Single-grid Multi-level Solvers for Coupled PDE Systems

耦合偏微分方程系统的单网格多级求解器

• 批准号: 1217142
• 负责人: Jinchao Xu
• 金额: $ 45.04万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1217142&HistoricalAwards=false
• 关键词: Single; grid; Multi; level; Solvers

The goal of this project is to develop and study a special class of multilevel methods that combine techniques from the Geometric Multigrid (GMG) and Algebraic Multigrid (AMG) methodologies, which we refer to as the "single-grid multilevel method" (SGML). The focus is discretized partial differential equations, for which detailed information on the underlying geometric grid is generally available to the user. The research team is designing solvers that use information from the finest grid (hence termed the single-grid method) to select a simple and fixed coarsening that allows for explicit control of the overall grid and operator complexities of the multilevel solver. The central new idea that we are investigating concerns the design and analysis of algorithms for adaptive construction of the MG relaxation scheme when used as a smoother. In contrast to existing AMG methods, in which the smoother is fixed and coarsening is the key component in the setup phase, SGML will construct the smoother in the setup phase to complement its simple geometry-based coarsening process. It should be noted that the algebraic construction of the smoother can also benefit from using properties of the geometric grid, for example, to obtain a suitable partitioning of the unknowns in parallel. The SGML approach (together with the many of the promising algebraic techniques for constructing the MG interpolations developed over the last decade) is also under consideration. The PI and co-PIs, though, are focusing on the SGML method because of its ability to explicitly control complexity, which in turn allows for (nearly) optimal load balancing and predictable communication patterns, such that the method is well suited for parallel computing. Overall, the iterative solvers under development are designed to be implemented in open source parallel codes and made available to the scientific computing community. This will provide a computational framework for future algorithm research and development in related areas as well as powerful tools for simulation. In summary, the proposed methodology constructs solvers using all the information available to increase the efficiency of numerical modeling and simulation of physical phenomena on parallel multi-core computing architectures. Educational activities include the training of graduate students.

该项目的目的是开发和研究一类特殊的多级方法，这些方法结合了几何多机（GMG）和代数多式（AMG）方法论，我们将其称为“单网格多级方法”（SGML）。焦点是离散的部分微分方程，对于用户，有关基础几何网格的详细信息通常可供用户使用。研究团队正在设计使用最优质网格（因此称为单网格方法）的信息来选择简单固定的粗化，从而可以明确控制多级求解器的整体网格和操作员的复杂性。我们正在调查的主要新想法涉及算法的设计和分析，用于自适应构建MG松弛方案时，将其用作更平滑。与现有的AMG方法相反，在设置阶段中，固定更平稳而固定是关键组成部分，SGML将在设置阶段构建更平滑的内容，以补充其简单的基于几何基于基于几何的粗糙过程。应当指出的是，更平滑的代数构造也可以从使用几何网格的属性中受益，例如，在并行中获得适当的未知数分区。 SGML方法（以及许多有希望的代数技术用于构建过去十年中开发的MG插值）。但是，PI和CO-PIS的重点是SGML方法，因为它具有明确控制复杂性的能力，进而允许（几乎）最佳的负载平衡和可预测的通信模式，因此该方法非常适合并行计算。总体而言，正在开发的迭代求解器以开源并行代码实现，并提供给科学计算社区。这将为未来的算法研究和开发提供一个计算框架，并为模拟的强大工具提供了一个计算框架。总而言之，所提出的方法使用所有可用的信息来构建求解器，以提高并行多核计算体系结构上物理现象的数值建模效率和仿真。教育活动包括对研究生的培训。