Projection-tree reduced order modeling for fast N-body computations
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This work presents a data-driven reduced-order modeling framework to accelerate the computations of N-body dynamical systems and their pair-wise interactions. The proposed framework differs from traditional acceleration methods, like the Barnes-Hut method, which requires online tree building of the state space, or the fast-multipole method, which requires rigorous a priori analysis of governing kernels and online tree building. Our approach combines Barnes-Hut hierarchical decomposition, dimensional compression via the least-squares Petrov-Galerkin (LSPG) projection, and hyper-reduction by way of the Gauss-Newton with approximated tensor (GNAT) approach. The resulting projection-tree reduced order model (PTROM) enables a drastic reduction in operational count complexity by constructing sparse hyper-reduced pairwise interactions of the N-body dynamical system. As a result, the presented framework is capable of achieving an operational count complexity that is independent of N, the number of bodies in the numerical domain. Capabilities of the PTROM method are demonstrated on the two-dimensional fluid-dynamic Biot-Savart kernel within a parametric and reproductive setting. Results show the PTROM is capable of achieving over 2000× wall-time speed-up with respect to the full-order model, where the speed-up increases with N. The resulting solution delivers quantities of interest with errors that are less than 0.1% with respect to full-order model.
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