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Formulation of a preconditioned Algorithm for the Conjugate Gradient Squared Method in Accordance with Its Logical Structure

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Applied Mathematics 2015年第8期6卷 1389-1406页

作者： Shoji Itoh Masaaki Sugihara Division of Science School of Science and Engineering Tokyo Denki University Saitama Japan Department of Physics and Mathematics College of Science and Engineering Aoyama Gakuin University Kanagawa Japan

In this paper, we propose an improved preconditioned algorithm for the conjugate gradient squared method (improved PCGS) for the solution of linear equations. Further, the logical structures underlying the formation of this preconditioned algorithm are demonstrated via a number of theorems. This improved PCGS algorithm retains some mathematical properties that are associated with the CGS derivation from the bi-conjugate gradient method under a non-preconditioned system. A series of numerical comparisons with the conventional PCGS illustrate the enhanced effectiveness of our improved scheme with a variety of preconditioners. This logical structure underlying the formation of the improved PCGS brings a spillover effect from various bi-Lanczos-type algorithms with minimal residual operations, because these algorithms were constructed by adopting the idea behind the derivation of CGS. These bi-Lanczos-type algorithms are very important because they are often adopted to solve the systems of linear equations that arise from large-scale numerical simulations.

关键词： Linear Systems Krylov Subspace Method Bi-Lanczos Algorithm preconditioned System PCGS

A theoretical analysis on efficiency of some Newton-PCG methods

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Science China Mathematics 2005年第8期48卷 1046-1064页

作者： DENG Naiyang, ZHANG Jianzhong & ZHONG Ping China Agricultural University, Beijing 100083, China City University of Hong Kong, Hong Kong, China China Agricultural University Beijing China City University of Hong Kong Hong Kong China

In this paper, we study the efficiency issue of inexact Newton-type methods for smooth unconstrained optimization problems under standard assumptions from theoretical point of view by discussing a concrete Newton-PCG algorithm. In order to compare the algorithm with Newton's method, a ratio between the measures of their approximate efficiencies is investigated. Under mild conditions, it is shown that first, this ratio is larger than 1, which implies that the Newton-PCG algorithm is more efficient than Newton's method, and second, this ratio increases when the dimension n of the problem increases and tends to infinity at least at a rate In n/In 2 when n→∞, which implies that in theory the Newton-PCG algorithm is much more efficient for middle- and large-scale problems. These theoretical results are also supported by our preliminary numerical experiments.

关键词： unconstrained optimization, Newton's method, preconditioned conjugate gradient method, efficiency.

Can Newton method be surpassed

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Chinese Science Bulletin 1999年第2期44卷 132-134页

作者： Naiyang Deng Zhaozhi Wang China Agr Univ Beijing 100083 Peoples R China

A local algorithm is proposed for unconstrained optimization problem. Compared with the traditional Newton method with Choleski factorization, this algorithm has the same quadratic convergence. But its computation cos... 详细信息

关键词： Newton method Choleski factorization preconditioned conjugate gradient method.

DOMAIN DECOMPOSITION PRECONDITIONERS FOR SECOND-ORDER HYPERBOLIC EQUATIONS ON L-SHAPED REGIONS

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Numerical Mathematics A Journal of Chinese Universities(English Series) 1995年第1期4卷 54-62页

作者：金小庆王朝光 Faculty of Science and Technology University of MacaoMacao. Department of Mathematics University of Hong KongHong Kong.

Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations on L-shaped region are considered. We analyse the use of domain deocmposilion preconditioner.s for the solution of linear systems via the preconditioned conjugate gradient method. For the constant-coefficient second-order hyperbolic equaions with initial and Dirichlet boundary conditions,we prove that the conditionnumber of the preconditioned interface system is bounded by 2+x2 2+0.46x2 where x is the quo-tient between the lime and space steps. Such condition number produces a convergence rale that is independent of gridsize and aspect ratios. The results could be extended to parabolic equations.

关键词： Domain decomposition hyperbolie equation capacitance matrix condition number preconditioned conjugate gradient method

SINE TRANSFORM PRECONDITIONERS FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS

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Numerical Mathematics A Journal of Chinese Universities(English Series) 1993年第1期2卷 116-123页

作者：金小庆 Faculty of Science and Technology University of Macao Macao.

In this paper, we are concerned with the numerical solution of second-order partial differential equations. We analyse the use of the Sine Transform precondilioners for the solution of linear systems arising from the discretization of p.d.e. via the preconditioned conjugate gradient method. For the second-order partial differential equations with Dirichlel boundary conditions, we prove that the condition number of the preconditioned system is O(1) while the condition number of the original system is O(m 2) Here m is the number of interior gridpoints in each direction. Such condition number produces a linear convergence rale.

关键词： Sine Transform finite difference method second-order partial differential equation condition number preconditioned conjugate gradient method

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近景区域网平差的预处理共轭梯度稀疏解法

国土资源遥感 2015年第1期27卷 44-47页

作者：徐振亮李艳焕闫利晏磊北京大学空间信息集成与3S工程应用北京市重点实验室北京100871 辽宁工程技术大学审计处阜新123009 武汉大学测绘学院武汉430079

针对大规模、近病态法的近景区域网平差法方程快速解算问题,提出基于预处理共轭梯度(preconditioned conjugate gradient,PCG)法的稀疏解算方法。首先,通过选择与法方程系数矩阵对应的对角平方根矩阵作为预处理矩阵,以改变待估参数向量... 详细信息

针对大规模、近病态法的近景区域网平差法方程快速解算问题,提出基于预处理共轭梯度(preconditioned conjugate gradient,PCG)法的稀疏解算方法。首先,通过选择与法方程系数矩阵对应的对角平方根矩阵作为预处理矩阵,以改变待估参数向量的坐标基,进而改善法方程系数矩阵性态,达到利用PCG提高收敛速度和解算精度的目的;然后,通过应用稀疏矩阵提高平差法方程系数矩阵的储存与求解效率。实验结果证明,该方法不影响摄影测量中区域网平差中多类、多尺度参数同时解算的收敛域,不但具有很高的解算精度,而且速度较快。

关键词：预处理稀疏矩阵预处理共轭梯度(PCG) 空中三角测量光束法平差从运动到结构 preconditioned conjugate gradient(PCG)

A Finite Volume Unstructured Mesh Method for Fractional-in-space Allen-Cahn Equation

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Chinese Quarterly Journal of Mathematics 2017年第4期32卷 345-354页

作者： CHEN Ai-min LIU Fa-wang School of Mathematics and Statistics Henan University Kaifeng475004 China Institute of Applied Mathematics Henan University Kaifeng 475004 China Laboratory of Data Analysis Technology Henan University Kaifeng 475004 China School of Mathematical Sciences Queensland Universityof Technology GPO Box 2434 Brisbane Qld. 4001 Australia

Fractional-in-space Allen-Cahn equation containing a very strong nonlinear source term and small perturbation shows metastability and a quartic double well *** a finite volume unstructured triangular mesh method, the present paper solves the twodimensional fractional-in-space Allen-Cahn equation with homogeneous Neumann boundary condition on different irregular domains. The efficiency of the method is presented through numerical computation of the two-dimensional fractional-in-space Allen-Cahn equation on different domains.

关键词： fractional-in-space Allen-Cahn equation finite volume method matrix transfertechnique preconditioned Lanczos method