Sparse Direct Linear Solvers: An Introduction

The minisymposium on sparse direct solvers included 11 talks on the state of the art in this area. The talks covered a wide spectrum of research activities in this area. The papers in this part of the proceedings are expanded, revised, and corrected versions of some the papers that appeared in the CD-ROM proceedings that were distributed at the conference. Not all the talks in the minisymposium have corresponding papers in these proceedings. This introduction explains the significance of the area itself. The introduction also briefly presents, from the personal and subjective viewpoint of the organizer, the significance and contribution of the talks. Sparse direct linear solvers solve linear systems of equations by factoring the sparse coefficient matrix into a product of permutation, triangular, and diagonal (or block diagonal) matrices. It is also possible to factor sparse matrices into products that include orthogonal matrices, such as sparse QR factorizations, but such factorizations were not discussed in the minisymposium. Sparse direct solvers lie at the heart of many software applications, such as finite-elements analysis software, optimization software, and interactive computational engines line Matlab and Mathematica. For most classes of matrices, sparse direct linear solvers scale super-linearly. That is, the cost of solving a linear system with n unknowns grows faster than n. This has led many to search for alternatives solvers with better scaling, mostly in the form of iterative solvers. For many classes of problems, there are now iterative solvers that scale better than direct solvers, and iterative solvers are now also widely deployed in software applications. But iterative solvers have not completely displaced iterative solvers, at least not yet. For some classes of problems, fast and reliable iterative solvers are difficult to construct. In other cases, the size and structure of linear system that application currently solve are such that direct solvers are simply faster. When applications must solve many linear systems with the same coefficient matrix, the amortized cost of the factorization is low. As a result of these factors, sparse direct solvers remain widely used, and research on them remains active. The talk that I think was the most important in the minisymposium was delivered by Jennifer Scott, and was based on joint work with Nick Gould and Yifan Hu. Jennifer’s talk, on the evaluation of sparse direct solvers for symmetric linear systems, described a large-scale study in which she and her colleagues carefully compared several solvers. A comprehensive, objective, and detailed comparison of existing techniques (concrete software packages in this case) is an essential tool for both researchers working within the field and users of the technology. But such studies are difficult to carry out, and are