Seminars and Colloquia by Series

Multiscale Modeling and Computation of Optically Manipulated Nano Devices

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 7, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Di LiuMichigan State University

We present a multiscale modeling and computational scheme for optical-
mechanical responses of nanostructures. The multi-physical nature of
the problem is a result of the interaction between the electromagnetic
(EM) field, the molecular motion, and the electronic excitation. To
balance accuracy and complexity, we adopt the semi-classical approach
that the EM field is described classically by the Maxwell equations,
and the charged particles follow the Schr ̈oidnger equations quantum
mechanically. To overcome the numerical challenge of solving the high
dimensional multi-component many- body Schr ̈odinger equations, we
further simplify the model with the Ehrenfest molecular dynamics to
determine the motion of the nuclei, and use the Time- Dependent
Current Density Functional Theory (TD-CDFT) to calculate the
excitation of the electrons. This leads to a system of coupled
equations that computes the electromagnetic field, the nuclear
positions, and the electronic current and charge densities
simultaneously. In the regime of linear responses, the resonant
frequencies initiating the out-of-equilibrium optical-mechanical
responses can be formulated as an eigenvalue problem. A
self-consistent multiscale method is designed to deal with the well
separated space scales. The isomerization of Azobenzene is presented as a numerical example.

Applied differential geometry and harmonic analysis in deep learning regularization

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 23, 2019 - 13:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wei ZhuDuke University

Deep neural networks (DNNs) have revolutionized machine learning by gradually replacing the traditional model-based algorithms with data-driven methods. While DNNs have proved very successful when large training sets are available, they typically have two shortcomings: First, when the training data are scarce, DNNs tend to suffer from overfitting. Second, the generalization ability of overparameterized DNNs still remains a mystery. In this talk, I will discuss two recent works to “inject” the “modeling” flavor back into deep learning to improve the generalization performance and interpretability of the DNN model. This is accomplished by DNN regularization through applied differential geometry and harmonic analysis. In the first part of the talk, I will explain how to improve the regularity of the DNN representation by enforcing a low-dimensionality constraint on the data-feature concatenation manifold. In the second part, I will discuss how to impose scale-equivariance in network representation by conducting joint convolutions across the space and the scaling group. The stability of the equivariant representation to nuisance input deformation is also proved under mild assumptions on the Fourier-Bessel norm of filter expansion coefficients.

Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 16, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Andre WibisonoGeorgia Tech
Sampling is a fundamental algorithmic task. Many modern applications require sampling from complicated probability distributions in high-dimensional spaces. While the setting of logconcave target distribution is well-studied, it is important to understand sampling beyond the logconcavity assumption. We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution on R^n under isoperimetry conditions. We show a convergence guarantee in Kullback-Leibler (KL) divergence assuming the target distribution satisfies log-Sobolev inequality and the log density has bounded Hessian. Notably, we do not assume convexity or bounds on higher derivatives. We also show convergence guarantees in Rényi divergence assuming the limit of ULA satisfies either log-Sobolev or Poincaré inequality. Joint work with Santosh Vempala (arXiv:1903.08568).

Stability and instability issues for kinetic gravitational systems

Series
Applied and Computational Mathematics Seminar
Time
Friday, August 30, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mohammed LemouUniversité de Rennes 1 et ENS de Rennes

Please Note: Special time

I will start by giving a short overview of the history around stability and instability issues in gravitational systems driven by kinetic equations. Conservations properties and  families of non-homogeneous steady states will be first presented. A well-know conjecture in both astrophysics and mathematics communities was that  "all steady states of the gravitational Vlasov-Poisson system which are decreasing functions of the energy, are non linearly stable up to space translations".  We explain why the traditional variational approaches are not sufficient to answer this conjecture. An alternative approach, inspired by astrophysics literature, will be then presented and quantitative stability inequalities will be shown, therefore solving the above conjecture for Vlasov-Poisson systems. This have been achieved by using a refined notion for the rearrangement of functions and Poincaré-like  functional inequalities. For other systems like the so-called Hamiltonian Mean Field (HMF), the decreasing property of the steady states is no more sufficient to guarantee their stability. An additional explicit criteria is needed, under which their non-linear stability is proved. This criteria is sharp as  non linear instabilities can be constructed if it is not satisfied.

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

Please Note: special time

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

Please Note: special time

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.

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