Seminars and Colloquia by Series

Approximation of Functions Over Manifolds by Moving Least Squares

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 16, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Barak Sober – Tel Aviv University –
We approximate a function defined over a $d$-dimensional manifold $M ⊂R^n$ utilizing only noisy function values at noisy locations on the manifold. To produce the approximation we do not require any knowledge regarding the manifold other than its dimension $d$. The approximation scheme is based upon the Manifold Moving Least-Squares (MMLS) and is therefore resistant to noise in the domain $M$ as well. Furthermore, the approximant is shown to be smooth and of approximation order of $O(h^{m+1})$ for non-noisy data, where $h$ is the mesh size w.r.t $M,$ and $m$ is the degree of the local polynomial approximation. In addition, the proposed algorithm is linear in time with respect to the ambient space dimension $n$, making it useful for cases where d is much less than n. This assumption, that the high dimensional data is situated on (or near) a significantly lower dimensional manifold, is prevalent in many high dimensional problems. Thus, we put our algorithm to numerical tests against state-of-the-art algorithms for regression over manifolds and show its dominance and potential.

On the recovery of measures without separation conditions

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 2, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Weilin Li – University of Maryland, College Park –
We formulate super-resolution as an inverse problem in the space of measures, and introduce a discrete and a continuous model. For the discrete model, the problem is to accurately recover a sparse high dimensional vector from its noisy low frequency Fourier coefficients. We determine a sharp bound on the min-max recovery error, and this is an immediate consequence of a sharp bound on the smallest singular value of restricted Fourier matrices. For the continuous model, we study the total variation minimization method. We borrow ideas from Beurling in order to determine general conditions for the recovery of singular measures, even those that do not satisfy a minimum separation condition. This presentation includes joint work with John Benedetto and Wenjing Liao.

Computational hemodynamics for Computer Aided Clinical Trials: looking at the theory, struggling with the practice

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 25, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Alessandro Veneziani – Emory Department of Mathematics and Computer Science
When we get to the point of including the huge and relevant experience of finite element fluid modeling collected in over 25 years of experience in the treatment of cardiovascular diseases, the risk of getting “lost in translation” is real. The most important issues are the reliability that we need to guarantee to provide a trustworthy decision support to clinicians; the efficiency we need to guarantee to fit into the demand coming from a large volume of patients in Computer Aided Clinical Trials as well as short timelines required by special circumstances (emergency) in Surgical Planning. In this talk, we will report on some recent activities taken at Emory to make this transition possible. Reliability requirements call for an appropriate integration of measurements and numerical models, as well as for uncertainty quantification. In particular, image and data processing are critical to feeding mathematical models. However, there are several challenges still open, e.g. in simulating blood flow in patient-specific arteries after stent deployment; or in assessing the correct boundary data set to be prescribed in complex vascular districts. The gap between theory, in this case, is apparent and good simulation and assimilation practices in finite elements for clinical hemodynamics need to be drawn. The talk will cover these topics. For computational efficiency, we will cover some numerical techniques currently in use for coronary blood flow, like the Hierarchical Model Reduction or efficient methods for coping with turbulence in aortic flows. As Clinical Trials are currently one of the most important sources of information for medical research and practice, we envision that the suitable achievement of reliability and efficiency requirements will make Computer Aided Clinical Trials (specifically with a strong Finite-Elements-in-Fluids component) an important source of information with a significant impact on the quality of healthcare. This is a joint work with the scholars and students of the Emory Center for Mathematics and Computing in Medicine (E(CM)2), the Emory Biomech Core Lab (Don Giddens and Habib Samady), the Beta-Lab at the University of Pavia (F. Auricchio ). This work is supported by the US National Science Foundation, Projects DMS 1419060, 1412963 1620406, Fondazione Cariplo, Abbott Vascular Inc., and the XSEDE Consortium.

Data-driven discovery of governing equations and physical laws

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 18, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Nathan Kutz – University of Washington, Applied Mathematics
The emergence of data methods for the sciences in the last decade has been enabled by the plummeting costs of sensors, computational power, and data storage. Such vast quantities of data afford us new opportunities for data-driven discovery, which has been referred to as the 4th paradigm of scientific discovery. We demonstrate that we can use emerging, large-scale time-series data from modern sensors to directly construct, in an adaptive manner, governing equations, even nonlinear dynamics, that best model the system measured using modern regression techniques. Recent innovations also allow for handling multi-scale physics phenomenon and control protocols in an adaptive and robust way. The overall architecture is equation-free in that the dynamics and control protocols are discovered directly from data acquired from sensors. The theory developed is demonstrated on a number of canonical example problems from physics, biology and engineering.

Compressive Sensing with Redundant Dictionaries

Series
Applied and Computational Mathematics Seminar
Time
Friday, August 25, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Song Li – Zhejiang University
In this talk, i shall provide some optimal PIR bounds, which confirmed a conjecture on optimal RIP bound. Furtheremore, i shall also investigate some results on signals recovery with redundant dictionaries, which are also related to statistics and sparse representation.

End-to-end learning for time series classification

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 24, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. George Mohler – IUPUI Computer Science
In this talk we focus on classification problems where noisy sensor measurements collected over a time window must be classified into one or more categories. For example, mobile phone health and insurance apps take as input time series from the accelerometer, gyroscope and GPS radio of the phone and output predictions as to whether the user is still, walking, running, biking, driving etc. Standard approaches to this problem consist of first engineering features from statistics of the data (or a transform) over a window and then training a discriminative classifier. For two applications we show how these features can instead be learned in an end-to-end modeling framework with the advantages of increased accuracy and decreased modeling and training time. The first application is reconstructing unobserved neural connections from Calcium fluorescence time series and we introduce a novel convolutional neural network architecture with an inverse covariance layer to solve the problem. The second application is driving detection on mobile phones with applications to car telematics and insurance.

Limits of the instanton approach to chaotic systems

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 17, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Andre Souza – Georgia Tech –
In this talk we discuss how to find probabilities of extreme events in stochastic differential equations. One approach to calculation would be to perform a large number of simulations and gather statistics, but an efficient alternative is to minimize Freidlin-Wentzell action. As a consequence of the analysis one also determines the most likely trajectory that gave rise to the extreme event. We apply this approach to stochastic systems whose deterministic behavior exhibit chaos (Lorenz and Kuramoto-Sivashinsky equations), comment on the observed behavior, and discuss.

Non-euclidean virtual reality

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 10, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Elisabetta Matsumoto – GT Physics
The properties of euclidean space seem natural and obvious to us, to thepoint that it took mathematicians over two thousand years to see analternative to Euclid’s parallel postulate. The eventual discovery ofhyperbolic geometry in the 19th century shook our assumptions, revealingjust how strongly our native experience of the world blinded us fromconsistent alternatives, even in a field that many see as purelytheoretical. Non-euclidean spaces are still seen as unintuitive and exotic,but with direct immersive experiences we can get a better intuitive feel forthem. The latest wave of virtual reality hardware, in particular the HTCVive, tracks both the orientation and the position of the headset within aroom-sized volume, allowing for such an experience. We use this nacenttechnology to explore the three-dimensional geometries of theThurston/Perelman geometrization theorem. This talk focuses on oursimulations of H³ and H²×E.

Analysis of an ice-structure interaction model with a dynamic nonlinearity and random resetting

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 3, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Michael Muskulus – NTNU: Norwegian University of Science and Technology –
This talk addresses an important problem in arctic engineering due to interesting dynamic phenomena in a forced linear system. The nonautonomous system considered is representative of a whole class of engineering problems that are not approachable by standard techniques from dynamical system theory.The background are ice-induced vibrations of structures (e.g. wind turbines or measurement masts) in regions with active sea ice. Ice is a complex material and the mechanism for ice-induced vibrations is not fully clear at present. In particular, the conditions under which the observed, qualitatively different vibration regimes are active cannot be predicted accurately so far. A recent mathematical model developed by Delft University of Technology assumes that a number of parallel ice strips are pushing with a constant velocity against a flexible structure. The structure is modelled as a single degree of freedom harmonic oscillator. The contact force acts on the structure, but at the same time slows down the advancement of the ice, thereby introducing a dynamic nonlinearity in the otherwise linear system. When the local contact force becomes large enough, the ice crushes and the corresponding strip is reset to a random offset in front of the structure.This is the first mathematical model that exhibits all three different dynamic regimes that are observed in reality: for slow ice velocities the structure undergoes quasi-static sawtooth responses where all ice strips fail at the same time (a kind of synchronization phenomenon), for large ice velocities the structure response appears random, and for intermediate ice velocities the system exhibits vibrations at the structure eigenfrequency, commonly called frequency lock-in behavior. The latter type of vibrations causes a lot of damage to the structure and poses a safety and economic risk, so its occurrence needs to be predicted accurately.As I will show in this talk, the descriptive terms for the three vibration regimes are slightly misleading, as the mechanisms behind the observed behaviors are somewhat different than intuition suggests. I will present first results in analyzing the system and offer some explanations of the observed behaviors, as well as some simple criteria for the switch between the different vibration regimes.

Polynomial convergence rate to nonequilibrium steady-state

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 13, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Yao Li – University of Massachusetts Amherst –
In this talk I will present my recent result about the ergodic properties of nonequilibrium steady-state (NESS) for a stochastic energy exchange model. The energy exchange model is numerically reduced from a billiards-like deterministic particle system that models the microscopic heat conduction in a 1D chain. By using a technique called the induced chain method, I proved the existence, uniqueness, polynomial speed of convergence to the NESS, and polynomial speed of mixing for the stochastic energy exchange model. All of these are consistent with the numerical simulation results of the original deterministic billiards-like system.

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