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Download Special Issue Journal of Applied Mathematics Preconditioning Techniques for Sparse Linear Systems Guest Editors: Massimiliano Ferronato, Edmond Chow, and Kok-Kwang Phoon Preconditioning Techniques for Sparse Linear Systems Journal of Applied Mathematics Preconditioning Techniques for Sparse Linear Systems Guest Editors: Massimiliano Ferronato, Edmond Chow, and Kok-Kwang Phoon Copyright q 2012 Hindawi Publishing Corporation. All rights reserved. This is a special issue published in “Journal of Applied Mathematics.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board Said Abbasbandy, Iran Pablo Gonzalez-Vera,´ Spain Michael Meylan, Australia Mina B. Abd-El-Malek, Egypt Laurent Gosse, Italy Alain Miranville, France Mohamed A. Abdou, Egypt K. S. Govinder, South Africa Jaime E. Munoz Rivera, Brazil Subhas Abel, India Jorge Luis Gracia, Spain Javier Murillo, Spain Mostafa Adimy, France Yuantong Gu, Australia Roberto Natalini, Italy Carlos J. S. Alves, Portugal Zhihong Guan, China Srinivasan Natesan, India Mohamad Alwash, USA Nicola Guglielmi, Italy Khalida Inayat Noor, Pakistan Igor Andrianov, Germany Frederico G. Guimaraes,˜ Brazil Donal O’Regan, Ireland Francis T. K. Au, Hong Kong Maoan Han, China Martin Ostoja-Starzewski, USA Olivier Bahn, Canada Ferenc Hartung, Hungary Turgut Ozis¨ ¸, Turkey Roberto Barrio, Spain Luis Javier Herrera, Spain Claudio Padra, Argentina Alfredo Bellen, Italy Ying U. Hu, France Reinaldo Martinez Palhares, Brazil J. Biazar, Iran Zhilong L. Huang, China Juan Manuel Pena,˜ Spain Hester Bijl, The Netherlands Kazufumi Ito, USA Ricardo Perera, Spain James Robert Buchanan, USA Takeshi Iwamoto, Japan Malgorzata Peszynska, USA J. C. Butcher, New Zealand George Jaiani, Georgia James F. Peters, Canada Xiao Chuan Cai, USA Zhongxiao Jia, China Mark A. Petersen, South Africa Weiming Cao, USA Jong H. Kim, Republic of Korea Miodrag Petkovic,´ Serbia Alexandre Carvalho, Brazil Kazutake Komori, Japan Vu Ngoc Phat, Vietnam Song Cen, China Vadim A. Krysko, Russia Andrew Pickering, Spain Qianshun Chang, China Jin L. Kuang, Singapore Hector Pomares, Spain Ke Chen, UK Miroslaw Lachowicz, Poland Maurizio Porfiri, USA Xinfu Chen, USA Hak-Keung Lam, UK Mario Primicerio, Italy Cheng-Sheng Chien, Taiwan Tak-Wah Lam, Hong Kong Morteza Rafei, The Netherlands Jeng-Tzong Chen, Taiwan Peter G. L. Leach, Greece B. V. Rathish Kumar, India Md. Chowdhury, Malaysia Yongkun Li, China Jacek Rokicki, Poland C. Conca, Chile Wan-Tong Li, China Carla Roque, Portugal Alessandro Corsini, Italy Chong Lin, China Debasish Roy, India Livija Cveticanin, Serbia Leevan Ling, Hong Kong Marcelo A. Savi, Brazil Mohamed Darouach, France Mingzhu Liu, China Wolfgang Schmidt, Germany Patrick De Leenheer, USA Kang Liu, USA Eckart Schnack, Germany Eric De Sturler, USA Yansheng Liu, China Mehmet Sezer, Turkey Kai Diethelm, Germany Fawang Liu, Australia Naseer Shahzad, Saudi Arabia Vit Dolejsi, Czech Republic Chein-Shan Liu, Taiwan Fatemeh Shakeri, Iran Meng Fan, China Julian´ Lopez-G´ omez,´ Spain Hui-Shen Shen, China Ya Ping Fang, China Shiping Lu, China Jian Hua Shen, China Antonio J. M. Ferreira, Portugal Gert Lube, Germany Fernando Simoes,˜ Portugal Michel Fliess, France Nazim I. Mahmudov, Turkey A. A. Soliman, Egypt M. A. Fontelos, Spain O. D. Makinde, South Africa Yuri N. Sotskov, Belarus Luca Formaggia, Italy F. Marcellan,´ Spain Peter Spreij, The Netherlands Huijun Gao, China Guiomar Mart´ın-Herran,´ Spain Niclas Stromberg,¨ Sweden B. Geurts, The Netherlands Nicola Mastronardi, Italy Ray Kai Leung Su, Hong Kong Jitao Sun, China E. S. Van Vleck, USA Jinyun Yuan, Brazil Wenyu Sun, China Ezio Venturino, Italy Alejandro Zarzo, Spain Xianhua Tang, China Jesus Vigo-Aguiar, Spain Guisheng Zhai, Japan Marco Henrique Terra, Brazil Junjie Wei, China Zhihua Zhang, China Alexander Timokha, Norway Li Weili, China Jingxin Zhang, Australia Jung-Fa Tsai, Taiwan Martin Weiser, Germany Shan Zhao, USA Ch. Tsitouras, Greece Dongmei Xiao, China Xiaoqiang Zhao, Canada Kuppalapalle Vajravelu, USA Yuesheng Xu, USA Renat Zhdanov, USA Alvaro Valencia, Chile Suh-Yuh Yang, Taiwan J. Hoenderkamp, The Netherlands Contents Preconditioning Techniques for Sparse Linear Systems, Massimiliano Ferronato, Edmond Chow, and Kok-Kwang Phoon Volume 2012, Article ID 518165, 3 pages A Graph Approach to Observability in Physical Sparse Linear Systems, Santiago Vazquez-Rodriguez, Jesus´ A.´ Gomollon,´ Richard J. Duro, and Fernando Lopez´ Pena˜ Volume 2012, Article ID 305415, 26 pages A Relaxed Splitting Preconditioner for the Incompressible Navier-Stokes Equations, Ning-Bo Tan, Ting-Zhu Huang, and Ze-Jun Hu Volume 2012, Article ID 402490, 12 pages A Parallel Wavelet-Based Algebraic Multigrid Black-Box Solver and Preconditioner, Fabio Henrique Pereira and S´ılvio Ikuyo Nabeta Volume 2012, Article ID 894074, 15 pages Parallel Rayleigh Quotient Optimization with FSAI-Based Preconditioning, Luca Bergamaschi, Angeles Mart´ınez, and Giorgio Pini Volume 2012, Article ID 872901, 14 pages Comparison of Algebraic Multigrid Preconditioners for Solving Helmholtz Equations, Dandan Chen, Ting-Zhu Huang, and Liang Li Volume 2012, Article ID 367909, 12 pages An Alternative HSS Preconditioner for the Unsteady Incompressible Navier-Stokes Equations in Rotation Form,JiaLiu Volume 2012, Article ID 307939, 12 pages A Note on the Eigenvalue Analysis of the SIMPLE Preconditioning for Incompressible Flow, Shi-Liang Wu, Feng Chen, and Xiao-Qi Niu Volume 2012, Article ID 564132, 7 pages Applications of Symmetric and Nonsymmetric MSSOR Preconditioners to Large-Scale Biot’s Consolidation Problems with Nonassociated Plasticity, Xi Chen and Kok Kwang Phoon Volume 2012, Article ID 352081, 15 pages A Direct Eigenanalysis of Multibody System in Equilibrium, Cheng Yang, Dazhi Cao, Zhihua Zhao, Zhengru Zhang, and Gexue Ren Volume 2012, Article ID 638546, 12 pages Finite Element Preconditioning on Spectral Element Discretizations for Coupled Elliptic Equations, JongKyum Kwon, Soorok Ryu, Philsu Kim, and Sang Dong Kim Volume 2012, Article ID 245051, 16 pages A Modified SSOR Preconditioning Strategy for Helmholtz Equations, Shi-Liang Wu and Cui-Xia Li Volume 2012, Article ID 365124, 9 pages Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 518165, 3 pages doi:10.1155/2012/518165 Editorial Preconditioning Techniques for Sparse Linear Systems Massimiliano Ferronato,1 Edmond Chow,2 and Kok-Kwang Phoon3 1 Department of Civil, Environmental and Architectural Engineering, University of Padova, 35131 Padova, Italy 2 School of Computational Science and Engineering, College of Computing, Georgia Institute of Technology, Atlanta, GA 30332, USA 3 Department of Civil Engineering, National University of Singapore, Singapore 117576 Correspondence should be addressed to Massimiliano Ferronato, [email protected] Received 8 June 2012; Accepted 8 June 2012 Copyright q 2012 Massimiliano Ferronato et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The implementation of and the solution to large computational models is becoming quite a common effort in both engineering and applied sciences. The accurate and cost-effective numerical solution to the sequence of linearized algebraic systems of equations arising from these models often represents the main memory-demanding and time-consuming task. Direct methods for sparse linear systems often represent the de facto solver in several commercial codes on the basis of their robustness and reliability. However, these methods scale poorly with the matrix size, especially on three-dimensional problems. For such large systems, iterative methods based on Krylov subspaces can be a much more attractive option. A significant number of general-purpose Krylov subspace, or conjugate gradient- like, solvers have been developed during the 70s through the 90s. Interest in these solvers is growing in many areas of engineering and scientific computing. Nonetheless, to become really competitive with direct solvers, iterative methods typically need an appropriate preconditioning to achieve convergence in a reasonable number of iterations and time. The term “preconditioning” refers to “the art of transforming a problem that appears intractable into another whose solution can be approximated rapidly” L.N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, Philadelphia, 1997, while the “preconditioner” is the operator that is responsible for such a transformation. It is widely recognized that preconditioning is the key factor to increasing the robustness and the computational efficiency of iterative methods. Generally speaking, there are three basic requirements for a good preconditioner: i the preconditioned matrix should have a clustered eigenspectrum away from 0; ii the preconditioner should be as cheap to compute as possible; 2 Journal of Applied Mathematics iii its application to a vector should be cost-effective. It goes without saying that these are conflicting requirements, and an appropriate trade-off must be found for any specific problem at hand. Unfortunately, theoretical results are few, and frequently somewhat “empirical” algorithms may work surprisingly well despite the lack of a rigorous foundation. This is why finding a good preconditioner for
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