Broader Significance and Importance: The proposed project is devoted to providing reliable models and the state-of-the-art numerical methods for studying the wave-matter interactions as in geometrical optics and nano optics with practical applications arising from seismic imaging, medical imaging, nanotechnology, nanosciences, and so on. The proposed models and numerical methods are based on knowledge of geometrical optics, physical optics, nano optics, computational chemistry and scientific computing, aiming to provide numerical tools and guidance for applications. Interdisciplinary collaborations will be pursued to extend the practical applications of the proposed methods. The project also involves the integration of research and education in computational mathematics. Graduate and undergraduate students, and members from underrepresented groups will be encouraged to participate in the project to enhance their knowledge and research. The proposed project will further the training and education of students and encourage them to pursue future career in science, technology, engineering and mathematics (STEM).
The PI will develop reliable models and efficient numerical methods to simulate the wave-matter interactions as in geometrical optics and nano optics. When the size of the matter is much larger than the wavelength, the interactions are equivalent to simulating high frequency waves in nonhomogeneous media with the effect of matter averaged and embedded continuously as medium constants. The PI will combine geometrical optics and physical optics along with scientific computing to design and analyze efficient numerical methods. When the size of the matter reaches nanoscale, the interactions require to simulate the motion of the matter and the evolution of the waves simultaneously as in the study of the optical responses of nanostructures in nano optics, where the quantum effects of the matter must be considered. The PI will use the semi-classical theories as the guideline to derive simple models to characterize the interactions and propose multiscale methods to simulate the interactions. The proposed models and methods consist of the following core ideas: (1) for simulating high frequency waves, the Huygens-Kirchhoff integral with the Green functions serves as the formulation and mechanism for advancing the wave propagation, where geometrical optics provides asymptotic approximations of the Green functions. The phase and amplitude to approximate the Green functions can be computed efficiently with accurate methods for Hamilton-Jacobi equations. The oscillatory Huygens-Kirchhoff integral can be evaluated efficiently with multilevel algorithms based on low-rank matrix decompositions. And data compression techniques to compress the phase and amplitude ensure the feasibility of building the waves in both two and three-dimensional spaces. And (2) for studying the optical responses of nanostructures, the semi-classical theories that treat the waves classically with the Maxwell equations and retain the quantum mechanical description of the matter provide characterizations of the light-matter interactions. The Born-Oppenheimer approximation and the ab initio molecular dynamics can be utilized to resolve the difficulty of dealing with the many-body system quantum mechanically, which results in simple semi-classical models that are numerically trackable with proposed multiscale schemes.
