Seminars and Colloquia by Series

Asymptotic-preserving and positivity-preserving numerical methods for a class of stiff kinetic equations

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 10, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Jingwei Hu – Purdue

Kinetic equations play an important role in multiscale modeling hierarchy. It serves as a basic building block that connects the microscopic particle models and macroscopic continuum models. Numerically approximating kinetic equations presents several difficulties: 1) high dimensionality (the equation is in phase space); 2) nonlinearity and stiffness of the collision/interaction terms; 3) positivity of the solution (the unknown is a probability density function); 4) consistency to the limiting fluid models; etc. I will start with a brief overview of the kinetic equations including the Boltzmann equation and the Fokker-Planck equation, and then discuss in particular our recent effort of constructing efficient and robust numerical methods for these equations, overcoming some of the aforementioned difficulties. This is joint work with Ruiwen Shu (University of Maryland).

Data-driven computation of stochastic dynamics

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 3, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Yao Li – UMass Amherst

Consider a stochastic process (such as a stochastic differential equation) arising from applications. In practice, we are interested in many things like the invariant probability measure, the sensitivity of the invariant probability measure, and the speed of convergence to the invariant probability measure. Existing rigorous estimates of these problems usually cannot provide enough details. In this talk I will introduce a few data-driven computational methods that solve these problems for a class of stochastic dynamical systems, including but not limited to stochastic differential equations. All these methods are driven by the simulation data, and are less affected by the curse-of-dimensionality than traditional grid-based methods. I will demonstrate a few high (up to 100) dimensional examples in my talk.

Structure-preserving low multilinear rank approximation of antisymmetric tensors

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 18, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Erna Begovic Kovac – GT Math

The talk is concerned with low multilinear rank approximations to antisymmetric tensors, that is, multivariate arrays for which the entries change sign when permuting pairs of indices. Such tensors play a major role in quantum chemistry. We show which ranks can be attained by an antisymmetric tensor and discuss the adaption of existing approximation algorithms to preserve antisymmetry, most notably a Jacobi-type algorithm. Particular attention is paid to the special case when choosing the rank equal to the order of the tensor. It is shown that this case can be addressed with an unstructured rank-1 approximation. This allows for the straightforward application of the higher-order power method, for which we discuss effective initialization strategies. This is a joint work with Daniel Kressner (EPFL).

Boundary control of optimal mixing via fluid flows

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 11, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Weiwei Hu – University of Georgia –

We discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field via an active control of the flow velocity, governed by the Stokes or the Navier-Stokes equations, in a two dimensional open bounded and connected domain.  We consider the velocity field steered by a control input that acts tangentially on the boundary of the domain through the  Navier slip boundary conditions. This is motivated by mixing  within a cavity or vessel  by moving the walls or stirring at the boundaries. Our main objective is to design an optimal Navier slip boundary control  that optimizes mixing at a given final time. Non-dissipative scalars, both passive and active, governed by the transport equation will be discussed.  In the absence of diffusion, transport and mixing occur due to pure advection.  This essentially leads to a nonlinear control problem of a semi-dissipative system. We shall provide a rigorous proof of the existence of an optimal controller, derive the first-order necessary conditions for optimality, and present some preliminary results on the numerical implementation.

Nonstationary signal analysis and decomposition via Fast Iterative Filtering and Adaptive Local Iterative Filtering techniques. State of the art and open problems

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 4, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Antonio Cicone – University of L'Aquila

The analysis and decomposition of nonstationary and nonlinear signals in the quest for the identification
of hidden quasiperiodicities and trends is of high theoretical and applied interest nowadays.

Linear techniques like Fourier and Wavelet Transform, historically used in signal processing, cannot capture
completely nonlinear and non stationary phenomena.

For this reason in the last few years new nonlinear methods have been developed like the groundbreaking
Empirical Mode Decomposition algorithm, aka Hilbert--Huang Transform, and the Iterative Filtering technique.

In this seminar I will give an overview of this kind of methods and I will introduce two new algorithms,
the Fast Iterative Filtering and the Adaptive Local Iterative Filtering. I will review the main theoretical results
and outline the most intriguing open problems that still need to be tackled in the field.
Some examples of applications of these techniques to both artificial and real life signals
will be shown to give a foretaste of their potential and robustness.
 

Analysis and Applications of Nonsmooth Bifurcations

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 28, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 05
Speaker
Oleg Makarenkov – Univ Texas at Dallas –
In this talk I will first give a brief overview of how nonsmooth bifurcations (border-splitting, grazing, and fold-fold bifurcations) help to rigorously explain the existence of nonsmooth limit cycles in the models of anti-lock braking systems, power converters, integrate-and-fire neurons, and climate dynamics. I will then focus on one particular application that deals with nonsmooth bifurcations in dispersing billiards. In [Nonlinearity 11 (1998)] Turaev and Rom-Kedar discovered that every periodic orbit that is tangent to the boundary of the billiard produces an island of stability upon smoothening the boundary of the billiard. The result to be presented in the talk (joint work with Turaev) proves that any dispersing billiard admits such an arbitrary small perturbation that ensures the occurrence of a tangent periodic orbit.

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 Liu – Michigan 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 Zhu – Duke 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 Wibisono – Georgia 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 Lemou – Université 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.

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