Single-grid Multi-level Solvers for Coupled PDE Systems

本项目的目标是开发和研究一类特殊的多级方法，该方法将几何多重网格（GMG）和代数多重网格（AMG）方法中的技术联合收割机结合起来，我们称之为“单网格多级方法”（SGML）。重点是离散化偏微分方程，其基本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精细网格（因此称为单网格方法）信息的求解器，以选择简单且固定的粗化，从而可以显式控制多级求解器的整体网格和运算符复杂性。中心的新想法，我们正在调查的关注MG松弛计划的自适应建设时，作为一个平滑的算法的设计和分析。与现有的AMG方法相比，在现有的AMG方法中，平滑器是固定的，粗化是设置阶段的关键组成部分，SGML将在设置阶段构造平滑器，以补充其简单的基于几何的粗化过程。应该注意的是，平滑器的代数构造也可以受益于使用几何网格的属性，例如，以并行地获得未知数的适当划分。SGML方法（连同许多有前途的代数技术，用于构建MG插值在过去十年中开发）也正在考虑中。然而，PI和co-PI专注于SGML方法，因为它能够显式控制复杂性，这反过来又允许（接近）最佳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被设计为在开源并行代码中实现，并提供给科学计算社区。这将为未来相关领域的算法研究和开发提供一个计算框架，并为仿真提供有力的工具。总之，所提出的方法使用所有可用的信息来构造求解器，以提高并行多核计算架构上的物理现象的数值建模和模拟的效率。教育活动包括培养研究生。

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.