SIGEST

Abstract

In discussions of the tremendous advances in scientific computation over the past few decades, it has often been written and demonstrated that as large as the increases in computer speed have been, they have been matched by improvements from more efficient algorithms. For many problem classes, over the last two to four decades, each has led to speedups of many, many orders of magnitude. The SIGEST paper in this issue, from the SIAM Journal on Scientific Computing, is an example of this tremendous type of algorithmic improvement. It also is an excellent example of how increasingly significant algorithmic advances are obtained by drawing upon a broad range of sophisticated techniques. In “On Large-Scale Diagonalization Techniques for the Anderson Model of Localization” by O. Schenk, M. Bollhöfer, and R. Römer, the authors demonstrate a greatly improved computational approach for an important, common problem: computing a few interior eigenvalues and eigenvectors of large, sparse, real symmetric, and highly indefinite matrices. By “large” they mean matrices with millions to hundreds of millions of rows and columns. The computational improvements that are demonstrated extend to a factor of a thousand or more over previous approaches, while keeping memory requirements modest. The application class is the Anderson model of localization, an important problem in quantum physics with a wide range of applications, and it appears likely that the approach can be applied successfully to large symmetric indefinite eigenvalue problems more generally. An interesting aspect of the paper is that is draws heavily upon recent advances in solving large sparse symmetric indefinite systems of linear equations. Solution of such systems is central to each of the eigenvalue solvers that the paper considers, two Lanczos-based methods and the Jacobi–Davidson method. In considering how to solve the linear systems in a manner that exploits sparsity as well as symmetry, the paper combines a diverse range of topics that includes preprocessing the matrix with symmetric weighted matchings, solving the linear system with Krylov subspace methods, and accelerating the linear system solution with multilevel preconditioners based upon incomplete factorizations. This makes for a very fitting paper for SIAM Review, highlighting a variety of important topics in numerical computation. We expect that a wide range of SIAM readers will find this to be a very readable paper that provides an interesting glimpse into a range of modern research approaches.