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Monday, January 22, 2018 - 13:55 ,
Location: Skiles 005 ,
Dr. Lee, Kiryung ,
GT ECE ,
Organizer: Sung Ha Kang

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.

Monday, December 4, 2017 - 14:00 ,
Location: Skiles 005 ,
Tao Pang ,
Department of Mathematics, North Carolina State University ,
Organizer: Luca Dieci

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.

Monday, November 27, 2017 - 14:00 ,
Location: Skiles 005 ,
Zhiliang Xu ,
Applied and Computational Mathematics and Statistics Dept, U of Notre Dame ,
zxu2@nd.edu ,
Organizer: Yingjie Liu

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.

Monday, November 20, 2017 - 14:00 ,
Location: Skiles 005 ,
Yat Tin Chow ,
Mathematics, UCLA ,
ytchow@math.ucla.edu ,
Organizer: Prasad Tetali

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.

Monday, November 6, 2017 - 13:55 ,
Location: Skiles 005 ,
Prof. Kevin Lin ,
University of Arizona ,
klin@math.arizona.edu ,
Organizer: Molei Tao

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.

Monday, October 16, 2017 - 14:00 ,
Location: Skiles 005 ,
Dr. Barak Sober ,
Tel Aviv University ,
barakino@gmail.com ,
Organizer: Doron Lubinsky

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.

Monday, October 2, 2017 - 13:55 ,
Location: Skiles 005 ,
Weilin Li ,
University of Maryland, College Park ,
wl298@math.umd.edu ,
Organizer: Wenjing Liao

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.

Monday, September 25, 2017 - 13:55 ,
Location: Skiles 005 ,
Professor Alessandro Veneziani ,
Emory Department of Mathematics and Computer Science ,
Organizer: Martin Short

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.

Monday, September 18, 2017 - 13:55 ,
Location: Skiles 005 ,
Prof. Nathan Kutz ,
University of Washington, Applied Mathematics ,
Organizer: Martin Short

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.

Friday, August 25, 2017 - 13:55 ,
Location: Skiles 005 ,
Prof. Song Li ,
Zhejiang University ,
Organizer: Haomin Zhou

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.