## Seminars and Colloquia by Series

### Averaging for Vlasov and Vlasov-Poisson equations

Series
Applied and Computational Mathematics Seminar
Time
Thursday, August 29, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Philippe ChartierInria-Rennes/IRMAR/ENS Rennes

Our ambition is to derive asymptotic equations of the Vlasov-Poisson system in the strong magntic field regime. This work is thus an attempt to (re-)derive rigorously gyrokinetic equations and to design uniformly accurate methods for solving fast-oscillating kinetic equations, i.e. methods whose cost and accuracy do not depend the stiffness parameter. The main tools used to reach this objective are averaging and PDE techniques. In this talk, I will focus primarily on the first.

### Highly-oscillatory evolution equations with time-varying vanishing frequency: asymptotics and numerics

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, August 28, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mohammed LemouUniversité de Rennes 1 et ENS de Rennes

In asymptotic analysis and numerical approximation of highly-oscillatory evolution problems, it is commonly supposed that the oscillation frequency is either constant or, at least, bounded from below by a strictly positive constant uniformly in time. Allowing for the possibility that the frequency actually depends on time and vanishes at some instants introduces additional difficulties from both the asymptotic analysis and numerical simulation points of view. I will present a first step towards the resolution of these difficulties. In particular, we show that it is still possible in this situation to infer the asymptotic behavior of the solution at the price of more intricate computations and we derive a second order uniformly accurate numerical method.

### Large Eddy Simulation of Turbulent Sooting Flames: Subfilter Scale Modeling of Soot Sources and Species Transport

Series
Applied and Computational Mathematics Seminar
Time
Monday, August 26, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Suo YangUniversity of Minnesota – Twin Cities

Soot particles are major pollutants emitted from propulsion and power generation systems. In turbulent combustion, soot evolution is heavily influenced by soot-turbulence-chemistry interaction. Specifically, soot is formed during combustion of fuel-rich mixtures and is rapidly oxidized before being transported by turbulence into fuel-lean mixtures. Furthermore, different soot evolution mechanisms are dominant over distinct regions of mixture fraction. For these reasons, a new subfilter Probability Density Function (PDF) model is proposed to account for this distribution of soot in mixture fraction space. At the same time, Direct Numerical Simulation (DNS) studies of turbulent nonpremixed jet flames have revealed that Polycyclic Aromatic Hydrocarbons (PAH), the gas-phase soot precursors, are confined to spatially intermittent regions of low scalar dissipation rates due to their slow formation chemistry. The length scales of these regions are on the order of the Kolmogorov scale (i.e., the smallest turbulence scale) or smaller, where molecular diffusion dominates over turbulent mixing irrespective of the large-scale turbulent Reynolds number. A strain-sensitivity parameter is developed to identify such species. A Strain-Sensitive Transport Approach (SSTA) is then developed to model the differential molecular transport in the nonpremixed “flamelet” equations. These two models are first validated a priori against a DNS database, and then implemented within a Large Eddy Simulation (LES) framework, applied to a series of turbulent nonpremixed sooting jet flames, and validated via comparisons with experimental measurements of soot volume fraction.

### Stochastic-Statistical Modeling of Criminal Behavior

Series
Applied and Computational Mathematics Seminar
Time
Monday, August 19, 2019 - 13:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chuntian WangThe University of Alabama

Residential crime is one of the toughest issues in modern society. A quantitative, informative, and applicable model of criminal behavior is needed to assist law enforcement. We have made progress to the pioneering statistical agent-based model of residential burglary (Short et al., Math. Models Methods Appl., 2008) in two ways. (1) In one space dimension, we assume that the movement patterns of the criminals involve truncated Lévy distributions for the jump length, other than classical random walks (Short et al., Math. Models Methods Appl., 2008) or Lévy flights without truncation (Chaturapruek et al., SIAM J. Appl. Math, 2013). This is the first time that truncated Lévy flights have been applied in crime modeling. Furthermore (2), in two space dimensions, we used the Poisson clocks to govern the time steps of the evolution of the model, rather than a discrete time Markov chain with deterministic time increments used in the previous works. Poisson clocks are particularly suitable to model the times at which arrivals enter a system. Introduction of the Poisson clock not only produces similar simulation output, but also brings in theoretically the mathematical framework of the Markov pure jump processes, e.g., a martingale approach. The martingale formula leads to a continuum equation that coincides with a well-known mean-field continuum limit. Moreover, the martingale formulation together with statistics quantifying the relevant pattern formation leads to a theoretical explanation of the finite size effects. Our conjecture is supported by numerical simulations.

### On the Synchronization Myth for Lateral Pedestrian-Instability of Suspension Bridges

Series
Applied and Computational Mathematics Seminar
Time
Tuesday, June 25, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Igor BelykhGeorgia State

The pedestrian-induced lateral oscillation of London's Millennium bridge on the day it opened in 2000 has become a much cited paradigm of an instability caused by phase synchronization of coupled oscillators. However, a closer examination of subsequent theoretical studies and experimental observations have brought this interpretation into question.

To elucidate the true cause of instability, we study a model in which each pedestrian is represented by a simplified biomechanically-inspired two-legged inverted pendulum. The key finding is that synchronization between individual pedestrians is not a necessary ingredient of instability onset. Instead, the side-to-side pedestrian motion should on average lag that of the bridge oscillation by a fraction of a cycle. Using a multi-scale asymptotic analysis, we derive a mathematically rigorous general criterion for bridge instability based on the notion of effective negative damping. This criterion suggests that the initiation of wobbling is not accompanied by crowd synchrony and crowd synchrony is a consequence but not the cause of bridge instability.

### Near-Best Adaptive Approximation on Conforming Simplicial Partitions

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 22, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter BinevUniversity of South Carolina

The talk presents an extension for high dimensions of an idea from a recent result concerning near optimal adaptive finite element methods (AFEM). The usual adaptive strategy for finding conforming partitions in AFEM is ”mark → subdivide → complete”. In this strategy any element can be marked for subdivision but since the resulting partition often contains hanging nodes, additional elements have to be subdivided in the completion step to get a conforming partition. This process is very well understood for triangulations received via newest vertex bisection procedure. In particular, it is proven that the number of elements in the final partition is limited by constant times the number of marked cells. This motivated us [B., Fierro, Veeser, in preparation] to design a marking procedure that is limited only to cells of the partition whose subdivision will result in a conforming partition and therefore no completion step is necessary. We also proved that this procedure is near best in terms of both error of approximation and complexity. This result is formulated in terms of tree approximations and opens the possibility to design similar algorithms in high dimensions using sparse occupancy trees introduced in [B., Dahmen, Lamby, 2011]. The talk describes the framework of approximating high dimensional data using conforming sparse occupancy trees.

### Solving Inverse Problems on Networks: Graph Cuts, Optimization Landscape, Synchronization

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 15, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shuyang LingNew York University
Information retrieval from graphs plays an increasingly important role in data science and machine learning. This talk focuses on two such examples. The first one concerns the graph cuts problem: how to find the optimal k-way graph cuts given an adjacency matrix. We present a convex relaxation of ratio cut and normalized cut, which gives rise to a rigorous theoretical analysis of graph cuts. We derive deterministic bounds of finding the optimal graph cuts via a spectral proximity condition which naturally depends on the intra-cluster and inter-cluster connectivity. Moreover, our theory provides theoretic guarantees for spectral clustering and community detection under stochastic block model. The second example is about the landscape of a nonconvex cost function arising from group synchronization and matrix completion. This function also appears as the energy function of coupled oscillators on networks. We study how the landscape of this function is related to the underlying network topologies. We prove that the optimization landscape has no spurious local minima if the underlying network is a deterministic dense graph or an Erdos-Renyi random graph. The results find applications in signal processing and dynamical systems on networks.

### Interface of statistics and computing

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 8, 2019 - 13:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Xiaoming HuoGT ISyE

Inference (aka predictive modeling) is in the core of many data science problems. Traditional approaches could be either statistically or computationally efficient, however not necessarily both. The existing principles in deriving these models - such as the maximal likelihood estimation principle - may have been developed decades ago, and do not take into account the new aspects of the data, such as their large volume, variety, velocity and veracity. On the other hand, many existing empirical algorithms are doing extremely well in a wide spectrum of applications, such as the deep learning framework; however they do not have the theoretical guarantee like these classical methods. We aim to develop new algorithms that are both computationally efficient and statistically optimal. Such a work is fundamental in nature, however will have significant impacts in all data science problems that one may encounter in the society. Following the aforementioned spirit, I will describe a set of my past and current projects including L1-based relaxation, fast nonlinear correlation, optimality of detectability, and nonconvex regularization. All of them integrates statistical and computational considerations to develop data analysis tools.

### Shape dynamics of point vortices

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 1, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Tomoki OhsawaUT Dallas
We present a Hamiltonian formulation of the dynamics of the shape'' of N point vortices on the plane and the sphere: For example, if N=3, it is the dynamics of the shape of the triangle formed by three point vortices, regardless of the position and orientation of the triangle on the plane/sphere.For the planar case, reducing the basic equations of point vortex dynamics by the special Euclidean group SE(2) yields a Lie-Poisson equation for relative configurations of the vortices. Particularly, we show that the shape dynamics is periodic in certain cases. We extend the approach to the spherical case by first lifting the dynamics from the two-sphere to C^2 and then performing reductions by symmetries.

### Global Convergence of Neuron Birth-Death Dynamics

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, March 6, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joan Bruna Estrach New York University
Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions. In this talk, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We implement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean-field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of parameters. We illustrate our algorithms with two models to provide intuition for the mechanism through which convergence is accelerated. Joint work with G. Rotskoff (NYU), S. Jelassi (Princeton) and E. Vanden-Eijnden (NYU).