General-Domain, Scalable, Accelerated Spectral Partial Differential Equation Solvers and Applications in Simulation and Design

该项目旨在为一系列重要的科学和工程领域的模拟和设计开发有效的计算方法。将开发促进电磁纳米器件设计的方法；流体动力学和弹性动力学问题的求解与优化辐射转移在乳腺癌检测和可视化中的应用。因此，这项工作影响了社会感兴趣的各个领域，包括飞机、船舶和水下航行器的效率优化、地震预测、光学设备和通信系统的设计、医疗计划等。目前的努力将导致培养博士后、研究生和本科生水平的学生和研究人员。这些活动通常对所有参与者都是有益的，包括学生和博士后，以及提出具体工程应用的工业和实验室研究人员。由于这项工作，一些参与的研究生发现了在数学教学和指导方面的才能，大多数参与的本科生继续攻读研究生院或四年制大学，并酌情寻求在数学相关领域的工作。在为期三年的项目中，每年将资助一名研究生、一名本科生和一名博士后。尽管在过去的几十年里，数值分析和计算方法取得了巨大的进步，甚至利用了随之而来的计算机能力的提高，但在PDE模拟和设计的一般领域仍然存在许多挑战。近年来，加速谱方法与经典思想在局部渐近分析和摄动理论中的结合应用，在解决科学和工程中的现实问题方面取得了重大进展。所提出方法的预期通用性、计算效率和并行可扩展性部分源于对PDE问题、解和Green函数结构中的重要特征的专门计算，包括解在空间和时间上的奇异、快速变化和/或振荡特征，以及在实际应用中经常出现的域和边界的完整几何复杂性、材料组成等。沿着这些路线，在这项努力中要进行的工作是寻求发展可扩展的频域格林函数方法，用于直接和迭代的积分方程求解，而不依赖于快速傅里叶变换（从而导致并行可扩展算法），以及考虑相关的理论和计算边缘奇点和时域问题，光子学和辐射传递中的新型结构优化方法，以及基于神经网络的冲击波问题的谱解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project seeks development of effective computational methods for simulation and design in a range of important areas of science and engineering. Methodologies will be developed that facilitate design of electromagnetic nanodevices; solution and optimization of fluid-dynamics and elasto-dynamics problems; and radiative transfer with applications to breast-cancer detection and visualization. This work thus impacts a variety of areas of societal interest, including efficiency optimization for aircraft, ships and underwater vehicles, seismic forecasting, design of optical devices and communication systems, medical treatment planning, etc. The present effort will result in training of students and researchers at post-doctoral, graduate, and undergraduate levels. These activities are typically rewarding for all involved, including the students and postdocs, as well as the industrial and lab researchers who propose specific engineering applications. As a result of this work, some of the participating graduate students discover a talent for teaching and advising in mathematics, and most of the participating undergraduate students go on to graduate schools or four-year colleges, as appropriate, and seek to pursue work on mathematically related fields. This project will support one graduate student, one undergraduate student and one postdoc each year of the three year project. Notwithstanding great advances in numerical analysis and computational methods over the last few decades, and even leveraging the accompanying improvement in computer power, a number of challenges remain in the general field of PDE simulation and design. Significant progress in recent years has arisen, in the context of realistic problems in science and engineering, from combined use of accelerated spectral methods and classical ideas in local asymptotic analysis and perturbation theory. The desired generality, computational efficiency and parallel scalability of the proposed approaches result in part from specialized accounting of important features in the structures of the PDE problems, solutions, and Green functions, including the singular, rapidly varying and/or oscillatory character of solutions both in space and time, as well as the full geometric complexity of domains and boundaries, material composition, etc., that often arise in realistic applications. Along these lines, the work to be pursued in this effort seeks development of scalable frequency-domain Green function methods for direct and iterative integral-equation solvers without recourse to Fast-Fourier transforms (thus leading to parallel scalable algorithms), as well as consideration of related theoretical and computational edge-singularity and time-domain problems, novel structural optimization methods in photonics and radiative transfer, and neural net-informed spectral solution of shock-wave problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.