Scalability analysis of different parallel solvers for 3D fractional power diffusion problems

Čiegis, Raimondas; Starikovičius, Vadimas; Margenov, Svetozar; Kriauzienė, Rima

doi:10.1002/cpe.5163

Cited by 7 publications

(6 citation statements)

References 34 publications

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“…[74, e.g. Formulas ( 22), (23)]. Here we shall present the main idea, while the interested reader can find all details and relevant numerical experiments in [74].…”

Section: Methods Based On Integral Representation Of Amentioning

confidence: 99%

“…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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“…[74, e.g. Formulas ( 22), (23)]. Here we shall present the main idea, while the interested reader can find all details and relevant numerical experiments in [74].…”

Section: Methods Based On Integral Representation Of Amentioning

confidence: 99%

Section: Methods Based On Integral Representation Of Amentioning

confidence: 99%

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

Harizanov¹,

Lazarov²,

Margenov³

2020

Preprint

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“…For large discrete linear systems approximating discrete fractional elliptic equations, parallel multigrid solvers are used. The efficiency of these solvers is studied in [25] and the references therein. We refer to recommendations given in this paper.…”

Section: Preconditioning Of the Coupled Problems: Condition Number Es...mentioning

confidence: 99%

Rational Approximations in Robust Preconditioning of Multiphysics Problems

2022

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Multiphysics or multiscale problems naturally involve coupling at interfaces which are manifolds of lower dimensions. The block-diagonal preconditioning of the related saddle-point systems is among the most efficient approaches for numerically solving large-scale problems in this class. At the operator level, the interface blocks of the preconditioners are fractional Laplacians. At the discrete level, we propose to replace the inverse of the fractional Laplacian with its best uniform rational approximation (BURA). The goal of the paper is to develop a unified framework for analysis of the new class of preconditioned iterative methods. As a final result, we prove that the proposed preconditioners have optimal computational complexity O(N), where N is the number of unknowns (degrees of freedom) of the coupled discrete problem. The main theoretical contribution is the condition number estimates of the BURA-based preconditioners. It is important to note that the obtained estimates are completely analogous for both positive and negative fractional powers. At the end, the analysis of the behavior of the relative condition numbers is aimed at characterizing the practical requirements for minimal BURA orders for the considered Darcy–Stokes and 3D–1D examples of coupled problems.

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“…Last but not least, we point to a comparison of solvers with respect to their parallelization properties. An extensive analysis of various parallel solvers for stationary fractional power elliptic problems was done in [38,39].…”

Section: Introductionmentioning

confidence: 99%

A Comparison of Discrete Schemes for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

Čiegis

Dapšys

2021

Mathematics

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The main aim of this article is to analyze the efficiency of general solvers for parabolic problems with fractional power elliptic operators. Such discrete schemes can be used in the cases of non-constant elliptic operators, non-uniform space meshes and general space domains. The stability results are proved for all algorithms and the accuracy of obtained approximations is estimated by solving well-known test problems. A modification of the second order splitting scheme is presented, it combines the splitting method to solve locally the nonlinear subproblem and the AAA algorithm to solve the nonlocal diffusion subproblem. Results of computational experiments are presented and analyzed.

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Scalability analysis of different parallel solvers for 3D fractional power diffusion problems

Cited by 7 publications

References 34 publications

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

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

Rational Approximations in Robust Preconditioning of Multiphysics Problems

A Comparison of Discrete Schemes for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

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