Single-grid Multi-level Solvers for Coupled PDE Systems

本项目的目标是开发和研究一类结合了几何多重网格(GMG)和代数多重网格(AMG)方法的特殊类型的多层方法，我们将其称为单网格多层方法(SGML)。重点是离散化的偏微分方程组，用户通常可以获得关于基本几何网格的详细信息。研究小组正在设计解算器，它使用来自最精细网格的信息(因此被称为单网格法)来选择一种简单而固定的粗化，从而允许显式控制整体网格和多级解算器的运算符复杂性。我们正在研究的中心新想法涉及当用作光滑器时自适应构造MG松弛方案的算法的设计和分析。现有的AMG方法中，光滑器是固定的，粗化是设置阶段的关键，而SGML将在设置阶段构造光滑器，以补充其简单的基于几何的粗化过程。应当注意，平滑器的代数构造还可以受益于使用几何网格的属性，例如，以并行方式获得未知数的适当划分。SGML方法(以及过去十年发展起来的用于构造MG内插的许多有前途的代数技术)也在考虑之中。然而，PI和共同PI将重点放在SGML方法上，因为它能够显式地控制复杂性，这反过来又允许(几乎)最佳的负载平衡和可预测的通信模式，从而使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旨在以开放源码并行代码实现，并可供科学计算社区使用。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，并为仿真提供强大的工具。总之，提出的方法利用所有可用的信息来构造求解器，以提高并行多核计算体系结构上物理现象的数值建模和模拟的效率。教育活动包括对研究生的培训。

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.