Design and Analysis of Algorithms for High-Performance Scientific Computing

物理系统的计算模拟是一种重要的科学和工业工具。最近模拟的改进来自于异构系统的高效并行算法的发展，模拟多种材料的问题，具有不同的材料特性，和/或耦合到额外的物理定律。在数学上，这些系统被建模为代表物理守恒和能量定律的偏微分方程（PDEs）耦合系统，变量和非线性系数反映了异质性。有限元离散化将这些连续统方程转化为有限维线性和非线性系统；这些系统的求解是许多仿真算法中资源密集型的核心计算任务。我的长期研究计划侧重于开发和分析有效的并行算法，以解决由这些离散化产生的线性，线性化和非线性系统。我的方法遵循多重网格方法，其中使用分层分解来确保迭代解决过程的最佳复杂性。近年来，这包括对结构网格多网格方法的强烈关注，这种方法可以自然地实现高并行效率。值得注意的是，我的研究小组已经开发了最先进的模拟工具，用于带电流体（磁流体动力学）和向列和手性液晶的流动。在这项工作的同时，我还承担了预测算法分析工具的开发，以帮助设计和优化这种情况下的求解器。此外，我还为快速发展的并行实时仿真领域贡献了基本算法和分析工具。本提案的研究目标是为这些领域的高性能科学计算开发改进的方法。一个具有挑战性的物理系统，微晶液晶，将推动这项研究，从依赖于一个辅助变量的自由能提供新的挑战。同时，我们将对局部傅里叶分析提出一个稳健的优化观点，这是优化单片多网格方法算法参数的最佳实践工具。这将使我们能够设计和分析日益复杂的系统，从传统的高CPU时间中解放出来，以对耦合有限元离散化的单片算法进行强力分析。最后，我将继续开发时空系统的算法和分析工具，重点是多网格时间约简算法和相应的半代数模式分析工具。在该项目的所有阶段，HQP在算法设计和分析以及高性能计算环境中的编程方面的培训将是一个中心主题，提供可应用于学术界和工业界的计算科学和工程方面的关键技能。

Computational simulation of physical systems is a significant scientific and industrial tool. Recent improvements in simulation have come from the development of efficient parallel algorithms for heterogeneous systems, simulating problems with multiple materials, with varying material properties, and/or coupling to additional physical laws. Mathematically, these systems are modeled as coupled systems of partial differential equations (PDEs) representing physical conservation and energy laws, with variable and nonlinear coefficients reflecting the heterogeneity. Finite-element discretizations transform these continuum equations into finite-dimensional linear and non-linear systems; the solution of these systems is the core resource-intensive computational task in many simulation algorithms. My long-term research program focuses on the development and analysis of efficient parallel algorithms for solving the linear, linearized, and non-linear systems that result from these discretizations. My approach follows the multigrid methodology, where a hierarchical decomposition is used to ensure optimal complexity of the iterative solution process. In recent years, this has included a strong focus on structured-grid multigrid methods, which can naturally achieve high parallel efficiency. Of note, my research group has developed state-of-the-art simulation tools for flows of charged fluids (magnetohydrodynamics) and nematic and chiral liquid crystals. Concurrent with this work, I have undertaken the development of predictive algorithmic analysis tools, to help design and optimize solvers in this setting. Furthermore, I have contributed fundamental algorithms and analysis tools to the rapidly growing field of parallel-in-time simulation. The research goals of this proposal are the development of improved methodologies for high-performance scientific computing in these areas. A challenging physical system, smectic liquid crystals, will drive this research, providing new challenges from the dependence of free energy on an auxiliary variable. Concurrently, we will develop a robust optimization viewpoint on local Fourier analysis, the best-practices tool for optimizing algorithmic parameters for monolithic multigrid methods. This will allow us to design and analyse systems of increasing complexity, freed from the traditional high CPU times required for brute-force analysis of monolithic algorithms for coupled finite-element discretizations. Finally, I will continue to develop both algorithms and analysis tools for space-time systems, with a focus on the multigrid reduction-in-time algorithm and corresponding semi-algebraic mode analysis tool. At all stages of this project, the training of HQP in algorithmic design and analysis, as well as programming in high-performance computing environments, will be a central theme, providing key skills in computational science and engineering that can be applied in both academia and industry.