The primary goal of this collaborative proposal is to develop theoretically based algebraic multigrid (AMG) solvers for Hermitian (and, where possible, non-Hermitian) positive-definite problems. The team aims to improve understanding of the performance of the family of AMG algorithms and, with this improved knowledge, to develop AMG methods that offer provable, computable, a priori information on the algorithm's performance. The project team represents a close collaboration of experts in this area, each of whom has made contributions in the field. Over the past several years, the team has begun to work collectively on developing new multilevel solvers and rigorous theoretical results for the convergence and complexity analysis thereof. Together, the team will have the capability to take a step toward answering some of the fundamental research questions associated with these two essential aspects of the analysis and design of efficient algorithms.
We expect the work proposed here to: (1) directly impact computational simulation codes currently employing multi-level solvers, by providing faster and more reliable computational tools for the numerical computations at the core of physical simulations; and (2) allow for simulation of phenomena for which suitable solvers are currently unavailable. The results from the proposed research will, thus, have a direct impact on scientific and engineering problems, including those from energy, through both the simulation of particle physics and processing of data from oil reservoir models, biophysics, in surgical simulation, and the environment, in climate prediction and contaminant remediation models. The algorithms to be investigated here are already in use in many of these fields, but are often considered to be "expert-only" tools. The goal of this proposal is to develop more reliable and robust versions of these tools. The proposed research will have a strong educational impact as well, as it provides for a solid base for training of graduate students in the modern theoretical and practical aspects of numerical methods for modeling of applications arising in science and engineering.
