Numerics for nonlocal potentials and highly-oscillatory PDEs

Nonlocal interactions are ubiquitous in (quantum) physic, their numerical evaluation requiring both accuracy and efficiency. Typically they have convolution integral structure, like the Newtonian potential 1/x. The project aims at developing efficient and accurate numerical algorithms for the evaluation of nonlocal potentials and investigating the numerics of high-order averaging schemes for highly oscillatory system, e.g. strongly confined systems of quantum particles involving some longrange interactions that are modeled by nonlocal potentials. The project partners are Leslie Greengard (Courant Inst., NYU), Shidong Jiang (NJIT New Jersey), Florian Méhats (IRMAR at U. Rennes) and Norbert J. Mauser (WPI at U. Wien). Especially for strongly confined system great numerical challenges occur since the pronounced anisotropy induces highly oscillatory behavior in space and time. Here the expertise of Méhats and Mauser is helpful. The singularity of convolution interaction kernel needs to be dealt with carefully, and also the (slow decay) long-range effect demands appropriate truncation. We develop such methods and their rigorous numerical analysis. The nonuniform Fast Fourier Transform (NUFFT) and Gaussian-Sum (GS) methods, recently introduced and developed further by the PI and the project partners, are state-of-the-art solvers. Together with Greengard, Jiang and Mauser these methods will be improved, extended and implemented. Fast oscillations usually impose severe step-size restriction so as to capture the long-time dynamics of such stiff systems. So far, many numerical schemes, based on asymptotic analysis, have been developed in order to alleviate the formidable computational cost imposed by the fast oscillations. Geometric integrators, including stroboscopic averaging method (SAM) and multi-revolution composition methods (MRCM) are promising candidates for general highly oscillatory systems, e.g. strongly confined kinetic equations. Here the expertise of the pioneers Méhats and Philippe Chartier in Rennes is crucial. Moreover, for strongly confined systems that involve nonlocal potential, both problems occur simultaneously. Major project goals are: Analytical and Numerical study of the two-component rotating dipolar BEC. Comprehensive study of nonlocal potential solvers, with further extensions. High-order averaging method for kinetic equations. Analysis and numerics for confined quantum systems with magnetic fields. 1

This project was devoted to the development, improvement, analysis and applications of fast algorithms for nonlocal convolution-type potentials, which are quite common in quantum physics, quantum chemistry, fluid dynamics, materials science, electronic structure calculations, kinetic theory, biology and cosmology etc. We have successfully developed such convolution-type nonlocal potentials with optimal efficiency, accuracy (which can be as accurate as machine precision), and flexibility to a vast variety of other potentials on (un)structured grids. Preliminary application to three dimensional dipolar BEC has shown amazing improvements, e.g. a typical run now takes only 2 mins on a personal laptop instead of the 12 hours on a parallel cluster with prior solvers. Combined with parallel computing techniques, computer simulations of time dependent problems in 3 real space dimensions are possible that save scientists and engineers a lot of computational / experimental resources. The project was successfully implemented at Courant Institute at NYU, the number 1 ranked institute in Applied Math, at IRMAR Rennes, one of the strongest French centers in math outside Paris, with a return phase at the WPI Wien, an interdisciplinary center of excellence.