Neumann fractional diffusion problems: BURA solution methods and algorithms

Harizanov, Stanislav; Kosturski, Nikola; Margenov, Svetozar; Vutov, Yavor

doi:10.1016/j.matcom.2020.07.018

Cited by 5 publications

(5 citation statements)

References 19 publications

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“…At the same time, it was noticed that when iterative PCG solvers are used in BURA implementation even for relatively small degrees k, the number of iterations is significantly different, depending on the values of d i (see [3] and the references therein).…”

Section: Discussionmentioning

confidence: 99%

“…The BURA method is originally introduced in [5], see [6] and the references therein for some further developments. The method is generalized to the case of Neumann boundary conditions in [3]. The present study is focused on the efficient implementation of the method for large-scale problems, where preconditioned conjugate gradient (PCG) iterative solver of optimal complexity is applied to the arising auxiliary sparse SPD linear systems.…”

Section: Introductionmentioning

confidence: 99%

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Reduced Sum Implementation of the BURA Method for Spectral Fractional Diffusion Problems

Harizanov¹,

Kosturski²,

Lirkov³

et al. 2021

Preprint

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The numerical solution of spectral fractional diffusion problems in the form A α u = f is studied, where A is a selfadjoint elliptic operator in a bounded domain Ω ⊂ R d , and α ∈ (0, 1]. The finite difference approximation of the problem leads to the system A α u = f , where A is a sparse, symmetric and positive definite (SPD) matrix, and A α is defined by its spectral decomposition. In the case of finite element approximation, A is SPD with respect to the dot product associated with the mass matrix. The BURA method is introduced by the best uniform rational approximation of degree k of t α in [0, 1], denoted by r α,k . Then the approximation u k ≈ u has the formthus requiring the solving of k auxiliary linear systems with sparse SPD matrices. The BURA method has almost optimal computational complexity, assuming that an optimal PCG iterative solution method is applied to the involved auxiliary linear systems. The presented analysis shows that the absolute values of first di k ′ i=1 can be extremely large. In such a case the condition number of A − diI is practically equal to one. Obviously, such systems do not need preconditioning. The next question is if we can replace their solution by directly multiplying f with −ci/ di. Comparative analysis of numerical results is presented as a proof-of-concept for the proposed RS-BURA method.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Reduced Sum Implementation of the BURA Method for Spectral Fractional Diffusion Problems

Harizanov¹,

Kosturski²,

Lirkov³

et al. 2021

Preprint

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“…In the multidimensional case, this means that such a preconditioned iterative solver is applied. In the numerical tests presented in our previous papers [9,11,23], the Boomer-AMG implementation from HYPRE [14] of the algebraic multigrid (AMG) preconditioner in the PCG framework is used for this purpose.…”

Section: The Bura Methodsmentioning

confidence: 99%

“…where C BURA 0.5,k [•] is determined following (20), and (23). The following assumptions will be used in the analysis of the introduced BURA preconditioners:…”

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

confidence: 99%

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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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