Optimized Domain Decomposition Methods for Wave Propagation in Complex Media

The propagation of electromagnetic or elastic waves in a medium is distorted by interfaces where the medium is discontinuous. This happens in many settings of great practical importance, involving devices for communications, optics, remote sensing, and geophysical exploration. To understand these interactions between waves and the complex structures of the media through which they move, which often involve multiple scatterers as well as multiple layers with different material properties, requires resolving the complicated reflections and transmissions that waves undergo in such environments. This in turn requires large-scale numerical simulations. The investigator develops and analyzes high-performance, efficient, accurate, and rapidly convergent algorithms for this class of problems. His recent work with colleagues resulted in the development of an efficient computational strategy that incorporates windowed Green's functions within the boundary integral equation approach for the simulation of interaction of waves with infinitely extending interfaces. This computational framework enables simulation of transmission and reflection of waves by periodic media at high frequencies. This project builds on these methods to enable high-fidelity simulations of waves propagating in engineering structures such as thin film solar cells and metasurfaces. Graduate students participate in the research.

The investigator develops a family of algorithms that focus mainly on optimized Schwarz domain decomposition (DD) methods, incorporate carefully designed quasi-optimal transmission operators, and are amenable to simple yet effective preconditioning strategies. Combining the merits of direct and iterative solvers, this class of methods has emerged as a leading contender for solution of high-frequency wave propagation in complex media. The computational methodology underlying this work is based on boundary integral solvers that, whenever applicable, can produce solutions to partial differential equations with high-order accuracy and no numerical dispersion. The project leverages recent advances introduced by the investigator and collaborators in the boundary integral equation treatment of infinitely extending media (including periodic media) and dielectric composite media, combined with the modularity and parallelism inherent to DD methods, to enable simulations of realistic engineering structures such as complex photonic or electronic devices. The work affects a variety of areas of societal interest, including communication, remote sensing, seismology, and optics. A major part of the project with wide applications is the development of fast, highly accurate solvers for periodic metamaterials, and use of the resulting numerical tools in detailed investigation and design of photonic structures and metamaterials. Graduate students participate in the research and are trained in the field of high-performance scientific computing. Software packages for solution of boundary integral equations that can be used for teaching and research purposes are made available.

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.