Integrated Geometric and Algebraic Multigrid Methods

该项目的主要目标是创建一个框架，用于开发更强大和用户友好的线性方程组求解器，这些方程组在科学，工程和工业应用中无处不在。研究人员将进行综合分析和几何多重网格和代数多重网格方法的发展。几何多重网格（GMG）方法形成了一类多级求解器，用于求解由某些类型的离散偏微分方程（PDE）产生的线性方程组。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底层网格几何或PDE信息的任何依赖。因此，GMG方法是解决一类更受限制的线性系统的有效工具，其性能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没有相同的数学严谨性。这个研究项目将采用功能分析作为一个自然的框架，在一个统一的设置研究GMG和AMG方法。由此产生的理论将建立强大的指导原则，分析和开发强大的，高效的，可扩展的多级求解器。正在开发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，该项目还提供了博士学位。学生有机会参加各种教育和研究活动，在其中他们将接受高级培训，参加会议，并与来自工业和能源部实验室的研究人员合作。本研究项目探讨了一个统一的框架，分析GMG和AMG方法在功能分析设置。该框架为理解这些多级求解器中涉及的算子、空间之间的关系和其他核心成分提供了坚实的基础，例如粗空间及其各自基础的构建，以及适用于各种现有AMG方法的收敛分析。此外，对于源自离散偏微分方程的问题，可以无缝集成用于构建辅助基于网格的预处理器和每个级别上的高效平滑器的几何信息的集成，从而允许更灵活和积极的代数粗化。更多的好处，将几何信息实现在接近最佳的负载平衡和可预测的通信模式的并行实现。对于更一般的线性系统，该项目研究使用（放松的）压缩感知技术来保持粗糙空间算子的稀疏性，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，以控制计算复杂性。预计这些技术将是有用的，在解决问题的非quasiuniform的底层网格或矩阵的问题，是不对称和正定（SPD）。为了验证所开发的方法的有效性，所得到的求解器将被应用到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.