Convergence of Waveform Relaxation Methods for Differential-Algebraic Systems

Jackiewicz, Zdzisław; Kwapisz, M.

doi:10.1137/s0036142992233098

Cited by 59 publications

(32 citation statements)

References 17 publications

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“…This formulation is very attractive for parallel implementations since the data exchange rate is minimum. On the contrary, if the sub-domains are not weakly coupled the algorithm can suffer from degraded convergence or even divergence [5], [6], [7], [8]. Other variants of this method can be found in literature depending on how often and in which order the interface variables are updated, for instance the additive or multiplicative Schwartz procedures [3].…”

Section: A21 Schwarz Alternating Methodsmentioning

confidence: 99%

Dynamic Simulation of Large-Scale Power Systems Using a Parallel Schur-Complement-Based Decomposition Method

Aristidou

Fabozzi

Cutsem

2014

IEEE Trans. Parallel Distrib. Syst.

116

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Article:Aristidou, P orcid.org/0000-0003- 4429-0225, Fabozzi, D and Van Cutsem, T (2014) Dynamic simulation of large-scale power systems using a parallel schur-complement-based decomposition method. IEEE Transactions on Parallel and Distributed Systems, 25 (10). pp. 2561 -2570 . ISSN 1045 https://doi.org/10.1109/TPDS.2013.252 © 2013 IEEE. This is an author produced version of a paper published in IEEE Transactions on Parallel and Distributed Systems. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Uploaded in accordance with the publisher's self-archiving policy.eprints@whiterose.ac.uk https://eprints.whiterose.ac.uk/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. APPENDIX A DOMAIN DECOMPOSITION METHODSDDMs were originally used due to the lack of memory in computing systems: data needed for smaller portions of a problem could fit entirely to the memory while for the whole problem they could not. They lost their appeal as larger and cheaper memory became available, only to resurface in the era of parallel computing. These methods are inherently suited for execution on parallel architectures and many parallel implementations have been presented on multi-core computers, clusters and lately Graphics Processing Units (GPUs) [1], [2]. They are mainly distinguished by three features: subdomain partitioning, problem solution over sub-domains and sub-domain interface variable processing [3]. A.1 Sub-domain PartitioningSub-domain partitioning has to be chosen based on the desired sub-domain characteristics for the given problem. This includes choosing the number of sub-domains, the type of partitioning, and the level of overlap between the sub-domains. Each of these choices depend on a variety of factors such as size, type, and geometry of the problem domain, the number of parallel processors, communication cost, and the actual system being solved.When considering spatial domain problems, such as PDEs, the decomposition is usually g...

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Section: A21 Schwarz Alternating Methodsmentioning

confidence: 99%

Dynamic Simulation of Large-Scale Power Systems Using a Parallel Schur-Complement-Based Decomposition Method

Aristidou

Fabozzi

Cutsem

2014

IEEE Trans. Parallel Distrib. Syst.

116

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“…Superlinear convergence of waveform relaxation iterations was shown in the monograph [1] and many papers, see, e.g., [2,3] for many various acceleration methods; [12,29,28] for applications to delay differential equations; and [11] for applications to differential-algebraic problems. However, since (1.1) is nonlinear, the results presented in these references do not apply to the sequence u (k) (t) defined by (3.2) and the boundedness and convergence results of Theorems 4.1 and 5.1 are not yet known in the literature.…”

Section: Convergence Of Waveform Relaxation Iterationsmentioning

confidence: 98%

“…These techniques in the context of functional-differential equations are discussed in [12,29,28]. Waveform relaxation methods for differential-algebraic systems are examined in [11].…”

Section: Waveform Relaxation Iterationsmentioning

confidence: 99%

Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations

Jackiewicz

Zubik‐Kowal

2006

Applied Numerical Mathematics

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“…Convergence of approximate iterations for (1) is proved, for example, in papers [3], [6] (see also [1]). In [5], an approximate solution is constructed by some numerical methods.…”

Section: Sectionmentioning

confidence: 99%