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Volume 12, Issue 2
Restrictive Preconditioning for Convection-Diffusion Distributed Control Problems

Wei Feng, Zeng-Qi Wang, Ruo-Bing Zhong & Galina V. Muratova

East Asian J. Appl. Math., 12 (2022), pp. 233-246.

Published online: 2022-02

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  • Abstract

The restrictive preconditioning technique is employed in the preconditioned conjugate gradient and preconditioned Chebyshev iteration methods for the saddle point linear systems arising in convection-diffusion control problems. Utilizing an appropriate approximation of Schur complement, one obtains preconditioned matrix with eigenvalues located in the interval [1/2,1]. The convergence rate of the methods is studied. Unlike the restrictively preconditioned conjugate gradient method, the restrictively preconditioned Chebyshev iteration method is more tolerant to the inexact execution of the preconditioning. This indicates that the preconditioned Chebyshev iteration method is more practical when dealing with large scale linear systems. Theoretical and numerical results demonstrate that the iteration count of the solvers used do not depend the mesh size, the regularization parameter and on the Peclet number.

  • AMS Subject Headings

65F10, 65N22

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-12-233, author = {Feng , WeiWang , Zeng-QiZhong , Ruo-Bing and Muratova , Galina V.}, title = {Restrictive Preconditioning for Convection-Diffusion Distributed Control Problems}, journal = {East Asian Journal on Applied Mathematics}, year = {2022}, volume = {12}, number = {2}, pages = {233--246}, abstract = {

The restrictive preconditioning technique is employed in the preconditioned conjugate gradient and preconditioned Chebyshev iteration methods for the saddle point linear systems arising in convection-diffusion control problems. Utilizing an appropriate approximation of Schur complement, one obtains preconditioned matrix with eigenvalues located in the interval [1/2,1]. The convergence rate of the methods is studied. Unlike the restrictively preconditioned conjugate gradient method, the restrictively preconditioned Chebyshev iteration method is more tolerant to the inexact execution of the preconditioning. This indicates that the preconditioned Chebyshev iteration method is more practical when dealing with large scale linear systems. Theoretical and numerical results demonstrate that the iteration count of the solvers used do not depend the mesh size, the regularization parameter and on the Peclet number.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.080621.030921}, url = {http://global-sci.org/intro/article_detail/eajam/20252.html} }
TY - JOUR T1 - Restrictive Preconditioning for Convection-Diffusion Distributed Control Problems AU - Feng , Wei AU - Wang , Zeng-Qi AU - Zhong , Ruo-Bing AU - Muratova , Galina V. JO - East Asian Journal on Applied Mathematics VL - 2 SP - 233 EP - 246 PY - 2022 DA - 2022/02 SN - 12 DO - http://doi.org/10.4208/eajam.080621.030921 UR - https://global-sci.org/intro/article_detail/eajam/20252.html KW - Convection-diffusion distributed control problem, restrictive preconditioning, conjugate gradient method, Chebyshev semi-iteration method. AB -

The restrictive preconditioning technique is employed in the preconditioned conjugate gradient and preconditioned Chebyshev iteration methods for the saddle point linear systems arising in convection-diffusion control problems. Utilizing an appropriate approximation of Schur complement, one obtains preconditioned matrix with eigenvalues located in the interval [1/2,1]. The convergence rate of the methods is studied. Unlike the restrictively preconditioned conjugate gradient method, the restrictively preconditioned Chebyshev iteration method is more tolerant to the inexact execution of the preconditioning. This indicates that the preconditioned Chebyshev iteration method is more practical when dealing with large scale linear systems. Theoretical and numerical results demonstrate that the iteration count of the solvers used do not depend the mesh size, the regularization parameter and on the Peclet number.

Wei Feng, Zeng-Qi Wang, Ruo-Bing Zhong & Galina V. Muratova. (2022). Restrictive Preconditioning for Convection-Diffusion Distributed Control Problems. East Asian Journal on Applied Mathematics. 12 (2). 233-246. doi:10.4208/eajam.080621.030921
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