W. D. Gropp, D. K. Kaushik, D. E. Keyes, D. A. Knoll, and B. F. Smith

The optimal number of levels to employ in parallel multilevel iterative algorithms, such as the various additive/multiplicative flavors of Schwarz-preconditioned Krylov iteration, in the sense of time-to-solution, is governed by a trade-off between the number of iterations required and the parallel complexity and processor efficiency of the average iteration. As processors, interconnection networks, gridfunction blocksizes, and coefficient and stencil structures vary, the sweetspot of this trade-off generally also varies. Therefore, the ability to tune a library of Schwarz methods is valuable. In recent domain decomposition meetings, 1-level methods have been advocated for multiple-component transient fluid dynamical simulations, since effective multiple-component coarse grid operators are elusive and relatively expensive to apply. In addition, when the linear system being solved arises as one of a progression of inexact Newton steps, strong asymptotic convergence is less important than an initial reduction of about an order of magnitude. However, recent experience with single-component radiation transport simulations shows a relevant role for two-level methods, and the possibility beckons of operator-split preconditioners that are two-level (or more) in each scalar component. Our contribution will include evaluations of the performance of some one-, two-, and multi-level methods in applications that illustrate the trade-off in levels, and rudimentary theoretical models incorporating convergence estimates for idealized problems and generic architectural parameters.