AMCS Seminar Series

A distinguishing feature of vehicular traffic flow is that it may exhibit significant wave patterns. We first demonstrate that those frustrating (when stuck in traffic) traffic features possess an intriguing structural beauty (when seen from the outside), rendering phantom traffic jams to be mathematical analogs of detonation waves. We then show how a few well-controlled automated vehicles can mitigate traffic instabilities and waves, first in theory and simulation, then in real-world traffic experiments. These culminate in the CIRCLES (Congestion Impacts Reduction via CAV-in-the-loop Lagrangian Energy Smoothing) project: the largest field test of deployed control vehicles on a fully instrumented highway, carried out by a consortium of mathematicians, engineers, industry partners, and government agencies.

Key challenges in studying conformational changes in biomolecules and atomic clusters via molecular dynamics simulations are high dimensionality, complexity of the system, and a broad range of timescales. The time step is determined by the smallest timescale, about 1fs, while the waiting times to observe transitions of interest may be of the order of microseconds or milliseconds. These challenges are often addressed by introducing collective variables (CVs), i.e., functions of atomic coordinates whose dynamics capture the processes happening on slow timescales.

We investigate the problem of learning CVs with two goals: (1) to minimize the error in transition rates in the reduced model, and (2) to preserve the symmetries present in the potential energy function of the system. We propose a computational algorithm to learn collective variables based on the theory of effective dynamics (Legoll and Lelievre, 2010; Duong et al., 2018). The algorithm involves a symmetry-preserving feature map, learning the residence manifold via diffusion maps, and learning CVs via autoencoders. We present three case studies: normal butane C4H10, Lennard-Jones-7 (LJ7) in 2D, and Lennard-Jones-8 (LJ8) in 3D.
Joint work with Shashank Sule, Jiaxin (Margot) Yuan, Arnav Mehta, and Yeuk Yin Lam

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Over the past decade, significant progress has been made by the scientific community in developing low-rank methods for solving time-dependent problems. In particular, the reduced storage of low-rank solutions allows us to address the curse of dimensionality often associated with solving high-dimensional problems, especially in kinetic simulations. Naturally, scientists also desire structure preservation and conservation incorporated into the low-rank framework. In this talk, I will overview my recent work developing low-rank integrators that are also implicit, high-order accurate in time, and structure-preserving. A particular emphasis will be placed on convection-diffusion equations, with the Vlasov-Fokker-Planck equation acting as our motivating example.