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 types of problems, such as eigenvalue calculations, can occur too. A tremendous impact will be made if the performance of these sparse matrix solvers can be improved. 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. This is the reason for the tremendous current interest in preconditioning techniques for large sparse matrix problems: preconditioners dramatically improve the robusteness and performance of iterative algorithms.
In spite of much progress on preconditioning techniques over the last few years, iterative solvers are still not always completely reliable and efficient. The proposed conference, the fourth in a series of biannual meetings on preconditioning, intends to address the complex issues related to the solution of general sparse matrix problems in large-scale real applications and in industrial settings. The issues related to sparse matrix software that are of interest to application scientists and industrial users are often fairly different from those on which the academic community is focused. For example, for an application scientist or an industrial user, improving robustness may be far more important than finding a method that would gain speed. Memory usage is also an important consideration, but is seldom accounted for in academic research on sparse matrix solvers. As a last example, linear systems solved in applications are almost always part of some nonlinear iteration (e.g., Newton) or optimization loop. It is important to consider the coupling between the linear and nonlinear parts, instead of focusing on the linear systems alone. The speakers of this conference will discuss some of the latest developments in the field of preconditioning techniques for sparse matrix problems. The conference will allow participants from academia, industry and government labs to exchange findings in this area and to explore possible new directions in light of emerging paradigms, such as parallel processing and object-oriented programming.
