Collaborative Research: Algebraic Multigrid Methods: Multilevel Theory and Practice

这项合作计划的主要目标是开发基于理论的代数多重网格（AMG）求解器，用于Hermitian（以及在可能的情况下，非Hermitian）正定问题。 该团队的目标是提高对AMG算法家族性能的理解，并利用这些改进的知识，开发AMG方法，提供可证明的，可计算的，先验的算法性能信息。 该项目小组是这一领域专家的密切合作，他们每个人都在这一领域作出了贡献。 在过去的几年里，该团队已经开始共同努力开发新的多层求解器，并在收敛性和复杂性分析方面取得了丰硕的理论成果。 我们期望本文提出的工作能够：（1）通过为物理模拟核心的数值计算提供更快、更可靠的计算工具，直接影响目前采用多级求解器的计算模拟代码;以及（2）允许对当前没有合适的求解器的现象进行模拟。 因此，拟议研究的结果将对科学和工程问题产生直接影响，包括通过粒子物理模拟和油藏模型、生物物理模拟和环境、气候预测和污染治理模型的数据处理产生的能源问题。 这里要研究的算法已经在这些领域中的许多领域中使用，但通常被认为是“专家专用”的工具。 该提案的目标是开发这些工具的更可靠和更强大的版本。 拟议的研究将有很强的教育影响，以及，因为它提供了一个坚实的基础，培养研究生在现代理论和实践方面的数值方法建模ofapplications出现在科学和工程。

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.