Special Section On Preconditioning Techniques

Abstract

At the Third International Conference on Preconditioning Techniques for Large Sparse Matrix Problems in Scientific and Industrial Applications, held in Napa, California, on October 27--29, 2003, Henk A. van der Vorst, Editor-in-Chief of the SIAM Journal on Matrix Analysis and Applications (SIMAX), expressed an interest in publishing in SIMAX a collection of articles that would focus on current research on preconditioning techniques for matrix computation. This special section on preconditioning techniques is the result of subsequent discussions with SIAM and SIMAX. Manuscripts were solicited through various channels, particularly from the speakers at the third preconditioning conference. The term preconditioning generally refers to the additional step designed to improve the performance of a matrix algorithm, such as the rate of convergence of a Krylov method. Preconditioning arises in a variety of matrix computation. However, the papers in this special section provide a snapshot of the ongoing research activities on preconditioning techniques specifically for the solution of systems of linear equations. Multilevel and multigrid preconditioning techniques are the focus of the papers by Notay; de Sterk, Yang, and Heys; and Vassilevski and Zikatanov. Bank, Wan, and Qu have considered correction and interpolation techniques for improving the convergence of a multigrid method for solving convection-diffusion equations. Bridson and Greif have used a combination of multiple preconditioners in an attempt to improve the convergence of an iterative method. Giraud and Gratton have employed first-order perturbation theory to derive the first-order perturbations on the eigenvalues of matrices preconditioned by some forms of spectral preconditioners, and thereby obtained some insight into the design of such preconditioners. Lee, Raghavan, and Ng have investigated the problem of improving the quality of incomplete factorization through row and column reordering techniques. Some of papers are problem specific. The paper by Benzi and Ng considers preconditioning techniques for Toeplitz least squares problems, while that by Botchev and Golub focuses on preconditioning techniques for saddle point problems. Le Borne and Grasedyck have employed sparse hierarchical matrix approximations as preconditioners in iterative methods for solving convection-dominated elliptic partial differential equations. Bardsley and Nagy have investigated preconditioning strategies for solving ill-posed inverse problems arising from image analysis. The editor-in-chief of this special section would like to thank Henk A. van der Vorst for the encouragement, as well as Mitch Chernoff and his staff in the SIAM office for the advice and support in putting the special section together. Michele Benzi, Edmond Chow, Yousef Saad, Valeria Simoncini, and Wei-Pai Tang served as guest editors; their effort was much appreciated. Last but not the least, the editor-in-chief would like to thank all the authors for submitting their manuscripts to the special section, and all the anonymous referees for their help in reviewing the manuscripts.