On coarse grid correction methods in Krylov subspaces
type: Item
instance of: scholarly article
title: On coarse grid correction methods in Krylov subspaces
zbMATH Open document ID: 1406.65019
author: V. P. Il'in
author: Ya. L. Gur'eva
published in: Journal of Mathematical Sciences (New York)
publication date: 2018-08-17T00:00:00Z
review text: The authors consider the numerical solution of large, sparse, and ill-conditioned systems of linear algebraic equations (SLAEs) originated from a boundary value problem with Dirichlet boundary conditions for a convection-diffusion equation based on a finite volume method of second order derived by the second author [J. Math. Sci., New York 224, No. 6, 900--910 (2017; Zbl 1373.65027); translation from Zap. Nauchn. Semin. POMI 453, 131--147 (2016)]. They emphasize that the method can be also applied to SLAEs arising from finite element approximations with sparse coefficient matrices. In the present paper, the authors propose an algorithm of coarse grid correction in two-level Krylov subspace processes with the aim to accelerate iterative domain decomposition methods (DDMs) with parallelization applied to the above mentioned problem. The two-level method consists of two processes, an outer process of block iterations over subdomains and an inner one for the synchronous solution of the relative small linear subsystems (SMLs) in the subdomains. Each of the SMLs is solved in an own multi-core processor using multithreaded computation on a shared memory. The inner Krylov iterations are traditionally carried out using an incomplete triangular factorization for preconditioning. That results in difficulties in the parallelization which is overcome by the proposed coarse-grid correction as a preconditioning method for the Krylov iterations with slow data exchanges. Applying the DDM, three types of basis functions are defined in the subdomains: the zeroth-order (so-called shelves), the second-order (caps), and the third-order basis functions. An approximate solution \(\vec{u}^n\) of the linear system is refined by the basis functions and the corrected approximate solution \(\vec{\tilde{u}}^n\) is calculated by minimizing the residual norm using the least squares method (LSM). The authors outline that also the singular value decomposition can be used instead of the LSM. The auth…
reviewed by: Georg Hebermehl
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F08
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65F50
Mathematics Subject Classification ID: 65Y05
Mathematics Subject Classification ID: 65N08
Mathematics Subject Classification ID: 65N55
zbMATH DE Number: 6920100
zbMATH Keywords: convection-diffusion equation
zbMATH Keywords: finite volume method
zbMATH Keywords: basis function of the zeroth, first, and second orders
zbMATH Keywords: large systems of linear algebraic equations
zbMATH Keywords: Krylov subspace method
zbMATH Keywords: two-level preconditioning
zbMATH Keywords: domain decomposition
zbMATH Keywords: coarse grid correction
zbMATH Keywords: minimization of the residual
zbMATH Keywords: least squares
zbMATH Keywords: singular value decomposition method
MaRDI profile type: Publication
full work available at URL: s10958-018-3907-9
OpenAlex ID: W2809862102
cites work: Q4414855
cites work: A Deflated Version of the Conjugate Gradient Algorithm
DOI: 10.1007/S10958-018-3907-9
description: scientific article; zbMATH DE number 6920100