Introduction to the Special Issue on Iterative Methods for Solving Systems of Algebraic Equations
Authors: Howard Elman
Venue: SIAM Journal on Scientific Computing (1998)

This is a special issue of SIAM Journal on Scientific Computing devoted to iterative methods for solving systems of algebraic equations. The development of efficient methods is of fundamental importance in modern computational modeling, as the solution of large systems of equations often represents the dominant computational cost in mathematical simulations. In contrast to direct methods based on Gaussian elimination, iterative methods tend to have storage requirements proportional to the size of the problem. Consequently, they have the potential for handling significantly larger problems than direct methods. Iterative methods are the only option when fine resolution is required, particularly for three-dimensional models. Similarly, they are the only choice for nonlinear problems, and such solvers in the nonlinear case are frequently combined with inner iterations for linear subproblems. The papers in this volume represent a snapshot of the current state of the art in iterative solution algorithms. The results in these papers were originally presented at the Copper Mountain meeting on iterative methods held in April 1996. This was the eleventh annual meeting in the Copper Mountain series and the fourth devoted exclusively to iterative methods. There were 200 attendees at the conference, and more than 150 presentations were made. The papers included here (identified by first author) can be loosely grouped into five categories: basic Krylov subspace methods (Spotz, Frommer, Field, Druskin, Ressel); multigrid methods (Bandy, Oosterlee, Yavneh); preconditioners (Campos, Chan, Dayar, DeLillo, Klawonn, Moulton, Sarin); eigenvalue problems (Stathopoulos); nonlinear and industrial models (Cai, Banoczi, Kelley, Knoll, Pernice). This collection points to some significant trends in the study of iterative methods, including the maturation of algorithm technology for new "model problems" such as saddle-point and fourth-order problems; the increasing complexity of problems, such as the full potential equation in fluid dynamics, being handled by iterative methods; and the integration of parallel computation and software development into many of the research efforts. I wish to thank the following members of the Copper Mountain program committee who acted as special editors for this issue: Loyce AdamsRoland Freund Anne GreenbaumSteve McCormick Seymour ParterNick Trefethen Henk van der VorstHomer Walker Olof Widlund Through their efforts, submissions were carefully refereed and brought to print on schedule. I also want to thank Tom Manteuffel and Steve McCormick, the organizers of the Copper Mountain conferences, for setting a consistently high standard for the meetings. Special thanks are due to Fred Howes of the Department of Energy, Michael Steuerwalt of the National Science Foundation, and Eric Pritcher from Cray Research for generous support of the meeting. Our hope is that the diverse topics covered in this issue make it clear that the development and analysis of iterative methods continues to be a challenging and fruitful area of research. Howard Elman University of Maryland, College Park