| Because linear systems are ubiquitous in science and engineering applications, improving the robustness of linear solvers continues to be of interest. We focus on the restarted generalized minimum residual (GMRES) method, as it is arguably one of the most popular iterative methods for solving large, sparse, nonsymmetric systems of linear equations.; Two approaches to improving the performance, i.e. time to solution, of an iterative linear solver algorithm are particularly viable. First, algorithmic changes that improve convergence properties result in faster convergence due to fewer overall floating-point operations. Second, modifications to an algorithm that reduce the movement of data through memory greatly impact performance because of the growing gap between CPU performance and memory access time. Ideally, a balance is achieved between improving the efficiency of an iterative linear solver from a memory-usage standpoint and maintaining favorable numerical properties.; We discuss the restarted GMRES algorithm in the context of both approaches to improving performance. In particular, we describe some interesting observed properties of the convergence of restarted GMRES as well as additional factors that affect its convergence behavior. We then present a new augmented method that accelerates the convergence of restarted GMRES by reducing the number of iterations required for convergence. Finally, we extend the new augmented method to a block method to demonstrate the feasibility of improving the performance of a common linear solver via a combination of algorithmic modifications that reduce data movement and a memory-efficient implementation. |
