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

该项目旨在为科学和工程的一系列重要领域的模拟和设计开发有效的计算方法。将开发方法，促进电磁纳米器件的设计;流体动力学和弹性动力学问题的解决方案和优化;以及辐射转移与乳腺癌检测和可视化的应用。因此，这项工作影响了各种社会利益的领域，包括飞机，船舶和水下航行器的效率优化，地震预报，光学设备和通信系统的设计，医疗规划等目前的努力将导致在博士后，研究生和本科生水平的学生和研究人员的培训。这些活动通常对所有参与者都有好处，包括学生和博士后，以及提出特定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数学教学和咨询的人才，大多数参与的本科生继续进入研究生院或四年制大学，并寻求从事数学相关领域的工作。该项目将支持一个研究生，一个本科生和一个博士后三年的项目每年。尽管在过去的几十年里，数值分析和计算方法取得了很大的进步，甚至利用了计算机能力的相应提高，但在PDE模拟和设计的一般领域仍然存在许多挑战。近年来，在科学和工程中的实际问题的背景下，结合使用加速谱方法和局部渐近分析和微扰理论中的经典思想，已经出现了重大进展。所提出的方法的期望的通用性、计算效率和并行可扩展性部分地来自于PDE问题、解和绿色函数的结构中的重要特征的专门考虑，包括解在空间和时间上的奇异、快速变化和/或振荡特性，以及域和边界、材料组成等的全部几何复杂性，这在实际应用中经常出现。沿着这些路线，在这项努力中要追求的工作是寻求开发用于直接和迭代积分方程求解器的可缩放频域绿色函数方法，而不求助于快速傅立叶变换（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异性和时域问题，光子学和辐射传输中的新型结构优化方法，该奖项反映了NSF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.