Collaborative Research: Algebraic Multigrid Methods: Multilevel Theory and Practice

合作研究：代数多重网格方法：多层次理论与实践

• 批准号: 0810982
• 负责人: Ludmil Zikatanov
• 金额: $ 19.86万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-10-01 至 2012-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0810982&HistoricalAwards=false
• 关键词: Collaborative; Research; Algebraic; Multigrid; Methods

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.

该合作提案的主要目标是开发基于理论的代数多重网格（AMG）求解器，用于解决厄米（以及可能的情况下，非厄米）正定问题。 该团队的目标是提高对 AMG 算法系列性能的理解，并利用这些改进的知识来开发 AMG 方法，提供有关算法性能的可证明、可计算的先验信息。 该项目团队由该领域的专家组成，每个专家都在该领域做出了贡献。 在过去的几年里，该团队开始共同致力于开发新的多级求解器，并为其收敛性和复杂性分析提供严格的理论结果。 总之，该团队将有能力朝着回答与高效算法的分析和设计的这两个基本方面相关的一些基础研究问题迈出一步。我们期望这里提出的工作能够：（1）直接影响目前采用多技术的计算模拟代码。 -级求解器，为物理模拟核心的数值计算提供更快、更可靠的计算工具； (2) 允许模拟目前没有合适的求解器的现象。 因此，拟议研究的结果将通过粒子物理模拟和油藏模型数据处理、手术模拟中的生物物理学以及气候预测中的环境数据处理，对科学和工程问题产生直接影响，包括能源问题。和污染物修复模型。 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。 该提案的目标是开发这些工具的更可靠、更强大的版本。 拟议的研究也将产生强大的教育影响，因为它为研究生在科学和工程应用建模的数值方法的现代理论和实践方面的培训奠定了坚实的基础。