Collaborative Research: Algebraic Multigrid Methods: Multilevel Theory and Practice

这个合作方案的主要目标是开发基于理论的代数多重网格(AMG)求解器，用于厄米特(以及可能的情况下，非厄米特)正定问题。该方法旨在提高对AMG算法家族的性能的理解，并利用这种改进的知识来开发AMG方法，以提供关于算法性能的可证明、可计算的先验信息。项目组代表了该领域专家的密切合作，他们每个人都在该领域做出了贡献。在过去的几年里，该团队已经开始共同努力，开发新的多级求解器，并为其收敛和复杂性分析提供严格的理论结果。总而言之，该团队将有能力朝着回答与分析和设计高效算法的这两个基本方面相关的一些基本研究问题迈出一步。我们预计这里提出的工作将：(1)通过为物理模拟核心的数值计算提供更快和更可靠的计算工具，直接影响目前采用多级解算器的计算模拟代码；以及(2)允许模拟目前无法获得合适解算器的现象。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处理来自油藏模型、生物物理、外科模拟和环境的数据，以及气候预测和污染物处理模型中的数据。这里要研究的算法已经在这些领域中的许多领域使用，但通常被认为是“仅限专家”的工具。这项提议的目标是开发这些工具的更可靠和更强大的版本。拟议的研究也将产生强大的教育影响，因为它为研究生提供了一个坚实的基础，用于培训在科学和工程应用中出现的数值模拟的现代理论和实践方面。

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.