## Seminars and Colloquia by Series

### Numerical methods for solving nonlinear PDEs from homotopy methods to machine learning

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 12, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Wenrui HaoPenn State University

Many systems of nonlinear PDEs are arising from engineering and biology and have attracted research scientists to study the multiple solution structure such as pattern formation. In this talk, I will present several methods to compute the multiple solutions of nonlinear PDEs. In specific, I will introduce the homotopy continuation technique to compute the multiple steady states of nonlinear differential equations and also to explore the relationship between the number of steady-states and parameters. Then I will also introduce a randomized Newton's method to solve the nonlinear system arising from neural network discretization of the nonlinear PDEs. Several benchmark problems will be used to illustrate these ideas.

### Optimal-Transport Bayesian Sampling in the Era of Deep Learning

Series
Applied and Computational Mathematics Seminar
Time
Monday, August 24, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Bluejeans (online) https://bluejeans.com/197711728
Speaker
Prof. Changyou ChenUniversity at Buffalo

Deep learning has achieved great success in recent years. One aspect overlooked by traditional deep-learning methods is uncertainty modeling, which can be very important in certain applications such as medical image classification and reinforcement learning. A standard way for uncertainty modeling is by adopting Bayesian inference. In this talk, I will share some of our recent work on scalable Bayesian inference by sampling, called optimal-transport sampling, motivated from the optimal-transport theory. Our framework formulates Bayesian sampling as optimizing a set of particles, overcoming some intrinsic issues of standard Bayesian sampling algorithms such as sampling efficiency and algorithm scalability. I will also describe how our sampling framework be applied to uncertainty and repulsive attention modeling in state-of-the-art natural-language-processing models.

https://bluejeans.com/197711728

### Langevin dynamics with manifold structure: efficient solvers and predictions for conformational transitions in high dimensions

Series
Applied and Computational Mathematics Seminar
Time
Monday, June 22, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/963540401
Speaker
Dr. Yuan GaoDuke University

We work on Langevin dynamics with collected dataset that distributed on a manifold M in a high dimensional Euclidean space. Through the diffusion map, we learn the reaction coordinates for N which is a manifold isometrically embedded into a low dimensional Euclidean space. This enables us to efficiently approximate the dynamics described by a Fokker-Planck equation on the manifold N. Based on this, we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of N and enjoys the weighted l^2 convergence to the Fokker-Planck equation. The proposed upwind scheme leads to a Markov chain with transition probability between the nearest neighbor points, which enables us to directly conduct manifold-related computations such as finding the optimal coarse-grained network and the minimal energy path that represents chemical reactions or conformational changes. To acquire information about the equilibrium potential on manifold N, we apply a Gaussian Process regression algorithm to generate equilibrium potentials for a new physical system with new parameters. Combining with the proposed upwind scheme, we can calculate the trajectory of the Fokker-Planck equation on N based on the generated equilibrium potential. Finally, we develop an algorithm to pullback the trajectory to the original high dimensional space as a generative data for the new physical system. This is a joint work with Nan Wu and Jian-Guo Liu.

### Adaptive Tracking and Parameter Identification

Series
Applied and Computational Mathematics Seminar
Time
Monday, May 11, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/614972446/
Speaker
Prof. Michael Malisoff Louisiana State University

Please Note: Virtual seminar held on BlueJeans

Adaptive control problems arise in many engineering applications in which one needs to design feedback controllers that ensure tracking of desired reference trajectories while at the same time identify unknown parameters such as control gains. This talk will summarize the speaker's work on adaptive tracking and parameter identification, including an application to curve tracking problems in robotics. The talk will be understandable to those familiar with the basic theory of ordinary differential equations. No prerequisite background in systems and control will be needed to understand and appreciate this talk.

### Adaptive Tracking and Parameter Identification (cancelled due to COVID-19)

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 23, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael MalisoffLSU

Adaptive control problems arise in many engineering applications in which one needs to design feedback controllers that ensure tracking of desired reference trajectories while at the same time identify unknown parameters such as control gains. This talk will summarize the speaker's work on adaptive tracking and parameter identification, including an application to curve tracking problems in robotics. The talk will be understandable to those familiar with the basic theory of ordinary differential equations. No prerequisite background in systems and control will be needed to understand and appreciate this talk.

### Wobbling of pedestrian bridges

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 9, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guillermo GoldszteinGeorgia Tech

On June 10, 2000, the Millennium Bridge in London opened to the public. As people crossed the bridge, it wobbled. The sway of the bridge was large enough that prompted many on the bridge to hold on to the rails. Three days later, the bridge closed. It reopened only after modifications to prevent the wobbling were made, eighteen months later. We develop and study a model motivated by this event

### Optimization over the Diffeomorphism Group Using Riemannian BFGS with Application

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, March 4, 2020 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dr. Darshan Bryner Naval Surface Warfare Center, Panama City Division

Please Note: This is a part of IEEE Signal Processing Society Lecture Series, organized by Dr. Alessio Medda (alessiomedda@ieee.org). PLEASE RSVP to https://events.vtools.ieee.org/m/222947

The set of diffeomorphisms of the unit interval, or “warping functions,” plays an important role in many in functional data analysis applications. Most prominently, the problem of registering, or aligning, pairs of functions depends on solving for an element of the diffeomorphism group that, when applied to one function, optimally aligns it to the other.
The registration problem is posed as the unconstrained minimization of a cost function over the infinite dimensional diffeomorphism function space. We make use of its well-known Riemannian geometry to implement an efficient, limited memory Riemannian BFGS optimization scheme. We compare performance and results to the benchmark algorithm, Dynamic Programming, on several functional datasets. Additionally, we apply our methodology to the problem of non-parametric density estimation and compare to the benchmark performance of MATLAB’s built-in kernel density estimator ‘ksdensity’.

### Data-Driven Structured Matrix Approximation by Separation and Hierarchy

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 24, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Difeng CaiEmory University, Department of Mathematics

The past few years have seen the advent of big data, which brings unprecedented convenience to our daily life. Meanwhile, from a computational point of view, a central question arises amid the exploding amount of data: how to tame big data in an economic and efficient way. In the context of matrix computations, the question consists in the ability to handle large dense matrices. In this talk, I will first introduce data-sparse hierarchical representations for dense matrices. Then I will present recent development of a new data-driven algorithm called SMASH to operate dense matrices efficiently in the most general setting. The new method not only outperforms existing algorithms but also works in high dimensions. Various experiments will be provided to justify the advantages of the new method.

### The Elastica Model for Image Restoration: An Operator-Splitting Approach

Series
Applied and Computational Mathematics Seminar
Time
Friday, February 21, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Roland GlowinskiUniversity of Houston, Hong Kong Baptist University

The most popular model for Image Denoising is without any doubt the ROF (for Rudin-OsherFatemi) model. However, since the ROF approach has some drawbacks (the stair-case effect being one of them) practitioners have been looking for alternatives. One of them is the Elastica model, relying on the minimization in an appropriate functional space of the energy functional $J$ defined by

$$J(v)=\varepsilon \int_{\Omega} \left[ a+b\left| \nabla\cdot \frac{\nabla v}{|\nabla v|}\right|^2 \right]|\nabla v| d\mathbf{x} + \frac{1}{2}\int_{\Omega} |f-v|^2d\mathbf{x}$$

where in $J(v)$: (i) $\Omega$ is typically a rectangular region of $R^2$ and $d\mathbf{x}=dx_1dx_2$. (ii) $\varepsilon, a$ and $b$ are positive parameters. (iii) function $f$ represents the image one intends to denoise.

Minimizing functional $J$ is a non-smooth, non-convex bi-harmonic problem from Calculus of  Variations. Its numerical solution is a relatively complicated issue. However, one can achieve this task rather easily by combining operator-splitting and finite element approximations. The main goal of this lecture is to describe such a methodology and to present the results of numerical experiments which validate it.