Applications for ultrascale systems

Margenov, Svetozar; Rauber, Thomas; Atanassov, Emanouil; Almeida, Francisco; Blanco, Vicente; Čiegis, Raimondas; Cabrera, Alberto; Frashëri, Neki; Harizanov, Stanislav; Kriauzien, Rima; Rünger, Gudula; Segundo, Pablo San; Starikovicius, Adimas; Szabo, Sandor; Zaválnij, Bogdán

doi:10.1049/pbpc024e_ch6

Cited by 1 publication

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“…We stress that in this implementation, the distribution of the 91 solves between the nodes is optimized, taking into account the different number of PCG iterations for each of them, needed to reach the stopping criteria of 10 −10 . Various aspects of the parallel implementation of the surveyed methods are discussed in [22,23,52] 2 . A scalability analysis of the PEX, k ′ -Q and BURA methods is presented in [23], where the test problem in Ω = (0, 1) 3 with a CheckerBoard right-hand-side is considered with up to 512 3 unknowns.…”

Section: Methods Based On Integral Representation Of Amentioning

confidence: 99%

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A Survey on Numerical Methods for Spectral Space-Fractional Diffusion Problems

Harizanov¹,

Lazarov²,

Margenov³

2020

Preprint

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The survey is devoted to numerical solution of the equation A α u = f , 0 < α < 1, where A is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in R d . The fractional power A α is a non-local operator and is defined though the spectrum of A. Due to growing interest and demand in applications of sub-diffusion models to physics and engineering, in the last decade, several numerical approaches have been proposed, studied, and tested. We consider discretizations of the elliptic operator A by using an N -dimensional finite element space V h or finite differences over a uniform mesh with N grid points. In the case of finite element approximation we get a symmetric and positive definite operatorThe numerical solution of this equation is based on the following three equivalent representations of the solution: (1) Dunford-Taylor integral formula (or its equivalent Balakrishnan formula, (2.5)), ( 2) extension of the a second order elliptic problem in Ω × (0, ∞) ⊂ R d+1 [17, 55] (with a local operator) or as a pseudo-parabolic equation in the cylinder (x, t) ∈ Ω × (0, 1), [70, 29], (3) spectral representation (2.6) and the best uniform rational approximation (BURA) of z α on [0, 1], [37,40]. Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of A −α h . In this paper we present the main ideas of these methods and the corresponding algorithms, discuss their accuracy, computational complexity and compare their efficiency and robustness.

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Section: Methods Based On Integral Representation Of Amentioning

confidence: 99%

“…A less commonly used approach to a comparison analysis of parallel efficiency of EEX, PEX, k ′ -Q and BURA methods is proposed in [52]. The presented results are based on using up to 32 nodes of the supercomputer, with a setting of up to 16 cpu cores per node 3 .…”

Section: Methods Based On Integral Representation Of Amentioning

confidence: 99%