Collaborative Research: Algebraic Multigrid Methods: Multilevel Theory and Practice

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.