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Monday, February 5, 2018 - 13:55 ,
Location: Skiles 005 ,
Mark A. Davenport ,
Georgia Institute of Technology ,
Organizer: Wenjing Liao

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.

Monday, January 29, 2018 - 13:55 ,
Location: Skiles 005 ,
Prof. Lou, Yifei ,
University of Texas, Dallas ,
Organizer: Sung Ha Kang

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.

Monday, January 29, 2018 - 13:55 ,
Location: Skiles 005 ,
Prof. Lou, Yifei ,
University of Texas, Dallas ,
Organizer: Sung Ha Kang
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.

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, 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.