Collaborative Research: On Some Fundamental Computational Issues in Simulating Interaction Models

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This project will investigate some fundamental computational issues for mathematical interaction models arising from scientific and engineering applications. Examples of these models include the electromagnetic and gravitational interactions in physics, interactions between molecules or cells in biology, and more general fractional differential equations and network models in material science, quantum theory, and social science. It is evident that modern interaction models are becoming more complex and demanding more accurate and efficient computational methods to handle the large scale interactions on high-performance computers. This project will (1) introduce novel representations of interaction models that are suitable for accelerated computation, (2) design efficient algorithms for interaction evaluations, (3) develop advanced open source tools, and (4) apply them to applications in nano-photonic devices, remote sensing, and medical imaging devices. This project will also train graduate students, including those from under-represented groups in STEM fields. This project will support one graduate per year for all 3 years on one campus and one graduate per year for years 2 and 3 on the other campus. In most interaction models, kernels usually depend on the spatial or temporal locations that may include the contributions from the interfaces between different materials. They may even depend on the given density functions at both the source and target locations. It is critical to find suitably compressed kernel representations for easier analysis and accelerated computation. This project will start from the optimal representations of the layered media Green’s functions for acoustic and electromagnetic waves that are spatially variant due to the contributions from the layer interfaces. These will be found through optimal integration contours and the corresponding discretized basis functions. The PIs will also generalize the partial-wave and plane-wave frame representations of the Laplace layer potentials to Yukawa, Helmholtz, and layered media potentials. The result will lead to better compressed density, kernel, and potential representations of more challenging non-local models in physics, biology, material science, social science, and image analysis. The PIs will develop effective numerical schemes for computing the compressible features and accelerating their algebraic operations by utilizing a multi-resolution framework to identify the interaction kernel features at different scales. The project also aims to create advanced open source software packages for the commonly used Laplace, Yukawa, and Helmholtz equations. Finally, through collaborations with application domain scientists and engineers, the numerical tools will be used in the design of optimal nano-photonic devices such as passive cooling devices, which may lead to a reduced carbon footprint and help socioeconomically disadvantaged communities lower their energy bills.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.