Collaborative Research: Algebraic Multigrid Methods: Multilevel Theory and Practice

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The primary goal of this collaborative proposal is to developtheoretically based algebraic multigrid (AMG) solvers for Hermitian(and, where possible, non-Hermitian) positive-definite problems. Theteam aims to improve understanding of the performance of the family ofAMG algorithms and, with this improved knowledge, to develop AMGmethods that offer provable, computable, a priori information on thealgorithm's performance. The project team represents a closecollaboration of experts in this area, each of whom has madecontributions in the field. Over the past several years, the team hasbegun to work collectively on developing new multilevel solvers andrigorous theoretical results for the convergence and complexityanalysis thereof. Together, the team will have the capability to takea step toward answering some of the fundamental research questionsassociated with these two essential aspects of the analysis and designof efficient algorithms.We expect the work proposed here to: (1) directly impact computationalsimulation codes currently employing multi-level solvers, by providingfaster and more reliable computational tools for the numericalcomputations at the core of physical simulations; and (2) allow forsimulation of phenomena for which suitable solvers are currentlyunavailable. The results from the proposed research will, thus, havea direct impact on scientific and engineering problems, includingthose from energy, through both the simulation of particle physics andprocessing of data from oil reservoir models, biophysics, in surgicalsimulation, and the environment, in climate prediction and contaminantremediation models. The algorithms to be investigated here arealready in use in many of these fields, but are often considered to be"expert-only" tools. The goal of this proposal is to develop morereliable and robust versions of these tools. The proposed researchwill have a strong educational impact as well, as it provides for asolid base for training of graduate students in the modern theoreticaland practical aspects of numerical methods for modeling ofapplications arising in science and engineering.