Parallel solvers for fractional power diffusion problems

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

doi:10.1002/cpe.4216

Cited by 19 publications

(18 citation statements)

References 30 publications

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“… The semi‐infinite cylinder is then truncated to allow numerical solution of elliptic problem in a bounded domain

normalΩ \times false[0, T false] \subset {double-struckR}^{d + 1}

. In our previous studies,)this method has led to a worse parallel scalability compared to the remaining methods and it is not used for 3D problems in this work. M2 Transformation to a pseudo‐parabolic problem . Here, the new additional dimension is defined by the pseudo‐time.…”

Section: State Of the Art In Numerical Solution Methodsmentioning

confidence: 99%

“…In the literature, several different numerical quadrature formulas are proposed to approximate integral . In our previous studies, we have used the quadrature formula based on a β ‐dependent graded mesh. However, initial computational experiments solving the current 3D test problem have shown superior results using another quadrature formula with uniformly distributed quadrature points y j = kj :

\begin{matrix} alignleft \\ align-1 U_{h}^{M 3} = \frac{2 k \sin (π β)}{π} \sum_{j = - m_{1}}^{m_{2}} e^{2 (β - 1) y_{j}} {(e^{- 2 y_{j}} I_{h} + L_{h})}^{- 1} f_{h}, & align-2 \end{matrix}

where m 1 = ⌈ π 2 /(4 βk 2 )⌉ and m 2 = ⌈ π 2 /(4(1 − β ) k 2 )⌉.…”

Section: Quadrature Methods (M3)mentioning

confidence: 99%

“…15,16 The semi-infinite cylinder is then truncated to allow numerical solution of elliptic problem in a bounded domain Ω × [0, T] ⊂ R d+1 . In our previous studies, 24,25 this method has led to a worse parallel scalability compared to the remaining methods and it is not used for 3D problems in this work.…”

Section: State Of the Art In Numerical Solution Methodsmentioning

confidence: 99%

“…A more detailed analysis of the developed parallel algorithms has been performed in the extended paper . Strong and weak parallel scalability has been analyzed for different partitionings of the problem domain.…”

Section: Introductionmentioning

confidence: 99%

“…A more detailed analysis of the developed parallel algorithms has been performed in the extended paper. 25 Strong and weak parallel scalability has been analyzed for different partitionings of the problem domain. The algebraic multigrid preconditioner BoomerAMG from the well-known HYPRE library 27,28 has been used in that study.…”

mentioning

confidence: 99%

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

Čiegis

Starikovičius

Margenov

et al. 2019

Concurrency and Computation

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Summary In this paper, we develop and investigate the parallel numerical algorithms for three different state‐of‐the‐art numerical methods for solving the non‐local problems described by fractional powers of elliptic operators. These methods transform the non‐local problem into some local differential problems of elliptic or parabolic type. A two‐level parallelization approach is applied to construct the efficient parallel algorithms using the domain decomposition and master‐slave methods, to deal with the increase in computational complexity. We show and compare the serial and parallel solution times that are required to achieve similar accuracy of the solution using different algorithms. Results of extensive convergence tests are presented solving a three‐dimensional test problem with known decrease of the solution's convergence rate depending on the fractional power coefficient. We analyze and discuss the non‐trivial question, which parallel algorithm is recommended to achieve certain accuracy for the given fractional power coefficient.

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“… The semi‐infinite cylinder is then truncated to allow numerical solution of elliptic problem in a bounded domain

normalΩ \times false[0, T false] \subset {double-struckR}^{d + 1}

Section: State Of the Art In Numerical Solution Methodsmentioning

confidence: 99%

\begin{matrix} alignleft \\ align-1 U_{h}^{M 3} = \frac{2 k \sin (π β)}{π} \sum_{j = - m_{1}}^{m_{2}} e^{2 (β - 1) y_{j}} {(e^{- 2 y_{j}} I_{h} + L_{h})}^{- 1} f_{h}, & align-2 \end{matrix}

where m 1 = ⌈ π 2 /(4 βk 2 )⌉ and m 2 = ⌈ π 2 /(4(1 − β ) k 2 )⌉.…”

Section: Quadrature Methods (M3)mentioning

confidence: 99%

Section: State Of the Art In Numerical Solution Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Čiegis

Starikovičius

Margenov

et al. 2019

Concurrency and Computation

Self Cite

View full text Add to dashboard Cite

show abstract

Parallel solvers for fractional power diffusion problems

Čiegis

Starikovičius

Margenov

et al. 2017

Concurrency and Computation

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Summary Mathematical models with fractional‐order differential operators are computationally expensive due to the non‐local nature of these operators. In this work, we construct and investigate parallel solvers for problems described by fractional powers of elliptic operators, like fractional diffusion. Three state‐of‐the‐art approaches are used to transform the non‐local fractional‐order differential problem into local partial differential equation problems formulated in a space of higher dimension. Numerical schemes and parallel algorithms are developed for all three approaches. The resulting parallel algorithms have very different properties. We investigate the weak and strong scalability of the developed parallel algorithms and compare their parallel performance.

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Limited‐memory polynomial methods for large‐scale matrix functions

Güttel,

Kressner,

Lund

2020

GAMM-Mitteilungen

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Matrix functions are a central topic of linear algebra, and problems requiring their numerical approximation appear increasingly often in scientific computing. We review various limited‐memory methods for the approximation of the action of a large‐scale matrix function on a vector. Emphasis is put on polynomial methods, whose memory requirements are known or prescribed a priori. Methods based on explicit polynomial approximation or interpolation, as well as restarted Arnoldi methods, are treated in detail. An overview of existing software is also given, as well as a discussion of challenging open problems.

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Parallel solvers for fractional power diffusion problems

Cited by 19 publications

References 30 publications

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

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

Parallel solvers for fractional power diffusion problems

Limited‐memory polynomial methods for large‐scale matrix functions

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