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Journal articles on the topic 'Conjugate Residual method'

Author: Grafiati

Published: 6 September 2023

Last updated: 7 September 2023

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1

Dahito, Marie-Ange, and Dominique Orban. "The Conjugate Residual Method in Linesearch and Trust-Region Methods." SIAM Journal on Optimization 29, no. 3 (January 2019): 1988–2025. http://dx.doi.org/10.1137/18m1204255.

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2

Axelsson, Owe, and M. Makarov. "On a generalized conjugate gradient orthogonal residual method." Numerical Linear Algebra with Applications 2, no. 5 (September 1995): 467–79. http://dx.doi.org/10.1002/nla.1680020507.

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3

Ogino, Masao, Shin-ichiro Sugimoto, Seigo Terada, Yanqing Bao, and Hiroshi Kanayama. "A Large-Scale Magnetostatic Analysis Using an Iterative Domain Decomposition Method Based on the Minimal Residual Method." Journal of Advanced Computational Intelligence and Intelligent Informatics 16, no. 4 (June 20, 2012): 496–502. http://dx.doi.org/10.20965/jaciii.2012.p0496.

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This paper describes a large-scale 3D magnetostatic analysis using the Domain Decomposition Method (DDM). To improve the convergence of the interface problem of DDM, a DDM approach based on the Conjugate Residual (CR) method or the MINimal RESidual (MINRES) method is proposed. The CR or MINRES method improved the convergence rate and showed more stable convergence behavior in solving the interface problem than the Conjugate Gradient (CG) method, and reduced computation time for a large-scale problem with about 10 million degrees of freedom.

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4

Li, Tao, Qing-Wen Wang, and Xin-Fang Zhang. "A Modified Conjugate Residual Method and Nearest Kronecker Product Preconditioner for the Generalized Coupled Sylvester Tensor Equations." Mathematics 10, no. 10 (May 18, 2022): 1730. http://dx.doi.org/10.3390/math10101730.

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This paper is devoted to proposing a modified conjugate residual method for solving the generalized coupled Sylvester tensor equations. To further improve its convergence rate, we derive a preconditioned modified conjugate residual method based on the Kronecker product approximations for solving the tensor equations. A theoretical analysis shows that the proposed method converges to an exact solution for any initial tensor at most finite steps in the absence round-off errors. Compared with a modified conjugate gradient method, the obtained numerical results illustrate that our methods perform much better in terms of the number of iteration steps and computing time.

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5

Washizawa, Teruyoshi. "On the Behavior of the Residual in Conjugate Gradient Method." Applied Mathematics 01, no. 03 (2010): 211–14. http://dx.doi.org/10.4236/am.2010.13025.

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6

Gurieva, Y. L. "Semi-conjugate residual method for iterative solving the Navier-Stokes problem." Optoelectronics, Instrumentation and Data Processing 43, no. 2 (April 2007): 177–81. http://dx.doi.org/10.3103/s8756699007020100.

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7

EGUCHI, Yuzuru, Laszlo FUCHS, and Genki YAGAWA. "A finite element analysis of unsteady flows utilizing conjugate residual method." Transactions of the Japan Society of Mechanical Engineers Series A 53, no. 485 (1987): 111–15. http://dx.doi.org/10.1299/kikaia.53.111.

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8

Sogabe, T., M. Sugihara, and S. L. Zhang. "An extension of the conjugate residual method to nonsymmetric linear systems." Journal of Computational and Applied Mathematics 226, no. 1 (April 2009): 103–13. http://dx.doi.org/10.1016/j.cam.2008.05.018.

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9

Toh, Kim-Chuan, and Masakazu Kojima. "Solving Some Large Scale Semidefinite Programs via the Conjugate Residual Method." SIAM Journal on Optimization 12, no. 3 (January 2002): 669–91. http://dx.doi.org/10.1137/s1052623400376378.

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10

Zhong, Hong-Xiu, Xian-Ming Gu, and Shao-Liang Zhang. "A Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides." Symmetry 11, no. 10 (October 16, 2019): 1302. http://dx.doi.org/10.3390/sym11101302.

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The block conjugate orthogonal conjugate gradient method (BCOCG) is recognized as a common method to solve complex symmetric linear systems with multiple right-hand sides. However, breakdown always occurs if the right-hand sides are rank deficient. In this paper, based on the orthogonality conditions, we present a breakdown-free BCOCG algorithm with new parameter matrices to handle rank deficiency. To improve the spectral properties of coefficient matrix A, a precondition version of the breakdown-free BCOCG is proposed in detail. We also give the relative algorithms for the block conjugate A-orthogonal conjugate residual method. Numerical results illustrate that when breakdown occurs, the breakdown-free algorithms yield faster convergence than the non-breakdown-free algorithms.

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11

Sogabe, Tomohiro, and Shao-Liang Zhang. "An Extension of the COCR Method to Solving Shifted Linear Systems with Complex Symmetric Matrices." East Asian Journal on Applied Mathematics 1, no. 2 (May 2011): 97–107. http://dx.doi.org/10.4208/eajam.260410.240510a.

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AbstractThe Conjugate Orthogonal Conjugate Residual (COCR) method [T. Sogabe and S.-L. Zhang, JCAM, 199 (2007), pp. 297-303.] has recently been proposed for solving complex symmetric linear systems. In the present paper, we develop a variant of the COCR method that allows the efficient solution of complex symmetric shifted linear systems. Some numerical examples arising from large-scale electronic structure calculations are presented to illustrate the performance of the variant.

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12

Li, Ling, and Yongxian Li. "Structure Preprocessing Method for the System of Unclosed Linear Algebraic Equations." Journal of Mathematics 2022 (December 27, 2022): 1–9. http://dx.doi.org/10.1155/2022/5435076.

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The complexity of open linear algebraic equations makes it difficult to obtain analytical solutions, and preprocessing techniques can be applied to coefficient matrices, which has become an effective method to accelerate the convergence of iterative methods. Therefore, it is important to preprocess the structure of open linear algebraic equations to reduce their complexity. Open linear algebraic equations can be divided into symmetric linear equations and asymmetric linear equations. The former is based on 2 × 2. The latter is preprocessed by the improved QMRGCGS method, and the applications of the two methods are analyzed, respectively. The experimental results show that when the step is 500, the pretreatment time of quasi-minimal residual generalized conjugate gradient square 2 method is 34.23 s, that of conjugate gradient square 2 method is 35.14 s, and that of conjugate gradient square method is 45.20 s, providing a new reference method and idea for solving and preprocessing non-closed linear algebraic equations.

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13

Zhang, Li-Tao, Xian-Yu Zuo, Tong-Xiang Gu, Ting-Zhu Huang, and Jiang-Hua Yue. "Conjugate residual squared method and its improvement for non-symmetric linear systems." International Journal of Computer Mathematics 87, no. 7 (June 2010): 1578–90. http://dx.doi.org/10.1080/00207160802401029.

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14

Carvalho, L. M., S. Gratton, R. Lago, and X. Vasseur. "A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting." SIAM Journal on Matrix Analysis and Applications 32, no. 4 (October 2011): 1212–35. http://dx.doi.org/10.1137/100786253.

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15

Mennouni, Abdelaziz, Nedjem Eddine Ramdani, and Khaled Zennir. "A new class of fredholm integral equations of the second kind with non symmetric kernel: solving by wavelets method." Boletim da Sociedade Paranaense de Matemática 39, no. 6 (January 1, 2021): 67–80. http://dx.doi.org/10.5269/bspm.41734.

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In this paper, we present an ecient modication of the wavelets method to solve a new class of Fredholm integral equations of the second kind with non symmetric kernel. This -analytical method based on orthonormal wavelet basis, as a consequence three systems are obtained, a Toeplitz system and two systems with condition number close to 1. Since the preconditioned conjugate gradient normal equation residual (CGNR) and preconditioned conjugate gradient normal equation error (CGNE) methods are applicable, we can solve the systems in O(2n log(n)) operations, by using the fast wavelet transform and the fast Fourier transform.

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16

OKAMOTO, NORIKO, REIJI TOMIKU, TORU OTSURU, and YOSUKE YASUDA. "NUMERICAL ANALYSIS OF LARGE-SCALE SOUND FIELDS USING ITERATIVE METHODS PART II: APPLICATION OF KRYLOV SUBSPACE METHODS TO FINITE ELEMENT ANALYSIS." Journal of Computational Acoustics 15, no. 04 (December 2007): 473–93. http://dx.doi.org/10.1142/s0218396x07003512.

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Krylov subspace iterative solvers are applied to large-scale finite element sound-field analyses of architectural rooms. First, convergence behaviors are compared among four iterative solvers. Results show that the Conjugate Orthogonal Conjugate Gradient (COCG) method offers the best characteristics for finite-element (FE) analysis from the viewpoint of robustness of convergence and computation time. Two investigations to reduce the computation time of the COCG method were carried out. Results show the following. (1) The mean residual of sound pressure levels between COCG method and direct method is less than 0.1 dB if the convergence criterion is set to 10-4 and the maximum residual of those between COCG method and direct method is less than 0.2 dB if the convergence criterion is set to 10-6. (2) The computation time of the COCG method with diagonal preconditioning is about 30% shorter than that of COCG method without preconditioning. Finally, sound pressure level distributions obtained using the authors' FEM are compared to those obtained using fast multipole BEM (FMBEM) and measurements.

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17

Dai, Jianhua Zhang and Hua. "Generalized Conjugate A-Orthogonal Residual Squared Method for Complex Non-Hermitian Linear Systems." Journal of Computational Mathematics 32, no. 3 (June 2014): 248–65. http://dx.doi.org/10.4208/jcm.1401-cr13.

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18

ABE, Kuniyoshi, and Seiji FUJINO. "340 Conjugate Residual Squared Method for Linear Systems with Symmetric Definite Coefficient Matrices." Proceedings of The Computational Mechanics Conference 2006.19 (2006): 77–78. http://dx.doi.org/10.1299/jsmecmd.2006.19.77.

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19

Pudov, S. G. "Parallel implementation of the semi-conjugate residual method for systems of linear algebraic equations." Optoelectronics, Instrumentation and Data Processing 43, no. 2 (April 2007): 153–58. http://dx.doi.org/10.3103/s8756699007020070.

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20

Gu, Xian-Ming, Ting-Zhu Huang, and Bruno Carpentieri. "BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems." Journal of Computational and Applied Mathematics 305 (October 2016): 115–28. http://dx.doi.org/10.1016/j.cam.2016.03.032.

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21

Ji, Jun. "CGG method for robust inversion and its application to velocity-stack inversion." GEOPHYSICS 71, no. 4 (July 2006): R59—R67. http://dx.doi.org/10.1190/1.2209547.

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The modified conjugate gradient (CG) method, called the conjugate guided gradient (CGG) method, is a robust iterative inversion method producing a parsimonious model estimation. The CG method for solving least-squares (LS) (i.e., [Formula: see text]-norm minimization) problems is modified to solve for different norms or different minimization criteria by guiding the gradient vector appropriately during iteration steps. Guiding is achieved by iteratively reweighting either the residual vector or the gradient vector during iteration steps like the iteratively reweighted least-squares IRLS method does. Robustness is achieved by weighting the residual vector and parsimonious model estimation is obtained by weighting the gradient vector. Unlike the IRLS method, however, the CGG method doesn’t change the corresponding forward operator of the problem and is implemented in a linear inversion template. Therefore, the CGG method requires less computation than the IRLS method. Since the solution in the CGG method is found in a least-squares sense along the gradient direction guided by the weights, this solution can be interpreted as the LS solution located in the guided gradient direction. Guiding the gradient gives us more flexibility in the choice of weighting parameters than the IRLS method. I applied the CGG method to velocity-stack inversion, and the results show that the CGG method gives a far more robust and parsimonious model estimation than the standard [Formula: see text]-norm solution, with results comparable to the [Formula: see text]-norm IRLS solution.

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22

Thorne, Jack. "On the automorphy of l-adic Galois representations with small residual image With an appendix by Robert Guralnick, Florian Herzig, Richard Taylor and Jack Thorne." Journal of the Institute of Mathematics of Jussieu 11, no. 4 (April 5, 2012): 855–920. http://dx.doi.org/10.1017/s1474748012000023.

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AbstractWe prove new automorphy lifting theorems for essentially conjugate self-dual Galois representations into GLn. Existing theorems require that the residual representation have ‘big’ image, in a certain technical sense. Our theorems are based on a strengthening of the Taylor–Wiles method which allows one to weaken this hypothesis.

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23

Jian, Ling, Shuqian Shen, and Yunquan Song. "Improving the Solution of Least Squares Support Vector Machines with Application to a Blast Furnace System." Journal of Applied Mathematics 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/949654.

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The solution of least squares support vector machines (LS-SVMs) is characterized by a specific linear system, that is, a saddle point system. Approaches for its numerical solutions such as conjugate methods Sykens and Vandewalle (1999) and null space methods Chu et al. (2005) have been proposed. To speed up the solution of LS-SVM, this paper employs the minimal residual (MINRES) method to solve the above saddle point system directly. Theoretical analysis indicates that the MINRES method is more efficient than the conjugate gradient method and the null space method for solving the saddle point system. Experiments on benchmark data sets show that compared with mainstream algorithms for LS-SVM, the proposed approach significantly reduces the training time and keeps comparable accuracy. To heel, the LS-SVM based on MINRES method is used to track a practical problem originated from blast furnace iron-making process: changing trend prediction of silicon content in hot metal. The MINRES method-based LS-SVM can effectively perform feature reduction and model selection simultaneously, so it is a practical tool for the silicon trend prediction task.

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24

Zhang, Li. "A derivative-free conjugate residual method using secant condition for general large-scale nonlinear equations." Numerical Algorithms 83, no. 4 (May 15, 2019): 1277–93. http://dx.doi.org/10.1007/s11075-019-00725-7.

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25

SIEJA, K., T. L. HA, P. QUENTIN, and A. BARAN. "PARTICLE NUMBER CONSERVING APPROACH TO CORRELATIONS." International Journal of Modern Physics E 16, no. 02 (February 2007): 289–97. http://dx.doi.org/10.1142/s0218301307005727.

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In the present work the so-called Higher Tamm-Dancoff Approximation method is presented for the generalized case of isovector and isoscalar residual interactions treated simultaneously. The role of different particle-hole excitations and of proton-neutron pairing correlations in the ground state of the self-conjugate 64 Ge nucleus is discussed.

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26

Zhang, Xiaodan, Dongxiao Liu, Guizhong Liu, Haiyang Gou, Rui Li, Zhiyu Zhang, and Lin Cui. "Research on the Improved Least Squares Reverse Time Migration Imaging Method." E3S Web of Conferences 236 (2021): 04017. http://dx.doi.org/10.1051/e3sconf/202123604017.

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An improved Least Squares Reverse Time Migration (LSRTM) method is proposed in the paper, which can effectively improve convergence speed and imaging accuracy. Firstly, the key techniques in the implementation of LSRTM are discussed. Secondly, a condition factor is introduced in the iteration process of conjugate gradient method. Finally, the imaging effect and performance of the algorithm are analyzed. The experiment results indicate that it can speed up the convergence speed and improve the convergence accuracy, so as to improve the imaging effect. Compared with the conventional LSRTM, the data residual of improved LSRTM can be reduced by about 5%.

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27

Rahman, M. M., Hakan Oztop, S. Mekhilef, R. Saidur, A. Chamkha, A. Ahsan, and Khaled S. Al-Salem. "A finite element analysis on combined convection and conduction in a channel with a thick walled cavity." International Journal of Numerical Methods for Heat & Fluid Flow 24, no. 8 (October 28, 2014): 1888–905. http://dx.doi.org/10.1108/hff-07-2013-0239.

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Purpose – The purpose of this paper is to examine the effects of thick wall parameters of a cavity on combined convection in a channel. In other words, conjugate heat transfer is solved. Design/methodology/approach – Galerkin weighted residual finite element method is used to solve the governing equations of mixed convection. Findings – The streamlines, isotherms, local and average Nusselt numbers are obtained and presented for different parameters. It is found heat transfer is an increasing function of dimensionless thermal conductivity ratio. Originality/value – The literature does not have mixed convection and conjugate heat transfer problem in a channel with thick walled cavity.

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28

Iakymchuk, Roman, Maria Barreda Vayá, Stef Graillat, José I. Aliaga, and Enrique S. Quintana-Ortí. "Reproducibility of parallel preconditioned conjugate gradient in hybrid programming environments." International Journal of High Performance Computing Applications 34, no. 5 (June 17, 2020): 502–18. http://dx.doi.org/10.1177/1094342020932650.

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The Preconditioned Conjugate Gradient method is often employed for the solution of linear systems of equations arising in numerical simulations of physical phenomena. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we propose two algorithmic solutions that originate from the ExBLAS project to enhance the accuracy of the solver as well as to ensure its reproducibility in a hybrid MPI + OpenMP tasks programming environment. One is based on ExBLAS and preserves every bit of information until the final rounding, while the other relies upon floating-point expansions and, hence, expands the intermediate precision. Instead of converting the entire solver into its ExBLAS-related implementation, we identify those parts that violate reproducibility/non-associativity, secure them, and combine this with the sequential executions. These algorithmic strategies are reinforced with programmability suggestions to assure deterministic executions. Finally, we verify these approaches on two modern HPC systems: both versions deliver reproducible number of iterations, residuals, direct errors, and vector-solutions for the overhead of less than 37.7% on 768 cores.

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29

Manteuffel, Thomas A., and James S. Otto. "On the roots of the orthogonal polynomials and residual polynomials associated with a conjugate gradient method." Numerical Linear Algebra with Applications 1, no. 5 (September 1994): 449–75. http://dx.doi.org/10.1002/nla.1680010503.

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30

Robichaud, Michel P., and Philippe A. Tanguy. "Finite element solution of three-dimensional incompressible fluid flow problems by a preconditioned conjugate residual method." International Journal for Numerical Methods in Engineering 24, no. 2 (February 1987): 447–57. http://dx.doi.org/10.1002/nme.1620240211.

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31

Mao, Weijian, and David Gubbins. "Simultaneous determination of time delays and stacking weights in seismic array beamforming." GEOPHYSICS 60, no. 2 (March 1995): 491–502. http://dx.doi.org/10.1190/1.1443786.

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An algorithm for the estimation of time delays and weights in an arbitrary‐single or three‐component seismic array is developed by the use of a linearized waveform inversion technique. This algorithm differs from conventional crosscorrelation methods in its ability to simultaneously obtain time delays and weights by minimizing residuals of all possible waveform fittings, and by its robustness in the presence of high random noise levels and local geological scattering. There are N stations in an array, and for each station, a beam is formed by a weighted linear combination of the remaining (N − 1) seismic traces. The time delays and weights are model parameters to be found by minimizing the sum of N objective functions. Two optimization algorithms for solving the least‐squares problem, singular‐value decomposition and conjugate gradient, are compared, and the conjugate gradient method is found to be satisfactory and faster for large arrays. The algorithm was tested using synthetic array data with high noise, real data from shots in a borehole to a linear array on land, and Ms 6.7 earthquake data recorded with a broadband three‐component array. The success with synthetic and real data shows the algorithm to be useful for seismic data stacking, residual static corrections, and phase picking when the data quality is poor.

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32

Kanayama, H. "An Industrial Application of Thermal Convection Analysis." International Journal of Computational Methods 13, no. 02 (March 2016): 1640005. http://dx.doi.org/10.1142/s0219876216400053.

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A coupling analysis of thermal convection problems is performed in this work. By approximating the material derivative along the trajectory of fluid particle, the characteristic curve (CC) method can be considered. The most attractive advantage of this method is the symmetry of the linear system, which enables some classic symmetric linear iterative solvers, like the conjugate gradient (CG) method or the minimal residual (MINRES) method, to be used to solve the interface problem of the domain decomposition system. An application to industrial problems is demonstrated to show the effectiveness of our approach.

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33

Kitagawa, Tsunehiro, Wataru Ohtani, Yoshiko Maeno, Kunio Fujiwara, and Yukio kimura. "Sensitive Enzyme Immunoassay of Colistin and Its Application to Detect Residual Colistin in Rainbow Itout Tissue." Journal of AOAC INTERNATIONAL 68, no. 4 (July 1, 1985): 661–64. http://dx.doi.org/10.1093/jaoac/68.4.661.

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Abstract An antibody against colistin (CL), an antibiotic effective for gramnegative bacteria, was produced in rabbits immunized with a colistinprotein conjugate. The conjugate was prepared by a novel and convenient procedure devised to couple an amino group of CL to thiol groups of bovine serum albumin (BSA) introduced by thiol exchange reduction of its disulfide bonds with dithiothreitol, using N-(m-maIeimidobenzoyloxy) succinimide (MBS) as a cross-linker. Enzyme labeling of CL with β-D-galactosidase was performed by utilizing another crosslinker, N(γ-maleimidobutyryIoxy)succinimide, by means of a convenient labeling method. A double antibody enzyme immunoassay of CL, which could determine as little as 30 ng/mL of CL, was developed using labeled CL and anti-CL antiserum. With this assay, drug levels were easily determined in fish tissue after CL administration. The enzyme immunoassay should provide a useful tool for detection and quantitation of residual drugs in foods and related products.

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34

Taner, M. Turhan, Donald E. Wagner, Edip Baysal, and Lee Lu. "A unified method for 2-D and 3-D refraction statics." GEOPHYSICS 63, no. 1 (January 1998): 260–74. http://dx.doi.org/10.1190/1.1444320.

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Most of the seismic data processing procedures are divided into 2-D, 2.5-D, crooked lines or 3-D versions dictated by the differences in the shot and receiver configurations. In this paper, we introduce a tomographic approach that overcomes these geometrical difficulties and provides stable statics solutions from picked first‐break times. We also show that the first‐break picks contain both the short and the long wavelength surface statics. The solutions are obtained by solving a set of generalized surface‐consistent delay‐time equations using the method of weighted least squares and conjugate gradient. While iterating, each first‐break pick is evaluated to ensure its consistency with the least‐squares solution. Based on consistency, we weight the traveltime picks and use them in the next iteration. These weights also serve as an indicator of anomalous picks to the user. We show that long wavelength solutions leave large residual errors in the least‐squares solutions. We also use the expected length of the Fresnel zone to differentiate between short and long wavelength static solutions. After removing the influence of long wavelength statics, we apply short wavelength statics to reduce the residual errors further. We demonstrate the validity of our unified method by applying it to actual data examples. The removal of both long and short wavelength statics improves the initial data set that produces a more consistent set of velocities and leaves only the short wavelength residual reflection statics, which are generally less than quarter wavelet period delays. This removes the most probable cause of the leg jump contamination and poor velocity estimates from the residual statics computations, especially from the 3-D data.

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35

Jin, Xiao-Qing, Fu-Rong Lin, and Zhi Zhao. "Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations." Communications in Computational Physics 18, no. 2 (July 30, 2015): 469–88. http://dx.doi.org/10.4208/cicp.120314.230115a.

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AbstractIn this paper, preconditioned iterative methods for solving two-dimensional space-fractional diffusion equations are considered. The fractional diffusion equation is discretized by a second-order finite difference scheme, namely, the Crank-Nicolson weighted and shifted Grünwald difference (CN-WSGD) scheme proposed in [W. Tian, H. Zhou and W. Deng, A class of second order difference approximation for solving space fractional diffusion equations, Math. Comp., 84 (2015) 1703-1727]. For the discretized linear systems, we first propose preconditioned iterative methods to solve them. Then we apply the D’Yakonov ADI scheme to split the linear systems and solve the obtained splitting systems by iterative methods. Two preconditioned iterative methods, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual (preconditioned CGNR) method, are proposed to solve relevant linear systems. By fully exploiting the structure of the coefficient matrix, we design two special kinds of preconditioners, which are easily constructed and are able to accelerate convergence of iterative solvers. Numerical results show the efficiency of our preconditioners.

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36

Mori, H., J. Kanno, and S. Tsuzuki. "A sparsity-oriented technique for power system small signal stability analysis with a precondition conjugate residual method." IEEE Transactions on Power Systems 8, no. 3 (1993): 1150–58. http://dx.doi.org/10.1109/59.260882.

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37

Davstad, Kjell. "A multigrid conjugate residual method for the numerical solution of the Hartree-Fock equation for diatomic molecules." Journal of Computational Physics 93, no. 2 (April 1991): 486. http://dx.doi.org/10.1016/0021-9991(91)90202-v.

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38

Davstad, Kjell. "A multigrid conjugate residual method for the numerical solution of the Hartree-Folk equation for diatomic molecules." Journal of Computational Physics 98, no. 1 (January 1992): 180. http://dx.doi.org/10.1016/0021-9991(92)90197-7.

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39

Davstad, Kjell. "A multigrid conjugate residual method for the numerical solution of the Hartree-Fock equation for diatomic molecules." Journal of Computational Physics 99, no. 1 (March 1992): 33–38. http://dx.doi.org/10.1016/0021-9991(92)90272-z.

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40

Bartels, Simon, Jon Cockayne, Ilse C. F. Ipsen, and Philipp Hennig. "Probabilistic linear solvers: a unifying view." Statistics and Computing 29, no. 6 (September 10, 2019): 1249–63. http://dx.doi.org/10.1007/s11222-019-09897-7.

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Abstract Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the matrix inverse. These approaches have typically focused on replicating the behaviour of the conjugate gradient method as a prototypical iterative method. In this work, surprisingly general conditions for equivalence of these disparate methods are presented. We also describe connections between probabilistic linear solvers and projection methods for linear systems, providing a probabilistic interpretation of a far more general class of iterative methods. In particular, this provides such an interpretation of the generalised minimum residual method. A probabilistic view of preconditioning is also introduced. These developments unify the literature on probabilistic linear solvers and provide foundational connections to the literature on iterative solvers for linear systems.

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41

Ali, Ayad Ramadhan, and Bayda Ghanim Fathi. "Hybridization Gradient Based Methods with Genetic Algorithm for Solving Systems of Linear Equations." Journal of duhok university 25, no. 2 (November 9, 2022): 41–49. http://dx.doi.org/10.26682/sjuod.2022.25.2.4.

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In this paper, we propose two hybrid gradient based methods and genetic algorithm for solving systems of linear equations with fast convergence. The first proposed hybrid method is obtained by using the steepest descent method and the second one by the Cauchy-Barzilai-Borwein method. These algorithms are based on minimizing the residual of solution which has genetic characteristics. They are compared with the normal genetic algorithm and standard gradient based methods in order to show the accuracy and the convergence speed of them. Since the conjugate gradient method is recommended for solving large sparse and symmetric positive definite matrices, we also compare the numerical results of our proposed algorithms with this method. The numerical results demonstrate the robustness and efficiency of the proposed algorithms. Moreover, we observe that our hybridization of the CBB method and genetic algorithm gives more accurate results with faster convergence than other mentioned methods in all given cases

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Chin-Lung Fong, David, and Michael Saunders. "CG Versus MINRES: An Empirical Comparison." Sultan Qaboos University Journal for Science [SQUJS] 16 (April 1, 2012): 44. http://dx.doi.org/10.24200/squjs.vol17iss1pp44-62.

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For iterative solution of symmetric systems the conjugate gradient method (CG) is commonly used when A is positive definite, while the minimum residual method (MINRES) is typically reserved for indefinite systems. We investigate the sequence of approximate solutions generated by each method and suggest that even if A is positive definite, MINRES may be preferable to CG if iterations are to be terminated early. In particular, we show for MINRES that the solution norms are monotonically increasing when A is positive definite (as was already known for CG), and the solution errors are monotonically decreasing. We also show that the backward errors for the MINRES iterates are monotonically decreasing.

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Cao, Ligang, and Yun Liu. "The Finite Element Numerical Modelling of 3D Magnetotelluric." Mathematical Problems in Engineering 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/890516.

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The ideal numerical simulation of 3D magnetotelluric was restricted by the methodology complexity and the time-consuming calculation. Boundary values, the variation of weighted residual equation, and the hexahedral mesh generation method of finite element are three major causes. A finite element method for 3D magnetotelluric numerical modeling is presented in this paper as a solution for the problem mentioned above. In this algorithm, a hexahedral element coefficient matrix for magnetoelluric finite method is developed, which solves large-scale equations using preconditioned conjugate gradient of the first-type boundary conditions. This algorithm is verified using the homogeneous model, and the positive landform model, as well as the low resistance anomaly model.

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Tang, L. Q., K. Pochiraju, C. Chassapis, and S. Manoochehri. "A Computer-Aided Optimization Approach for the Design of Injection Mold Cooling Systems." Journal of Mechanical Design 120, no. 2 (June 1, 1998): 165–74. http://dx.doi.org/10.1115/1.2826955.

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A methodology is presented for the design of optimal cooling systems for injection mold tooling which models the mold cooling as a nonlinear constrained optimization problem. The design constraints and objective function are evaluated using Finite Element Analysis (FEA). The objective function for the constrained optimization problem is stated as minimization of both a function related to part average temperature and temperature gradients throughout the polymeric part. The goal of this minimization problem is to achieve reduction of undesired defects as sink marks, differential shrinkage, thermal residual stress built-up, and part warpage primarily due to non-uniform temperature distribution in the part. The cooling channel size, locations, and coolant flow rate are chosen as the design variables. The constrained optimal design problem is solved using Powell’s conjugate direction method using penalty function. The cooling cycle time and temperature gradients are evaluated using transient heat conduction simulation. A matrix-free algorithm of the Galerkin Finite Element Method (FEM) with the Jacobi Conjugate Gradient (JCG) scheme is utilized to perform the cooling simulation. The optimal design methodology is illustrated using a case study.

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Akiniwa, Yoshiaki, and Hidehiko Kimura. "Determination of Residual Stress Distribution in Severe Surface Deformed Steel by Shot Peening." Materials Science Forum 571-572 (March 2008): 15–20. http://dx.doi.org/10.4028/www.scientific.net/msf.571-572.15.

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The compressive stress distribution below the specimen surface of a nanocrystalline medium carbon steel was investigated nondestructively by using high-energy X-rays from a synchrotron radiation source, SPring-8 (Super Photon ring-8 GeV) in the Japan Synchrotron Radiation Research Institute. A medium carbon steel plate was shot-peened with fine cast iron particles of the size of 50 μm. By using the monochromatic X-ray beam with three energy levels of 10, 30 and 72 keV, the stress values at the arbitrary depth were measured by the constant penetration depth method. The stress was calculated from the slope of the sin2ψ diagram. Measured stress corresponds to the weighted average associated with the attenuation of the X-rays in the material. The real stress distribution was estimated by using the optimization technique. The stress distribution was assumed by the third order polynomial in the near surface layer and the second order polynomial. The coefficients of the polynomials were determined by the conjugate gradient iteration. The predicted stress distribution agreed well with that measured by the conventional surface removal method.

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Nélias, Daniel, Eduard Antaluca, Vincent Boucly, and Spiridon Cretu. "A Three-Dimensional Semianalytical Model for Elastic-Plastic Sliding Contacts." Journal of Tribology 129, no. 4 (May 24, 2007): 761–71. http://dx.doi.org/10.1115/1.2768076.

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A three-dimensional numerical model based on a semianalytical method in the framework of small strains and small displacements is presented for solving an elastic-plastic contact with surface traction. A Coulomb’s law is assumed for the friction, as commonly used for sliding contacts. The effects of the contact pressure distribution and residual strain on the geometry of the contacting surfaces are derived from Betti’s reciprocal theorem with initial strain. The main advantage of this approach over the classical finite element method (FEM) is the computing time, which is reduced by several orders of magnitude. The contact problem, which is one of the most time-consuming procedures in the elastic-plastic algorithm, is obtained using a method based on the variational principle and accelerated by means of the discrete convolution fast Fourier transform (FFT) and conjugate gradient methods. The FFT technique is also involved in the calculation of internal strains and stresses. A return-mapping algorithm with an elastic predictor∕plastic corrector scheme and a von Mises criterion is used in the plasticity loop. The model is first validated by comparison with results obtained by the FEM. The effect of the friction coefficient on the contact pressure distribution, subsurface stress field, and residual strains is also presented and discussed.

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Lingen, Frederik Jan. "A Generalised Conjugate Residual method for the solution of non-symmetric systems of equations with multiple right-hand sides." International Journal for Numerical Methods in Engineering 44, no. 5 (February 20, 1999): 641–56. http://dx.doi.org/10.1002/(sici)1097-0207(19990220)44:5<641::aid-nme520>3.0.co;2-p.

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Li, Pengyang, Lingxia Zhou, Fangyuan Cui, Quandai Wang, Meiling Guo, Yanfei Zhai, Yan Li, and Dengxin Hua. "Elasto-plastic analysis of the contact region between a rigid ellipsoid and a semi-flat surface." Advances in Mechanical Engineering 10, no. 9 (September 2018): 168781401879739. http://dx.doi.org/10.1177/1687814018797397.

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When the load acting on a mechanical structure is greater than the yield strength of the material, the contact surface will undergo plastic deformation. Cumulative plastic deformation has an important influence on the lifespan of mechanical parts. This article presents a three-dimensional semi-analytical model based on the conjugate gradient method and fast Fourier transform algorithm, with the aim of studying the characteristic parameters of the contact region between a rigid ellipsoid and elasto-plastic half-space. Moreover, normal forces and tangential traction were considered, as well as the contact pressure resulting from various sliding speeds and friction coefficients. The contact pressure, effective plastic strain, von Mises stress, and residual stress were measured and shown to increase with increasing sliding velocity. Finally, when the friction coefficient, contact pressure, and effective plastic strain are increased, the von Mises stress is also shown to increase, whereas the residual stress decreases.

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Ji, Jun. "Tomographic velocity estimation with plane‐wave synthesis." GEOPHYSICS 62, no. 6 (November 1997): 1825–38. http://dx.doi.org/10.1190/1.1444284.

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In areas with structurally complex geology, tomographic velocity analysis is often required to estimate velocities. In this paper I describe an algorithm for tomographic velocity estimation that uses plane‐wave synthesis imaging as a prestack migration. The classical iterative two‐step process (measures the traveltime errors with the current velocity model and then update the velocity model) is performed as follows. The events are picked in the image space after prestack migration with surface‐oriented plane‐wave synthesis imaging. the traveltime deviations are measured through residual‐moveout (RMO) velocity analysis in common‐surface‐location (CSL) gathers obtained by reflector‐oriented plane‐wave synthesis imaging, and the velocity update is calculated by inverting the traveltime deviations through a conjugate gradient. The results from synthetic data indicate that the tomographic method successfully estimates interval‐velocity models that lead to depth‐migrated images with no residual moveout.

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Tong, P., D. Zhao, D. Yang, X. Yang, J. Chen, and Q. Liu. "Wave-equation based traveltime seismic tomography – Part 1: Method." Solid Earth Discussions 6, no. 2 (August 25, 2014): 2523–66. http://dx.doi.org/10.5194/sed-6-2523-2014.

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Abstract. In this paper, we propose a wave-equation based traveltime seismic tomography method with a detailed description of its step-by-step process. First, a linear relationship between the traveltime residual Δt = Tobs – Tsyn and the relative velocity perturbation δc(x) / c(x) connected by a finite-frequency traveltime sensitivity kernel K(x) is theoretically derived using the adjoint method. To accurately calculate the traveltime residual Δt, two automatic arrival-time picking techniques including the envelop energy ratio method and the combined ray and cross-correlation method are then developed to compute the arrival times Tsyn for synthetic seismograms. The arrival times Tobs of observed seismograms are usually determined by manual hand picking in real applications. Traveltime sensitivity kernel K(x) is constructed by convolving a forward wavefield u(t,x) with an adjoint wavefield q(t,x). The calculations of synthetic seismograms and sensitivity kernels rely on forward modelling. To make it computationally feasible for tomographic problems involving a large number of seismic records, the forward problem is solved in the two-dimensional (2-D) vertical plane passing through the source and the receiver by a high-order central difference method. The final model is parameterized on 3-D regular grid (inversion) nodes with variable spacings, while model values on each 2-D forward modelling node are linearly interpolated by the values at its eight surrounding 3-D inversion grid nodes. Finally, the tomographic inverse problem is formulated as a regularized optimization problem, which can be iteratively solved by either the LSQR solver or a non-linear conjugate-gradient method. To provide some insights into future 3-D tomographic inversions, Fréchet kernels for different seismic phases are also demonstrated in this study.

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