CAREER: Towards General-Purpose, High-Order Integral Equation Methods for Computer Simulation in Engineering: Analysis, Algorithm Design, and Applications

数值模拟已经成为几乎所有科学和工程领域的重要工具，从发动机设计到造船，从个性化医疗到城市规划。然而，在这些应用领域中出现的大规模，全局耦合（所谓的椭圆）计算问题的有效解决仍然是一个主要挑战。虽然积分方程（IE）数值模拟方法在应用于科学和工程中的常见问题时通常具有最优的低成本，但由于技术障碍，其影响仅限于少数应用。该项目的目的是采取重要步骤消除这些障碍。首先，该项目将开发新的数值和符号算法，以减少将IE方法应用于新类型应用时所需的方法设计和实现工作的数量。这将使这些方法的相关成本节约更广泛地实现。其次，该项目将设计、实施和分析并行算法，使国家的大规模计算资源能够与IE方法结合使用。它们提高的计算效率将有助于提高模型的保真度和精度。第三，该项目将扩展IE方法可以解决的问题集，包括体积（而不仅仅是表面）数据，同时使用高精度的几何表示。第四，为这些方法中的数值误差自动控制提供理论认识和实践方法。最后，该项目将展示新方法及其在流体动力学的数学和数值挑战背景下的应用。为了促进下一代国家劳动力对这类计算工具的理解，该项目将为正在形成阶段的中学生提供为期一天的体验。这一经验将表明，计算机建模和模拟可以通过测试简单、机械模型的预测能力来帮助理解世界。通过依赖于独立的、动手操作的计算机实验，这种体验将是互动的、视觉上的吸引人的，几乎不需要数学准备，并且很容易与现实世界的计算应用建立联系。该计划的重点是创造参与和兴趣，目标是促进数学和计算领域的职业和教育选择。计算机模拟的积分方程（IE）方法通常具有较低的成本，但由于技术障碍，其影响仅限于少数应用。本研究的目的是采取重要步骤消除这些障碍，通过提供：复杂几何中具有一般符号给定核的层势评估的快速多极方法的高阶奇异正交和基础，2。在大规模应用中使用IE方法的可扩展和高效的分布式内存并行算法；非齐次偏微分方程中复杂几何体积势的高阶数值计算方法的设计与分析。4 .基于后验误差估计的自适应网格自动细化理论与方法；演示了在不可压缩的Navier-Stokes方程中开发的方法和算法的能力，包括高阶有限元方法和IE耦合。

Numerical simulation has become an essential tool in nearly all areas of science and engineering, ranging from engine design to naval architecture, and from personalized medicine to city planning. Yet, the efficient solution of large-scale, globally coupled (so-called elliptic) computational problems arising in these application areas remains a major challenge. Although integral equation (IE) methods for numerical simulation typically have optimally low cost when applied to common problems in science and engineering, their impact has been limited to a small handful of applications by technical obstacles. The purpose of this project is to take important steps to remove these obstacles. First, the project will develop novel numerical and symbolic algorithms to reduce the amount of method design and implementation work required when adapting IE methods to new classes of applications. This will make the associated cost savings of these methods more broadly accessible. Second, the project will design, implement, and analyze parallel algorithms to enable the nation's large-scale computing resources to be used in conjunction with IE methods. Their increased computational efficiency will facilitate the study of models with increased fidelity and higher accuracy. Third, the project will extend the set of problems that can be attacked with IE methods to those including volume (and not just surface) data while using highly accurate geometric representations. Fourth, it will provide a theoretical understanding and practical methods for automatic control of numerical error in these methods. Lastly, the project will demonstrate the new methods and their use in the mathematically and numerically challenging context of fluid dynamics. To foster an understanding of the power of these kinds of computational tools in the next generation of the nation's workforce, this project will employ a day-long experience for students in their formative middle-school years. This experience will convey that computer modeling and simulation can help understand the world by testing the predictive power of simple, mechanistic models. Through its reliance on self-contained, hands-on computer experiments, the experience will be interactive and visually engaging, require little mathematics preparation, and easily establish connections with real-world applications of computing. The program focuses on creating engagement and interest, with the goal of promoting career and educational choices in mathematics and computing. Integral equation (IE) methods for computer simulation typically have optimally low cost, but their impact has been limited to a handful of applications by technical obstacles. The purpose of this research is to take important steps to remove these obstacles, by providing: 1. high-order singular quadrature and infrastructure for fast multipole methods for the evaluation of layer potentials with general, symbolically given kernels in complex geometry, 2. scalable and efficient distributed-memory parallel algorithms for the use of IE methods in large-scale applications, 3. design and analysis of high-order numerical methods for volume potentials in complex geometry as needed by inhomogeneous partial differential equations, 4. theory and methods for automatic adaptive mesh refinement based on a-posteriori error estimates, and 5. a demonstration of the capabilities of the developed methods and algorithms in the context of the incompressible Navier-Stokes equations, including high-order finite-element-method and IE coupling.