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

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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.