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ON BLOCK PRECONDITIONERS FOR PDE-CONSTRAINED OPTIMIZATION PROBLEMS

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Journal of Computational Mathematics 2014年第3期32卷 272-283页

作者： Xiaoying Zhang Yumei Huang School of Mathematics and Statistics Lanzhou University Lanzhou 730000 China

Recently, Bal proposed a block-counter-diagonal and a block-counter-triangular precon- ditioning matrices to precondition the gmres method for solving the structured system of linear equations arising from the Galerkin finite-element discretizations of the distributed control problems in （Computing 91 （2011） 379-395）. He analyzed the spectral properties and derived explicit expressions of the eigenvalues and eigenvectors of the preconditioned matrices. By applying the special structures and properties of the eigenvector matrices of the preconditioned matrices, we derive upper bounds for the 2-norm condition numbers of the eigenvector matrices and give asymptotic convergence factors of the preconditioned gmres methods with the block-counter-diagonal and the block-counter-triangular pre- conditioners. Experimental results show that the convergence analyses match well with the numerical results.

关键词： PDE-constrained optimization gmres method Preconditioner Condition number Asymptotic convergence factor.

An Augmented Approach for the Pressure Boundary Condition in a Stokes Flow

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Communications in Computational Physics 2006年第5期1卷 874-885页

作者： Zhilin Li Xiaohai Wan Kazufumi Ito Sharon R.Lubkin Center for Research in Scientific Computation&Department of Mathematics North Carolina State UniversityRaleighNC 27695-8205USA Biomathematics Graduate Program&Department of Mathematics North Carolina State UniversityRaleighNC 27695-8203USA

An augmented method is proposed for solving stationary incompressible Stokes equations with a Dirichlet boundary condition along parts of the *** this approach,the normal derivative of the pressure along the parts of the boundary is introduced as an additional variable and it is solved by the gmres iterative *** dimension of the augmented variable in discretization is the number of grid points along the boundary which is O(N).Each gmres iteration(or one matrix-vector multiplication)requires three fast Poisson solvers for the pressure and the *** our numerical experiments,only a few iterations are *** have also combined the augmented approach for Stokes equations involving interfaces,discontinuities,and singularities.

关键词： Incompressible Stokes equations pressure boundary condition augmented method interface problem immersed interface method fast Poisson solver gmres method

Iterative convergence of boundary-volume integral equation method

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Earthquake Science 2011年第5期24卷 391-400页

作者： Gengxin Yu Liyun Fu Institute of Geology and Geophysics Chinese Academy of Sciences Beijing 100029 China

The boundary-volume integral equation numerical technique can be a powerful tool for piecewise heterogeneous media, but it is limited to small problems or low frequencies because of great computational cost. Therefore, a restarted gmres method is applied to solve large-scale boundary-volume scattering problems in this paper to overcome the computational barrier. The iterative method is firstly applied to responses of dimensionless frequency to a semicircular alluvial valley filled with sediments, compared with the standard Gaussian elimination method. Then the method is tested by a heterogeneous multilayered model to show its applicability. Numerical experiments indicate that the preconditioned gmres method can significantly improve computational efficiency especially for large Earth models and high frequencies, but with a faster convergence for the left diagonal preconditioning.

关键词： boundary-volume integral equation generalized Lipmann-Schwinger integral equation gmres method diagonal preconditioner

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求解分数阶对流弥散方程的边界值法

高等学校计算数学学报 2016年第4期38卷 350-356页

作者：杨淑伶广东工业大学应用数学学院

1引言在现代地下水动力学研究中,对流弥散方程常用来刻画污染物等的运移过程.但在介质不纯或不规则的分形多孔介质情况下,穿透曲线的实测数据呈不对称的偏态分布,即存在提前穿透和拖尾的现象[1].研究表明,此时污染物的运移过程已从局域... 详细信息

1引言在现代地下水动力学研究中,对流弥散方程常用来刻画污染物等的运移过程.但在介质不纯或不规则的分形多孔介质情况下,穿透曲线的实测数据呈不对称的偏态分布,即存在提前穿透和拖尾的现象[1].研究表明,此时污染物的运移过程已从局域性的布朗运动改变为非局域性的Levy过程,考虑到分数阶导数与反常扩散过程在本质上相类似,因而分数阶对流弥散方程(FADE)被引入以取代传统的整数阶对流弥散方程[2].

关键词： fractional advection dispersion equation boundary value methods gmres method preconditioner

求解PageRank问题的gmres重启的松弛两步分裂法

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高等学校计算数学学报 2018年第4期40卷 331-345页

作者：邱志红顾传青上海大学理学院

1引言随着互联网信息的快速发展,网络搜索引擎是人们通过互联网获取信息的重要工具.搜索引擎的关键技术就是网页排序,其核心是搜索引擎算法的设计.斯坦福大学的Larry和Sergey Brin在1996年提出的PageRank算法,主要是用于网页评级的系统... 详细信息

1引言随着互联网信息的快速发展,网络搜索引擎是人们通过互联网获取信息的重要工具.搜索引擎的关键技术就是网页排序,其核心是搜索引擎算法的设计.斯坦福大学的Larry和Sergey Brin在1996年提出的PageRank算法,主要是用于网页评级的系统,该算法在Google搜索引擎中取得了很大的成功.所谓PageRank问题就是求解Google矩阵A的首特征值1所对应的特征向量,即线

关键词： PageRank damping factor PIO method gmres method