Seminars and Colloquia by Series

The Fast Slepian Transform

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 5, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mark A. Davenport – Georgia Institute of Technology
The discrete prolate spheroidal sequences (DPSS's) provide an efficient representation for discrete signals that are perfectly timelimited and nearly bandlimited. Due to the high computational complexity of projecting onto the DPSS basis - also known as the Slepian basis - this representation is often overlooked in favor of the fast Fourier transform (FFT). In this talk I will describe novel fast algorithms for computing approximate projections onto the leading Slepian basis elements with a complexity comparable to the FFT. I will also highlight applications of this Fast Slepian Transform in the context of compressive sensing and processing of sampled multiband signals.

Minimizing the Difference of L1 and L2 norms with Applications

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 29, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Lou, Yifei – University of Texas, Dallas
A fundamental problem in compressive sensing (CS) is to reconstruct a sparse signal under a few linear measurements far less than the physical dimension of the signal. Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, in which case conventional methods, such as L1 minimization, do not work well. In this talk, I will present a novel non-convex approach, which is to minimize the difference of L1 and L2 norms, denoted as L1-L2, in order to promote sparsity. In addition to theoretical aspects of the L1-L2 approach, I will discuss two minimization algorithms. One is the difference of convex (DC) function methodology, and the other is based on a proximal operator, which makes some L1 algorithms (e.g. ADMM) applicable for L1-L2. Experiments demonstrate that L1-L2 improves L1 consistently and it outperforms Lp (p between 0 and 1) for highly coherent matrices. Some applications will be discussed, including super-resolution, image processing, and low-rank approximation.

Model-Based Multichannel Blind Deconvolution: Mathematical Analysis and Nonconvex Optimization Algorithms

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 22, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Lee, Kiryung – GT ECE
There are numerous modern applications in data science that involve inference from incomplete data. Various geometric prior models such as sparse vectors or low-rank matrices have been employed to address the ill-posed inverse problems arising in these applications. Recently, similar ideas were adopted to tackle more challenging nonlinear inverse problems such as phase retrieval and blind deconvolution. In this talk, we consider the blind deconvolution problem where the desired information as a time series is accessed as indirect observations through a time-invariant system with uncertainty. The measurements in this case is given in the form of the convolution with an unknown kernel. Particularly, we study the mathematical theory of multichannel blind deconvolution where we observe the output of multiple channels that are all excited with the same unknown input source. From these observations, we wish to estimate the source and the impulse responses of each of the channels simultaneously. We show that this problem is well-posed if the channel impulse responses follow a simple geometric model. Under these models, we show how the channel estimates can be found by solving corresponding non-convex optimization problems. We analyze methods for solving these non-convex programs, and provide performance guarantees for each.

Portfolio Optimization Problems for Models with Delays

Series
Applied and Computational Mathematics Seminar
Time
Monday, December 4, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tao Pang – Department of Mathematics, North Carolina State University
In the real world, the historical performance of a stock may have impacts on its dynamics and this suggests us to consider models with delays. We consider a portfolio optimization problem of Merton’s type in which the risky asset is described by a stochastic delay model. We derive the Hamilton-Jacobi-Bellman (HJB) equation, which turns out to be a nonlinear degenerate partial differential equation of the elliptic type. Despite the challenge caused by the nonlinearity and the degeneration, we establish the existence result and the verification results.

Central and Central Discontinuous Galerkin (DG) Schemes on Overlapping Cells of Unstructured Grids for Solving Ideal MHD Equations with Globally Divergence-Free Magnetic Field

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 27, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zhiliang Xu – Applied and Computational Mathematics and Statistics Dept, U of Notre Dame –
In this talk, we will present new central and central DG schemes for solving ideal magnetohydrodynamic (MHD) equations while preserving globally divergence-free magnetic field on triangular grids. These schemes incorporate the constrained transport (CT) scheme of Evans and Hawley with central schemes and central DG methods on overlapping cells which have no need for solving Riemann problems across cell edges where there are discontinuities of the numerical solution. The schemes are formally second-order accurate with major development on the reconstruction of globally divergence-free magnetic field on polygonal dual mesh. Moreover, the computational cost is reduced by solving the complete set of governing equations on the primal grid while only solving the magnetic induction equation on the polygonal dual mesh.

A partial Laplacian as an infinitesimal generator on the Wasserstein space

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 20, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yat Tin Chow – Mathematics, UCLA –
In this talk, we will introduce a family of stochastic processes on the Wasserstein space, together with their infinitesimal generators. One of these processes is modeled after Brownian motion and plays a central role in our work. Its infinitesimal generator defines a partial Laplacian on the space of Borel probability measures, taken as a partial trace of a Hessian. We study the eigenfunction of this partial Laplacian and develop a theory of Fourier analysis. We also consider the heat flow generated by this partial Laplacian on the Wasserstein space, and discuss smoothing effect of this flow for a particular class of initial conditions. Integration by parts formula, Ito formula and an analogous Feynman-Kac formula will be discussed. We note the use of the infinitesimal generators in the theory of Mean Field Games, and we expect they will play an important role in future studies of viscosity solutions of PDEs in the Wasserstein space.

Implicit sampling in the small-noise limit

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 6, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Kevin Lin – University of Arizona –
Weighted direct samplers, sometimes also called importance samplers, are Monte Carlo algorithms for generating independent, weighted samples from a given target probability distribution. They are used in, e.g., data assimilation, state estimation for dynamical systems, and computational statistical mechanics. One challenge in designing weighted samplers is to ensure the variance of the weights, and that of the resulting estimator, are well-behaved. Recently, Chorin, Tu, Morzfeld, and coworkers have introduced a class of novel weighted samplers called implicit samplers, which possess a number of nice empirical properties. In this talk, I will summarize an asymptotic analysis of implicit samplers in the small-noise limit and describe a simple method to obtain a higher-order accuracy. I will also discuss extensions to stochastic differential equatons. This is joint work with Jonathan Goodman, Andrew Leach, and Matthias Morzfeld.

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.

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