asynchrony-tolerant finite difference schemes for partial differential equations

Aditya

Donzis

2017

Journal of Computational Physics

View full text Add to dashboard Cite

Synchronizations of processing elements (PEs) in massively parallel simulations, which arise due to communication or load imbalances between PEs, significantly affect the scalability of scientific applications. We have recently proposed a method based on finite-difference schemes to solve partial differential equations in an asynchronous fashion -synchronization between PEs is relaxed at a mathematical level. While standard schemes can maintain their stability in the presence of asynchrony, their accuracy is drastically affected. In this work, we present a general methodology to derive asynchrony-tolerant (AT) finite difference schemes of arbitrary order of accuracy, which can maintain their accuracy when synchronizations are relaxed. We show that there are several choices available in selecting a stencil to derive these schemes and discuss their effect on numerical and computational performance. We provide a simple classification of schemes based on the stencil and derive schemes that are representative of different classes. Their numerical error is rigorously analyzed within a statistical framework to obtain the overall accuracy of the solution. Results from numerical experiments are used to validate the performance of the schemes.

show abstract

“…In order to overcome this bottleneck, some work has focused on relaxing the synchronization requirements among the processors and perform so-called asynchronous numerical simulations. Early work in the literature dealt with asynchronous simulations but severely limited to lower orders of accuracy and restricted to certain class of PDEs [19,20,21,22]. A new and more generalized approach, extensible to arbitrarily high orders of accuracy, has been recently developed [23,24] to derive the so-called Asynchrony-Tolerant (AT) finite-difference schemes.…”

Section: Introductionmentioning

confidence: 99%

Direct Numerical Simulations of turbulent flows using high-order Asynchrony-Tolerant schemes: accuracy and performance

Kumari,

Donzis

2020

Preprint

View full text Add to dashboard Cite

Direct numerical simulations (DNS) are an indispensable tool for understanding the fundamental physics of turbulent flows. Because of their steep increase in computational cost with Reynolds number (R λ ), wellresolved DNS are realizable only on massively parallel supercomputers, even at moderate R λ . However, at extreme scales, the communications and synchronizations between processing elements (PEs) involved in current approaches become exceedingly expensive and are expected to be a major bottleneck to scalability. In order to overcome this challenge, we developed algorithms using the so-called Asynchrony-Tolerant (AT) schemes that relax communication and synchronization constraints at a mathematical level, to perform DNS of decaying and solenoidally forced compressible turbulence. Asynchrony is introduced using two approaches, one that avoids synchronizations and the other that avoids communications. These result in periodic and random delays, respectively, at PE boundaries. We show that both asynchronous algorithms accurately resolve the large-scale and small-scale motions of turbulence, including instantaneous and intermittent fields. We also show that in asynchronous simulations the communication time is a relatively smaller fraction of the total computation time, especially at large processor count, compared to standard synchronous simulations. As a consequence, we observe improved parallel scalability up to 262144 processors for both asynchronous algorithms.

show abstract

Delayed Difference Scheme for Large Scale Scientific Simulations

Cited by 13 publications

References 18 publications

Proxy-equation paradigm: A strategy for massively parallel asynchronous computations

Proxy-equation paradigm: A strategy for massively parallel asynchronous computations

High-order asynchrony-tolerant finite difference schemes for partial differential equations

Direct Numerical Simulations of turbulent flows using high-order Asynchrony-Tolerant schemes: accuracy and performance

Contact Info

Product

Resources

About