Volume 18, Issue 5
An Adaptive Combined Preconditioner with Applications in Radiation Diffusion Equations

Xiaoqiang Yue, Shi Shu, Xiaowen Xu & Zhiyang Zhou

Commun. Comput. Phys., 18 (2015), pp. 1313-1335.

Published online: 2018-04

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  • Abstract

The paper aims to develop an effective preconditioner and conduct the convergence analysis of the corresponding preconditioned GMRES for the solution of discrete problems originating from multi-group radiation diffusion equations. We firstly investigate the performances of the most widely used preconditioners (ILU(k) and AMG) and their combinations ($B_{co}$ and $\widetilde{B}_{co}$), and provide drawbacks on their feasibilities. Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing the features suitable for AMG using more detailed measurements on multiscale nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present an adaptive combined preconditioner $B^α_{co}$ involving an improved ILU(0) along with its convergence constraints. Numerical results demonstrate that $B^α_{co}$-GMRES holds the best robustness and efficiency. At last, we analyze the convergence of GMRES with combined preconditioning which not only provides a persuasive support for our proposed algorithms, but also updates the existing estimation theory on condition numbers of combined preconditioned systems.

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@Article{CiCP-18-1313, author = {}, title = {An Adaptive Combined Preconditioner with Applications in Radiation Diffusion Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {18}, number = {5}, pages = {1313--1335}, abstract = {

The paper aims to develop an effective preconditioner and conduct the convergence analysis of the corresponding preconditioned GMRES for the solution of discrete problems originating from multi-group radiation diffusion equations. We firstly investigate the performances of the most widely used preconditioners (ILU(k) and AMG) and their combinations ($B_{co}$ and $\widetilde{B}_{co}$), and provide drawbacks on their feasibilities. Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing the features suitable for AMG using more detailed measurements on multiscale nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present an adaptive combined preconditioner $B^α_{co}$ involving an improved ILU(0) along with its convergence constraints. Numerical results demonstrate that $B^α_{co}$-GMRES holds the best robustness and efficiency. At last, we analyze the convergence of GMRES with combined preconditioning which not only provides a persuasive support for our proposed algorithms, but also updates the existing estimation theory on condition numbers of combined preconditioned systems.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.091014.060315a}, url = {http://global-sci.org/intro/article_detail/cicp/11070.html} }
TY - JOUR T1 - An Adaptive Combined Preconditioner with Applications in Radiation Diffusion Equations JO - Communications in Computational Physics VL - 5 SP - 1313 EP - 1335 PY - 2018 DA - 2018/04 SN - 18 DO - http://doi.org/10.4208/cicp.091014.060315a UR - https://global-sci.org/intro/article_detail/cicp/11070.html KW - AB -

The paper aims to develop an effective preconditioner and conduct the convergence analysis of the corresponding preconditioned GMRES for the solution of discrete problems originating from multi-group radiation diffusion equations. We firstly investigate the performances of the most widely used preconditioners (ILU(k) and AMG) and their combinations ($B_{co}$ and $\widetilde{B}_{co}$), and provide drawbacks on their feasibilities. Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing the features suitable for AMG using more detailed measurements on multiscale nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present an adaptive combined preconditioner $B^α_{co}$ involving an improved ILU(0) along with its convergence constraints. Numerical results demonstrate that $B^α_{co}$-GMRES holds the best robustness and efficiency. At last, we analyze the convergence of GMRES with combined preconditioning which not only provides a persuasive support for our proposed algorithms, but also updates the existing estimation theory on condition numbers of combined preconditioned systems.

Xiaoqiang Yue, Shi Shu, Xiaowen Xu & Zhiyang Zhou. (2020). An Adaptive Combined Preconditioner with Applications in Radiation Diffusion Equations. Communications in Computational Physics. 18 (5). 1313-1335. doi:10.4208/cicp.091014.060315a
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