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.

The study of fluid-structure interaction spans many disciplines, from fluid dynamics and biological locomotion to mechanical engineering, geophysical phenomena, and applied mathematics. In this talk, I will demonstrate a few examples in which initially symmetric systems bifurcate into asymmetric and unstable states due to close coupling between structures and fluids. One example shows how a symmetric wing, when flapped in a fluid, can break symmetry to generate sustained unidirectional flight. Another reveals how a model continent in a laboratory experiment, interacting with a convective fluid mantle, can enter into oscillatory or trapped states.

Further, I will discuss a recent experiment inspired by the suspected super-rotation (differential rotation) of Earth’s inner core, showing how a symmetric structure can begin to spontaneously spin within a symmetric cavity filled with turbulent thermal convection. These phenomena were first discovered experimentally, and their results have been rigorously investigated through follow-up experiments and numerical simulations.

The study of reaction-diffusion equations on metric graphs has been drawing attention recently. Here, we focus on pattern dynamics on compact metric graphs. There are eight different types of compact metric graphs that are formed by two or three finite intervals. We consider systems of reaction-diffusion equations on these compact metric graphs with natural Kirchhoff boundary conditions.

Suppose further that the system exhibits Turing or wave instability. Then, we may observe pattern formation even on compact metric graphs. However, the bifurcation structures depend on the domain. It turns out that a solution on the simple circle or interval can be naturally related to that on the double-leaf metric graph. We are going to compare the variations of other solutions and examine how the bifurcation structures differ by considering the double-leaf metric graph.

In this talk, we present randomized algorithms to construct approximations for a set of solutions of parameter-dependent partial differential equations (PDEs), where the parameter set is high dimensional. It is well-known that the Proper Orthogonal Decomposition (POD)/Principal Component Analysis (PCA) and the greedy algorithm perform (quasi-)optimally in approximating such a set. However, both the POD and the greedy algorithm rely on a training set of finite cardinality that is chosen such that every point in the admissible parameter set is close to a point in the training set. Therefore, both algorithms suffer from the curse of dimensionality. We suggest breaking the curse of dimensionality by exploiting the concentration of measure phenomenon, which is also sometimes called the “blessing of dimensionality”.

In detail, we will present a randomized greedy algorithm that provides with high probability a certification for the whole parameter set rather than only for the parameters in the training set. Moreover, we will present a randomized POD and a corresponding error analysis that shows that for exponentially decaying eigenvalues of the randomized POD which uses the exact correlation operator (integral in the expectation) the approximation error between any solution corresponding to a parameter in the admissible parameter set and the approximation with the POD that uses a Monte-Carlo approximation converges exponentially as well.

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

The study of reaction-diffusion equations on metric graphs has been drawing attention recently. Here, we focus on pattern dynamics on compact metric graphs. There are eight different types of compact metric graphs that are formed by two or three finite intervals. We consider systems of reaction-diffusion equations on these compact metric graphs with natural Kirchhoff boundary conditions.

Suppose further that the system exhibits Turing or wave instability. Then, we may observe pattern formation even on compact metric graphs. However, the bifurcation structures depend on the domain. It turns out that a solution on the simple circle or interval can be naturally related to that on the double-leaf metric graph. We are going to compare the variations of other solutions and examine how the bifurcation structures differ by considering the double-leaf metric graph.

Information to follow…

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.

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