Collaborative Research: Algebraic Multigrid Methods: Multilevel Theory and Practice

这一合作建议的主要目标是为厄米特（以及可能的非厄米特）正定问题开发基于理论的代数多网格（AMG）求解器。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，提供关于算法性能的可证明、可计算、先验信息。项目小组代表了该领域专家的密切合作，他们每个人都在该领域作出了贡献。在过去的几年中，该团队开始共同致力于开发新的多层求解器，并为其收敛性和复杂性分析提供严格的理论结果。总之，该团队将有能力采取步骤，以回答一些基本研究问题，这些问题与分析和设计有效算法的这两个基本方面有关。我们希望这里提出的工作：(1)通过为物理模拟的核心数值计算提供更快、更可靠的计算工具，直接影响目前使用多级求解器的计算模拟代码；(2)允许对目前没有合适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，包括来自能源的问题，通过粒子物理学的模拟和来自油藏模型的数据处理，生物物理学，外科模拟，以及环境，气候预测和污染修复模型。这里要研究的算法已经在许多这些领域中使用，但通常被认为是“仅限专家”的工具。本建议的目标是开发这些工具的更可靠和健壮的版本。建议的研究也将具有强大的教育影响，因为它为培养研究生在科学和工程应用中数值方法建模的现代理论和实践方面提供了坚实的基础。

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.