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On the Convergence Acceleration and Parallel Implementation of Continuation in Disconnected Bifurcation Diagrams for Large-Scale Problems

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Parallel Computational Technologies (PCT 2019)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1063))

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Abstract

The automated construction of disconnected bifurcation diagrams for large-scale steady-state problems is a difficult task. Standard continuation methods are not suited for Krylov-type solvers when dealing with disconnected solutions and, to allow branch switching, they require that the vectors that span the linear operator kernel at bifurcation points be known. An alternative approach to Krylov-type solvers was suggested by Farrell, Beentjes, and Birkisson in 2016, based on the idea of solution deflation. In this paper, we modify the aforementioned method by changing the algorithm to accelerate convergence and rearranging it to be more suited for parallel architectures together with the pseudo-arclength continuation method. We provide a detailed explanation of the corresponding serial and parallel algorithms. Furthermore, we demonstrate the efficiency and correctness of the algorithms by constructing bifurcation diagrams for the 1D Bratu problem and the stationary Kuramoto–Sivashinsky equation considered in 1D and 2D.

This work was supported by the Russian Foundation for Basic Research (grant No. 18-29-10008 mk).

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Authors and Affiliations

  1. Federal Research Center “Computer Science and Control”, Institute for System Analysis, Russian Academy of Science, Moscow, Russia

    Nikolay M. Evstigneev

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  1. Nikolay M. Evstigneev
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Correspondence to Nikolay M. Evstigneev .

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  1. South Ural State University, Chelyabinsk, Russia

    Leonid Sokolinsky

  2. South Ural State University, Chelyabinsk, Russia

    Mikhail Zymbler

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Evstigneev, N.M. (2019). On the Convergence Acceleration and Parallel Implementation of Continuation in Disconnected Bifurcation Diagrams for Large-Scale Problems. In: Sokolinsky, L., Zymbler, M. (eds) Parallel Computational Technologies. PCT 2019. Communications in Computer and Information Science, vol 1063. Springer, Cham. https://doi.org/10.1007/978-3-030-28163-2_9

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  • DOI: https://doi.org/10.1007/978-3-030-28163-2_9

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