The need for large-scale numerical simulation is currently driving the development of the largest and most powerful parallel computers in the world. Activities at U.S. government laboratories, for example, are currently replacing underground nuclear testing with computational predictive simulation. A major challenge is to design software and algorithms that can efficiently utilize upwards of ten thousand parallel processors.
This talk addresses the problem of solving large, sparse linear (matrix) equations on these large parallel computers. These equations arise in all implicit methods for numerical simulation and usually form the bottleneck in the computation. We will introduce how these equations arise, and describe a novel method of approximating the inverse of a matrix by a sparse matrix. This is an elegant and inherently parallel method used to "precondition" the original set of equations in order to make them easier to solve by a parallel, iterative procedure. For perspective, we will intuitively describe other state-of-the-art methods, as well as computer science issues for achieving high computing performance.