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STOPPING CRITERIA FOR THE conjugate gradient algorithm IN HIGH-ORDER FINITE ELEMENT METHODS\ast

SIAM JOURNAL ON SCIENTIFIC COMPUTING 2025年第1期47卷 A238-A267页

作者： Guo, Yichen DE Sturler, Eric Warburton, Tim Virginia Tech Dept Math Blacksburg VA 24061 USA

We consider stopping criteria that balance algebraic and discretization errors for the conjugate gradient algorithm applied to high-order finite element discretizations of Poisson problems. First, we introduce a new stopping criterion that suggests stopping when the norm of the linear system residual is less than a small fraction of an error indicator derived directly from the residual. This indicator shares the same mesh size and polynomial degree scaling as the norm of the residual, resulting in a robust criterion regardless of the mesh size, the polynomial degree, and the shape regularity of the mesh. Second, for solving Poisson problems with highly variable piecewise constant coefficients, we introduce a sub domain-based criterion that recommends stopping when the norm of the linear system residual restricted to each subdomain is smaller than the corresponding indicator also restricted to that sub domain. Reliability and efficiency theorems for the first criterion are established. Numerical experiments, including tests with highly variable piecewise constant coefficients and a GPU-accelerated three-dimensional elliptic solver, demonstrate that the proposed criteria efficiently avoid both premature termination and oversolving.

关键词： stopping criteria conjugate gradient algorithm high-order finite element method

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An efficient modified conjugate gradient algorithm under Wolfe conditions with applications in compressive sensing

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 458卷

作者： Zhu, Zhibin Huang, Jiaqi Liu, Ying Ding, Yuehong Guilin Univ Elect Technol Sch Math & Comp Sci Guilin 541002 Peoples R China Ctr Appl Math Guangxi GUET Guilin 541002 Peoples R China Guangxi Coll Guilin 541002 Peoples R China Univ Key Lab Data Anal & Computat Guilin 541002 Peoples R China

This paper presents a new modified conjugate gradient (NMCG) algorithm which satisfies the sufficient descent property under any line search for unconstrained optimization problems. We analyze that the algorithm is global convergence under the Wolfe line search. We use the proposed algorithm NMCG to unconstrained optimization problems to prove its effectiveness. Furthermore, we also extend it to solve image restoration and sparse signal recovery problems in compressive sensing, and the results indicate that our algorithm is effective and competitive.

关键词： Unconstrained optimization conjugate gradient algorithm Wolfe line search Sufficient descent property Global convergence Compressive sensing

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conjugate gradient algorithm in the Four-Dimensional Variational Data Assimilation System in GRAPES

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Journal of Meteorological Research 2018年第6期32卷 974-984页

作者： Yongzhu LIU Lin ZHANG Zhihua LIAN National Meteorological Center Beijing 100081

Minimization algorithms are singular components in four-dimensional variational data assimilation(4DVar).In this paper,the convergence and application of the conjugate gradient algorithm(CGA),which is based on the Lanczos iterative algorithm and the Hessian matrix derived from tangent linear and adjoint models using a non-hydrostatic framework,are investigated in the 4DVar ***,the influence of the Gram-Schmidt orthogonalization of the Lanczos vector on the convergence of the Lanczos algorithm is *** results show that the Lanczos algorithm without orthogonalization fails to converge after the ninth iteration in the 4DVar minimization,while the orthogonalized Lanczos algorithm converges ***,the convergence and computational efficiency of the CGA and quasi-Newton method in batch cycling assimilation experiments are compared on the 4DVar platform of the Global/Regional Assimilation and Prediction System(GRAPES).The CGA is 40%more computationally efficient than the quasi-Newton method,although the equivalent analysis results can be obtained by using either the CGA or the quasi-Newton ***,the CGA based on Lanczos iterations is better for solving the optimization problems in the GRAPES 4DVar system.

关键词： numerical weather prediction Global/Regional Assimilation and Prediction System four-dimensional variation conjugate gradient algorithm Lanczos algorithm

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conjugate-gradient algorithm BASED ON MATRIX REPRESENTATION OF LINEAR-OPERATORS IN FINITE DIMENSIONAL HILBERT-SPACES

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ELECTRONICS LETTERS 1988年第7期24卷 405-406页

作者： VANRENSBURG, DJJ MCNAMARA, DA Dept. of Electron. & Comput. Eng. Pretoria Univ.

The matrix representation of a linear operator is used in the conjugate gradient algorithm to solve electromagnetic boundary value problems. The use of this approach obviates the difficult task of finding the adjoint ... 详细信息

关键词： conjugate gradient algorithm electromagnetic boundary value problems electromagnetic field theory finite dimensional Hilbert spaces matrix algebra matrix representation of linear operators

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A stopping criterion for the conjugate gradient algorithm in the framework of anisotropic adaptive finite elements

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COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING 2009年第4期25卷 339-355页

作者： Picasso, M. Ecole Polytech Fed Lausanne Inst Anal & Calcul Sci CH-1015 Lausanne Switzerland

We propose a simple stopping criterion for the conjugate gradient (CG) algorithm in the framework of anisotropic, adaptive finite elements for elliptic problems. The goal of the adaptive algorithm is to find a triangulation such that the estimated relative error is close to a given tolerance TOL. We propose to stop the CG algorithm whenever the residual vector has Euclidian norm less than a small fraction of the estimated error. This stopping criterion is based on a posteriori error estimates between the true solution u and the computed solution u(h)(n) (the superscript n stands for the CG iteration number, the subscript It for the typical mesh size) and on heuristics to relate the error between u(h) and u(h)(n) to the residual vector. Numerical experiments with anisotropic adaptive meshes show that the total number of CG iterations can be divided by 10 without significant discrepancy in the computed results. Copyright (C) 2008 John Wiley & Sons, Ltd.

关键词： conjugate gradient algorithm stopping criterion a posteriori error estimates anisotropic adaptive finite elements

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Flow search approach and new bounds for the m-step linear conjugate gradient algorithm

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2004年第1期120卷 53-71页

作者： Huang, HX Liang, ZA Pardalos, PM Tsinghua Univ Dept Math Sci Beijing 100084 Peoples R China Shanghai Univ Finance & Econ Dept Appl Math Shanghai Peoples R China Univ Florida Dept Ind & Syst Engn Gainesville FL 32611 USA

A flow search approach is presented in this paper. In the approach, each iterative process involves a subproblem, whose variables are the stepsize parameters. Every feasible solution of the subproblem corresponds to some serial search stages, the stepsize parameters in different search stages may interact mutually, and their optimal values are determined by evaluating the total effect of the interaction. The main idea of the flow search approach is illustrated via the minimization of a convex quadratic function. Based on the flow search approach, some properties of the m-step linear conjugate gradient algorithm are analyzed and new bounds on its convergence rate are also presented. Theoretical and numerical results indicate that the new bounds are better than the well-known ones.

关键词： flow search approach conjugate gradient algorithm m-step linear conjugate gradient algorithm convergence rate

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VECTORIZATION AND PARALLELIZATION OF THE conjugate-gradient algorithm ON HYPERCUBE-CONNECTED VECTOR PROCESSORS

MICROPROCESSING AND MICROPROGRAMMING

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MICROPROCESSING AND MICROPROGRAMMING 1990年第2期29卷 67-82页

作者： AYKANAT, C OZGUNER, F SCOTT, DS BILKENT UNIV DEPT COMP & INFORMAT SCI06572 MALTEPETURKEY INTEL SCI COMP BEAVERTONOR 97006 OHIO STATE UNIV DEPT ELECT ENGNCOLUMBUSOH 43210

Solution of large sparse linear systems of equations in the form A x = b constitutes a significant amount of the computations in the simulation of physical phenomena [1]. For example, the finite element discretization of a regular domain, with proper ordering of the variables x , renders a banded N × N coefficient matrix A . The conjugate gradient (CG) [2,3] algorithm is an iterative method for solving sparse matrix equations and is widely used because of its convergence properties. In this paper an implementation of the conjugate gradient algorithm, that exploits both vectorization and parallelization on a 2-dimensional hypercube with vector processors at each node (iPSC-VX/d2), is described. The implementation described here achieves efficient parallelization by using a version of the CG algorithm suitable for coarse grain parallelism [4,5] to reduce the communication steps required and by overlapping the computations on the vector processor with internode communication. With parallelization and vectorization, a speedup of 58 over a μVax II is obtained for large problems, on a two dimensional vector hypercube (iPSC-VX/d2).

关键词： Vectorization Parallelization conjugate gradient algorithm Hypercube-connected vector processors

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Hybrid spectrum conjugate gradient algorithm in electromagnetic tomography

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INSTRUMENTATION SCIENCE & TECHNOLOGY 2023年第4期51卷 435-446页

作者： Li, Liu Luo, Yue Zhao, Qian Wang, Zhanjun Shenyang Normal Univ Coll Phys Sci & Technol Shenyang Peoples R China Shenyang Normal Univ Comp & Math Teaching Dept Shenyang Peoples R China Shenyang Normal Univ Comp & Math Teaching Dept Shenyang 110034 Peoples R China

Electromagnetic tomography is a process detection technology based upon the principles of electromagnetic induction. The forward problem model and sensitivity distribution matrix of electromagnetic tomography are introduced as the basis of the inverse problem. The search direction and iterative parameters of the conjugate gradient algorithm are modified to improve the quality and convergence of image reconstruction. A new spectral parameter conjugate gradient algorithm is described to modify the search direction, which is used to control the angle between the old and new search directions. The search direction is determined according to the iteration of each step in order to find the optimal solution. Combining the advantages of the Fletcher-Reeves and Polak-Ribiere-Polyak algorithms in the nonlinear conjugate gradient algorithm, they are mixed in a specific proportion to obtain a new hybrid conjugate gradient algorithm. In order to verify the effectiveness of the modified conjugate gradient algorithm, three physical models of electromagnetic tomography system are constructed, and the modified conjugate gradient algorithm is compared with the traditional algorithm. The experimental results show that the reconstructed image quality of the modified spectral conjugate gradient algorithm is higher and has better numerical performance. The hybrid conjugate gradient algorithm highlights the advantages of the Fletcher-Reeves and Polak-Ribiere-Polyaks algorithms. The convergence speed is faster than the Polak-Ribiere-Polyak method, and the imaging quality is higher than the other algorithms.

关键词： Electromagnetic tomography (EMT) conjugate gradient algorithm image reconstruction

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An OSIC based reduced-rank MIMO equalizer using conjugate gradient algorithm

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IEICE TRANSACTIONS ON COMMUNICATIONS 2003年第9期E86B卷 2656-2664页

作者： Ho, CL Lin, GJ Lee, TS Natl Chiao Tung Univ Dept Commun Engn & Microelect Hsinchu Taiwan Natl Chiao Tung Univ Informat Syst Res Ctr Hsinchu Taiwan

A reduced complexity multiple-input multiple-output (MIMO) equalizer with ordered successive interference cancellation (OSIC) is proposed for combating intersymbol interference (ISI) and cochannel interference (CCI) over frequency-selective multipath channels. It is developed as a reduced-rank realization of the conventional MMSE decision feedback equalizer (DFE). In particular, the MMSE weight vectors at each stage of OSIC are computed based on the generalized sidelobe canceller (GSC) technique and reduced-rank processing is incorporated by using the conjugate gradient (CG) algorithm for reduced complexity implementation. The CG algorithm leads to a best low-rank representation of the GSC blocking matrix via an iterative procedure, which in turn gives a reduced-rank equalizer weight vector achieving the best compromise between ISI and CCI suppression. With the dominating interference successfully cancelled at each stage of OSIC, the number of iterations required for the convergence of the CG algorithm decreases accordingly for the desired signal. Computer simulations demonstrate that the proposed reduced-rank MIMO DFE can achieve nearly the same performance as the full-rank MIMO MMSE DFE with an effective rank much lower than the dimension of the signal-plus-interference subspace.

关键词： MIMO decision feedback equalizer ordered successive interference cancellation reduced rank conjugate gradient algorithm

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A conjugate gradient algorithm based on double parameter scaled Broyden-Fletcher-Goldfarb-Shanno update for optimization problems and image restoration

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NEURAL COMPUTING & APPLICATIONS 2022年第1期34卷 535-553页

作者： Luo, Dan Li, Yong Lu, Junyu Yuan, Gonglin Guangxi Univ Coll Math & Informat Sci Ctr Appl Math Guangxi Nanning 530004 Guangxi Peoples R China Baise Univ Sch Math & Stat Baise 533000 Guangxi Peoples R China Beibu Gulf Univ Coll Sci Qinzhou 535011 Guangxi Peoples R China

In this paper, we present a three-term conjugate gradient algorithm and three approaches are used in the designed algorithm: (i) A modified weak Wolfe-Powell line search technique is introduced to obtain alpha(k). (ii) The search direction d(k) is given by a symmetrical Perry matrix which contains two positive parameters, and the sufficient descent property of the generated directions holds independent of the MWWP line search technique. (iii) A parabolic will be proposed and regarded as the projection surface, the next point x(k+1) is generated by a new projection technique. The global convergence of the new algorithm under a MWWP line search is obtained for general functions. Numerical experiments show that the given algorithm is promising.

关键词： conjugate gradient algorithm Global convergence Line search

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