Integrated Geometric and Algebraic Multigrid Methods

该项目的主要目标是创建一个框架，用于开发更健壮和用户友好的线性方程组解算器，这些解算器在科学、工程和工业应用中无处不在。研究人员将进行几何多重网格和代数多重网格方法的综合分析和发展。几何多重网格（GMG）方法形成了一类多层求解器，用于求解由某些离散偏微分方程（PDEs）引起的线性方程组。代数多重网格（AMG）方法也是多层求解器，尽管这些技术避免了对有关底层网格几何或PDE的信息的任何依赖。因此，GMG方法是解决更有限的一类线性系统的有效工具，具有强大的理论支持，而AMG求解器适用于更一般的线性系统，尽管在证明其性能方面不具有相同的数学严密性。本研究项目将以功能分析为自然框架，统一研究GMG和AMG方法。由此产生的理论将为分析和开发健壮、高效和可扩展的多级求解器建立强有力的指导原则。开发中的迭代求解器将以开源并行代码实现，提供给更广泛的科学计算社区，为模拟提供强大的工具，并为未来的算法研究和开发奠定基础。此外，该项目为博士生提供了参与各种教育和研究活动的机会，在这些活动中，他们将接受高级培训，参加会议，并与工业界和能源部实验室的研究人员合作。本研究计划探讨在功能分析设定中分析GMG和AMG方法的统一框架。该框架为理解算子、空间之间的关系以及这些多层求解器中涉及的其他核心成分（例如粗空间的构造及其各自的基）以及适用于各种现有AMG方法的收敛分析提供了坚实的基础。此外，对于源自离散化偏微分方程的问题，可以将几何信息集成用于构建辅助网格预调节器和每个级别的高效平滑器，从而实现更灵活和更积极的代数粗化。合并几何信息的更多好处体现在并行实现的近乎最佳的负载平衡和可预测的通信模式中。对于更一般的线性系统，该项目研究使用（放松）压缩感知技术来保持粗空间算子的稀疏性，这些算子可以补充来自底层几何网格或矩阵邻接图的信息，以获得对计算复杂性的控制。预计这些技术将有助于解决非准均匀底层网格问题或非对称正定矩阵问题。为了验证所开发方法的有效性，所得到的求解器将应用于流固耦合问题和近奇异SPD问题。

A primary goal of this project is to create framework for developing more robust and user-friendly solvers for linear systems of equations, which are ubiquitous in science, engineering, and industrial applications. The investigators will carry out integrated analysis and development of geometric multigrid and algebraic multigrid methodologies. Geometric multigrid (GMG) methods form a class of multilevel solvers designed to solve linear systems of equations arising from certain classes of discretized partial differential equations (PDEs). Algebraic multigrid (AMG) methods are also multilevel solvers, though these techniques avoid any dependence on information regarding an underlying grid geometry or PDE. As a result, GMG methods are effective tools for solving a more restricted class of linear systems with a strong theoretical backing for their performance, whereas AMG solvers apply to more general linear systems, though do not share the same mathematical rigor in justifying their performance. This research project will employ functional analysis as a natural framework for studying GMG and AMG methods in a unified setting. The resulting theory will establish strong guiding principles for analyzing and developing robust, efficient, and scalable multilevel solvers. The iterative solvers under development will be implemented in open source parallel codes, made available to a broader scientific computing community, providing powerful tools for simulation and a foundation for future algorithm research and development. Moreover, this project provides Ph.D. students with opportunities to participate in a variety of education and research activities, in which they will receive advanced training, participate in conferences, and collaborate with researchers from industry and Department of Energy laboratories.This research project investigates a unifying framework for analyzing GMG and AMG methods in a functional analysis setting. This framework provides a strong foundation for understanding the operators, relationships between spaces, and other core ingredients involved in these multilevel solvers, such as the construction of coarse spaces and their respective bases, and a convergence analysis that applies to a broad variety of existing AMG methods. Furthermore, for problems originating from discretized PDEs, the integration of geometric information for constructing auxiliary grid-based preconditioners and highly effective smoothers on each level can be seamlessly integrated, allowing for more flexible and aggressive algebraic coarsening. More benefits of incorporating geometric information are realized in nearly optimal load balancing and predictable communication patterns for parallel implementations. For more general linear systems, the project studies the use of (relaxed) compressed sensing techniques to preserve sparsity for coarse space operators, which can be supplemented with information from an underlying geometric grid or adjacency graphs of the matrix to gain control over the computational complexity. It is expected that these techniques will be useful in solving problems with non-quasiuniform underlying grids or problems with matrices that are not symmetric and positive-definite (SPD). To verify the efficacy of the developed methodologies, the resulting solvers will be applied to fluid-structure interaction problems and nearly singular SPD problems.