广义最小残量法研究与应用近况综述
An Overview of Recent Developments and Applications of the GMRES Method
DOI: 10.12677/PM.2013.33027, PDF, HTML,  被引量 下载: 3,904  浏览: 12,977 
作者: 马晓飞*：太原科技大学应用科学学院
关键词: GMRES；历史发展；实际应用；GMRES; Historical Development; Practical Application
摘要: 用于求解大型非对称线性方程组的广义最小残量法(GMRES)以其迭代速度快的优点广泛应用于科学工程计算。本文就GMRES算法的研究近况，分别对其历史发展和实际应用进行概括性的介绍。先从纵向概括了该算法的起源，并介绍了该算法发展过程中有突出影响的变形算法以及近期发展情况，再从横向阐述了近年内它在各领域的应用、与各领域之间的联系和对各领域产生的影响。最后对GMRES算法的进一步发展及应用作出展望。
Abstract: The generalized minimum residual method (GMRES) is widely applied in the scientific and engi- neering computations due to its general merit of fast convergence. This paper presents a summary introduc- tion of the GMRES method for its historical development and practical applications, with an emphasis on its recent status. We start with a summary on the origin of the method, followed by some notable variants, to- gether with some recent developments. Then, we introduce some recent applications of the GMRES method in various research fields, pointing out its connection to and impact on these fields. Finally, we provide an outlook on the further development and applications of the GMRES method.
文章引用：马晓飞. 广义最小残量法研究与应用近况综述[J]. 理论数学, 2013, 3(3): 181-187. http://dx.doi.org/10.12677/PM.2013.33027

参考文献

[1] C. C. Paige, M. A. Saunders. Solution of sparse indefinite systems of linear equations. Society for Industrial and Applied Mathematics, 1975, 12(4): 617-629.
[2] Y. Saad, M. H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. Society for Industrial and Applied Mathematics, 1986, 7(3): 856-869.
[3] D. M. Young, K. C. Jea. Generalized conjugate-gradient acceleration ofnonsymmetrizable iterative methods. Linear Algebra and Its Applications, 1980, 34: 159-194.
[4] S. C. Eisenstat, H. C. Elman and M. H. Schultz. Variation aliterative methods for nonsymmetric systems of linear equations. Society for Industrial and Applied Mathematics, 1983, 20(2): 345-357.
[5] H. F. Walker. A simpler GMRES. Numerical Linear Algebra with Applications, 1994, 1(6): 571-581.
[6] H. A. Van Der Vorst, C. Vuik. GMRESR: A family of nested GMRES methods. Numeral Linear Algebra with Applications, 1994, 1(4): 369- 386.
[7] A. Essai. Weighted FOM and GMRES for solving nonsymmetric linear systems. Numerical Algorithms, 1998, 18(3-4): 277-292.
[8] 杨圣炜, 卢琳璋. 一种加权的Simpler GMRES算法[J]. 厦门大学学报, 2008, 47(4): 484-488.
[9] Z. Zhou, J. Zhao. An improved GMRES method augmented with eigenvectors. Journal of Nanjing University (Natural Science), 2001, 37(1): 1-11.
[10] E. H. Ayachour. A fast implementation for GMRES method. Journal of Computational and Applied Mathematics, 2003, 159(2): 269-283.
[11] H. S. Najafi, H. Zareamonghaddam. A new computational GMRES method. Applied Mathematics and Computation, 2008, 199(2): 527-534.
[12] M. Eiermann, O. G. Ernst and O. Schneider. Analysis of acceleration strategies for restarted minimum residual methods. Journal of Computational and Applied Mathematics, 2000, 123(1): 261-292.
[13] M. Embree. The tortoise and the hare restart GMRES. SIAM Review, 2003, 45(2): 256-266.
[14] A. H. Baker, E. R. Jessup and T. Z. V. Kolev. A simple strategy for varying the restart parameter in GMRES (m). Journal of Computational and Applied Mathematics, 2009, 230(2): 751-761.
[15] E. Vecharynski, J. Langou. The cycle-convergence of restarted GMRES for normal matrices is sublinear. Siam Journal on Scientific Computing, 2010, 32(1): 186-196.
[16] E. Vecharynski, J. Langou. Any decreasing cycle-convergence curve is possible for restarted GMRES, 2012. http://arXiv:0907.3573v1
[17] H. D. Sterck. Steepest descent preconditioning for nonlinear GMRES optimization, 2012. http://arXiv:1106.4426v2
[18] M. Huhtanen, A. Peramaki. Orthogonal polynomials of the r-linear generalized minimal residual method, 2012. http://arXiv:1111.5167v2
[19] J. Liesen, P. Tichy. The field of values bound on ideal GMRES, 2013.
http://arxiv.org/abs/1211.5969v1
[20] R. Campagna, L. D’Amore and A. Murli. An efficient algorithm for regularization of Laplace transform inversion in real case. Journal of Computation and Applied Mathematics, 2007, 210(1-2): 84-98.
[21] J. Huang, J. Jia, M. Minion. Arbitrary order Krylov deferred correction methods for differential algebraic equations. Journal of Computational Physics, 2007, 221(2): 739-760.
[22] J. Kou, Y. Li. A uniparametric LU-SGS method for systems of nonlinear equations. Applied Mathematics and Computation, 2007, 189(1): 235- 240.
[23] H. An, Z. Bai. A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations. Applied Numerical Mathematics, 2007, 57(3): 235-252.
[24] F. S. B. F. Oliveira. Application of a Krylov subspace iterative method in a multi-level adaptive technique to solve the mild-slope equation in nearshore regions. Applied Mathematics Modelling, 2007, 31(4): 655-662.
[25] B. Chang. A deterministic photon free method to solve radiation transfer equations. Journal of Computational Physics, 2007, 222(1): 71-85.
[26] A. Nejat, C. Ollivier-Gooch. Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations. Journal of Computational Physics, 2008, 227(4): 2366-2386.
[27] V. Hernandez, J. J. Ibanez, J. Peinado and E. Arias. A GMRES-based BDF method for solving differential Riccati equations. Applied Mathe-matics and Computational, 2008, 196(2): 613-626.
[28] A. Bouhamidi, K. Jbilou. A note on the numerical approximate solutions for generalized Sylvester matrix equations with applications. Applied Mathematics and Computational, 2008, 206(2): 687-694.
[29] M. Bellalij, K. Jbilou and H. Sadok. New convergence results on the global GMRES method for diagonalizable matrices. Journal of Computational and Applied Mathematics, 2008, 219(2): 350-358.
[30] R. Li, W. Zhang. The rate of convergence of GMRES on a tridiagonal toeplitz linear system. II. Linear Algebra and Its Applications, 2009, 431(12): 2425-2436.
[31] T. Airaksinen, A. Pennanen and J. Toivanen. A damping preconditioner for time-harmonic wave equations in fluid and elastic material. Journal of Computational Physics, 2009, 228(5): 1466-1479.
[32] L. T. Diosady, D. L. Darmofal. Preconditioning methods for discontinuous Galerkin solutions of the Navier-Stokes equations. Journal of Computational Physics, 2009, 228(11): 3917-3935.
[33] G. Pashos, M. E. Kavousanakis, A. N. Spyropoulos, J. A. Palyvos and A. G. Boudouvis. Simultaneous solution of large-scale linear systems and eigenvalue problems with a parallel GMRES method. Journal of Computational and Applied Mathematics, 2009, 227(1): 196-205.
[34] Y. Yan. The 3-D boundary element formulation of linear force-free magnetic fields with finite energy content in semi-infinite space. Solar Physics, 1995, 159(1): 97-113.
[35] Y. Li, Y. Yan, M. Devel, R. Langlet and G. Song. The GMRES method applied to the BEM extrapolation of solar force-free magnetic fields I. Constant alpha force-free fields. Astronomy & Astrophysics, 2007, 470: 1185-1191.
[36] 柳有权, 尹康学, 吴恩华. 大规模稀疏线性方程组的GMRES-GPU快速求解算法[J]. 计算机辅助设计与图形学学报, 2011, 23(4): 553- 560.

为你推荐