Themes and Motivation

The innermost computational kernel of many large-scale scientific applications and industrial numerical simulations is often a large sparse matrix problem, which typically consumes a significant portion of the overall computational time required by the simulation.  Many of the matrix problems are in the form of systems of linear equations, although other matrix problems, such as eigenvalue calculations, can occur too. A traditional approach to solving large sparse matrix equations is to use direct methods.  This approach is often preferred in industry because direct solvers are robust and effective for moderate size problems.  However, the unprecedented pace of the advance in technology has led to a dramatic growth in the size of the matrices to be handled.  For example, the storage requirement for three-dimensional simulations makes direct methods prohibitively expensive.  Iterative techniques are the only viable alternative.

Unfortunately, iterative methods lack the robustness of direct methods.  They often fail when the matrix is very ill-conditioned.  While a "bullet-proof" iterative method may not exist, effective and robust sparse matrix iterative solvers are becoming a vital part of large-scale scientific and industrial applications.  In the past decade or so, some emphasis has been devoted to exploring more "powerful" iterative solvers.  The performance of these methods is eventually related to the condition number of the coefficient matrix of the system. Many of the large linear systems arising in industry still challenge most linear equations solvers available.

Constructing a preconditioner to improve the condition number of a matrix was proposed a few decades ago.  These techniques did not have much impact initially due to the simplicity of their heuristics and the relatively small size of the matrices to be solved.  However, more and more computational experience indicates that a good preconditioner holds the key to an effective iterative solver.  The big impact of these simple techniques on the performance of an iterative method have attracted increased attentions in recent years.  Parallel computers also generate many new research topics in the study of preconditioning.  Many new promising techniques have been reported.

However, the theoretical basis for high performance preconditioners is still not well understood; many existing techniques still suffer from lack of robustness.  Promising ideas need still to be tested in real applications.

Of course, the issue of preconditioning does not arise only in the solution of linear equations.  For example, preconditioning techniques are equally important in the use of the Jacobi-Davidson method for solving eigenvalue problems.

This is the motivation for holding this conference specifically dedicated to the issues in preconditioning for large-scale scientific and industrial applications.  The conference will bring the researchers and application scientists in this field together to discuss the latest developments, progress made, and to exchange findings and explore possible new directions.