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求解双鞍点线性系统的一种改进维数分裂预处理子
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Abstract:
为了提高维数分裂(DS)预处理子和松弛维数分解(RDF)预处理子的性能，针对双鞍点问题，本文提出了一种改进维数分裂(IDS)预处理子，详细分析了预处理矩阵的谱性质并讨论了最优参数。数值实验结果验证了IDS预处理子的有效性。
In order to improve the performance of dimensional splitting (DS) preconditioner and relaxed dimensional factorization (RDF) preconditioner, an improved dimensional splitting (IDS) preconditioner is proposed for the double saddle point problem. The spectral properties of the preconditioned matrix are analyzed in detail and the optimal parameters are discussed. The effectiveness of IDS preconditioner is verified by numerical experiments.
| [1] | Elman, H.C., Silvester, D.J. and Wathen, A.J. (2014) Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press. |
| [2] | Bai, Z. (2010) Block Preconditioners for Elliptic PDE-Constrained Optimization Problems. Computing, 91, 379-395. https://doi.org/10.1007/s00607-010-0125-9 |
| [3] | Bai, Z. and Lu, K. (2021) Optimal Rotated Block-Diagonal Preconditioning for Discretized Optimal Control Problems Constrained with Fractional Time-Dependent Diffusive Equations. Applied Numerical Mathematics, 163, 126-146. https://doi.org/10.1016/j.apnum.2021.01.011 |
| [4] | de Sturler, E. and Liesen, J. (2005) Block-diagonal and Constraint Preconditioners for Nonsymmetric Indefinite Linear Systems. Part I: Theory. SIAM Journal on Scientific Computing, 26, 1598-1619. https://doi.org/10.1137/s1064827502411006 |
| [5] | Cao, Z. (2007) Positive Stable Block Triangular Preconditioners for Symmetric Saddle Point Problems. Applied Numerical Mathematics, 57, 899-910. https://doi.org/10.1016/j.apnum.2006.08.001 |
| [6] | Zhou, S., Yang, A. and Wu, Y. (2016) A Relaxed Block-Triangular Splitting Preconditioner for Generalized Saddle-Point Problems. International Journal of Computer Mathematics, 94, 1609-1623. https://doi.org/10.1080/00207160.2016.1226500 |
| [7] | Chaparpordi, S.H.A., Beik, F.P.A. and Salkuyeh, D.K. (2018) Block Triangular Preconditioners for Stabilized Saddle Point Problems with Nonsymmetric (1, 1)-Block. Computers & Mathematics with Applications, 76, 1544-1553. https://doi.org/10.1016/j.camwa.2018.07.006 |
| [8] | Aslani, H. and Salkuyeh, D.K. (2023) A Block Triangular Preconditioner for a Class of Three-By-Three Block Saddle Point Problems. Japan Journal of Industrial and Applied Mathematics, 40, 1015-1030. https://doi.org/10.1007/s13160-022-00561-8 |
| [9] | Cao, Z.H. (2006) A Class of Constraint Preconditioners for Nonsymmetric Saddle Point Matrices. Numerische Mathematik, 103, 47-61. https://doi.org/10.1007/s00211-006-0675-0 |
| [10] | Bai, Z., Ng, M.K. and Wang, Z. (2009) Constraint Preconditioners for Symmetric Indefinite Matrices. SIAM Journal on Matrix Analysis and Applications, 31, 410-433. https://doi.org/10.1137/080720243 |
| [11] | Benzi, M., Golub, G.H. and Liesen, J. (2005) Numerical Solution of Saddle Point Problems. Acta Numerica, 14, 1-137. https://doi.org/10.1017/s0962492904000212 |
| [12] | Benzi, M. and Guo, X. (2011) A Dimensional Split Preconditioner for Stokes and Linearized Navier-Stokes Equations. Applied Numerical Mathematics, 61, 66-76. https://doi.org/10.1016/j.apnum.2010.08.005 |
| [13] | Benzi, M., Ng, M., Niu, Q. and Wang, Z. (2011) A Relaxed Dimensional Factorization Preconditioner for the Incompressible Navier-Stokes Equations. Journal of Computational Physics, 230, 6185-6202. https://doi.org/10.1016/j.jcp.2011.04.001 |
| [14] | Ren, B., Chen, F. and Wang, X. (2022) Improved Splitting Preconditioner for Double Saddle Point Problems Arising from Liquid Crystal Director Modeling. Numerical Algorithms, 91, 1363-1379. https://doi.org/10.1007/s11075-022-01305-y |
| [15] | Horn, R.A. and Johnson, C.R. (2012). Matrix Analysis. 2nd Edition, Cambridge University Press. https://doi.org/10.1017/cbo9781139020411 |
| [16] | Ren, Z. and Cao, Y. (2015) An Alternating Positive-Semidefinite Splitting Preconditioner for Saddle Point Problems from Time-Harmonic Eddy Current Models. IMA Journal of Numerical Analysis, 36, 922-946. https://doi.org/10.1093/imanum/drv014 |
| [17] | Benzi, M., Deparis, S., Grandperrin, G. and Quarteroni, A. (2016) Parameter Estimates for the Relaxed Dimensional Factorization Preconditioner and Application to Hemodynamics. Computer Methods in Applied Mechanics and Engineering, 300, 129-145. https://doi.org/10.1016/j.cma.2015.11.016 |
| [18] | Tan, N., Huang, T. and Hu, Z. (2012) A Relaxed Splitting Preconditioner for the Incompressible Navier‐Stokes Equations. Journal of Applied Mathematics, 2012, Article ID: 402490. https://doi.org/10.1155/2012/402490 |
| [19] | Huang, Y. (2014) A Practical Formula for Computing Optimal Parameters in the HSS Iteration Methods. Journal of Computational and Applied Mathematics, 255, 142-149. https://doi.org/10.1016/j.cam.2013.01.023 |
| [20] | Elman, H.C., Ramage, A. and Silvester, D.J. (2007) Algorithm 866: IFISS, a Matlab Toolbox for Modelling Incompressible Flow. ACM Transactions on Mathematical Software, 33, Article 14. https://doi.org/10.1145/1236463.1236469 |
