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Monday, October 6, 2014 - 14:00 ,
Location: Skiles 005 ,
Dr. Maryam Yashtini ,
Georgia Tech Mathematics ,
Organizer: Martin Short

An alternating direction approximate Newton method (ADAN) is developedfor solving inverse problems of the form$\min \{\phi(Bu) +1/2\norm{Au-f}_2^2\}$,where $\phi$ is a convex function, possibly nonsmooth,and $A$ and $B$ are matrices.Problems of this form arise in image reconstruction where$A$ is the matrix describing the imaging device, $f$ is themeasured data, $\phi$ is a regularization term, and $B$ is aderivative operator. The proposed algorithm is designed tohandle applications where $A$ is a large, dense ill conditionmatrix. The algorithm is based on the alternating directionmethod of multipliers (ADMM) and an approximation to Newton's method in which Newton's Hessian is replaced by a Barzilai-Borwein approximation. It is shown that ADAN converges to a solutionof the inverse problem; neither a line search nor an estimateof problem parameters, such as a Lipschitz constant, are required.Numerical results are provided using test problems fromparallel magnetic resonance imaging (PMRI).ADAN performed better than the other schemes that were tested.

Monday, September 29, 2014 - 14:00 ,
Location: Skiles 005 ,
Dr. Manuela Manetta ,
Georgia Tech Mathematics ,
Organizer: Martin Short

The distance of a nxn stable matrix to the set of unstable matrices, the
so-called distance to instability, is a well-known measure of linear
dynamical system stability. Existing techniques compute this quantity
accurately but the cost is of the order of multiple SVDs of order n,
which makes the method suitable to middle size problems.
A new approach is presented, based on Newton's iteration applied to
pseudospectral abscissa, whose implementation is obtained by
discretization on differential equation for low-rank matrices,
particularly suited for large sparse matrices.

Monday, September 22, 2014 - 14:00 ,
Location: Skiles 005 ,
Dr. Chunmei Wang ,
Georgia Tech Mathematics ,
Organizer: Martin Short

Weak Galerkin finite element method is a new and efficient numerical
method for solving PDEs which was first proposed by Junping Wang and Xiu
Ye in 2011. The main idea of WG method is to introduce weak
differential operators and apply them to the corresponding variational
formulations to solve PDEs. In this talk, I will focus on the WG methods
for biharmonic equations, maxwell equations and div-curl equations.

Monday, September 8, 2014 - 14:00 ,
Location: Skiles 005 ,
Dr. Marta Canadell ,
Georgia Tech Mathematics ,
Organizer: Martin Short

We explain a method for the computation of normally hyperbolic invariant manifolds (NHIM) in discrete dynamical systems.The method is based in finding a parameterization for the manifold formulating a functional equation. We solve the invariance equation using a Newton-like method taking advantage of the dynamics and the geometry of the invariant manifold and its invariant bundles. The method allows us to compute a NHIM and its internal dynamics, which is a-priori unknown.We implement this method to continue the invariant manifold with respect to parameters, and to explore different mechanisms of breakdown. This is a joint work with Alex Haro.

Monday, April 28, 2014 - 14:00 ,
Location: Skiles 005 ,
Deanna Needell ,
Claremont McKenna College ,
Organizer: Martin Short

In this talk we will discuss results for robust signal reconstruction
from random observations via synthesis and analysis methods in
compressive signal processing (CSP). CSP is a new and exciting field
which arose as an efficient alternative to traditional signal
acquisition techniques. Using a (usually random) projection, signals
are measured directly in compressed form, and methods are then needed
to recover the signal from those measurements. Synthesis methods
attempt to identify the low-dimensional representation of the signal
directly, whereas analysis type methods reconstruct in signal space.
We also discuss special cases including provable near-optimal
reconstruction guarantees for total-variation
minimization and new techniques in super-resolution.

Monday, April 14, 2014 - 14:00 ,
Location: Skiles 005 ,
Professor Ke Chen ,
The University of Liverpool, UK ,
Organizer: Haomin Zhou

Mathematical imaging is not only a multidisciplinary research area but also a major cross-disciplinesubject within mathematical sciences as image analysis techniques involve analysis, optimization, differential geometry and nonlinear partial differential equations, computational algorithms and numerical analysis.In this talk I first review various models and techniques in the variational frameworkthat are used for restoration of images. Then I discuss more recent work on i) choice of optimal coupling parameters for the TV model,ii) the blind deconvolution and iii) high order regularization models.This talk covers joint work with various collaborators in imaging including J. P. Zhang, T.F. Chan, R. H. Chan, B. Yu, L. Sun, F. L. Yang (China), C. Brito (Mexico), N. Chumchob (Thailand), M. Hintermuller (Germany), Y. Q. Dong (Denmark), X. C. Tai (Norway) etc.

Monday, April 7, 2014 - 14:00 ,
Location: Skiles 005 ,
Ming-Jun Lai ,
University of Georgia ,
Organizer: Martin Short

I mainly discuss the following problem: given a set of scattered locations and nonnegative values, how can one construct a smooth interpolatory or fitting surface of the given data? This problem arises from the visualization of scattered data and the design of surfaces with shape control. I shall start explaining scattered data interpolation/fitting based on bivariate spline functions over triangulation without nonnegativity constraint. Then I will explain the difficulty of the problem of finding nonnegativity perserving interpolation and fitting surfaces and recast the problem into a minimization problem with the constraint. I shall use the Uzawa algorithm to solve the constrained minimization problem. The convergence of the algorithm in the bivariate spline setting will be shown. Several numerical examples will be demonstrated and finally a real life example for fitting oxygen anomalies over the Gulf of Mexico will be explained.

Monday, March 31, 2014 - 14:00 ,
Location: Skiles 005 ,
Benjamin Seibold ,
Temple University ,
Organizer: Martin Short

Initially homogeneous vehicular traffic flow can become inhomogeneous
even in the absence of obstacles. Such ``phantom traffic jams'' can be
explained as instabilities of a wide class of ``second-order''
macroscopic traffic models. In this unstable regime, small
perturbations amplify and grow into nonlinear traveling waves. These
traffic waves, called ``jamitons'', are observed in reality and have
been reproduced experimentally. We show that jamitons are analogs of
detonation waves in reacting gas dynamics, thus creating an
interesting link between traffic flow, combustion, water roll waves,
and black holes. This analogy enables us to employ the Zel'dovich-von
Neumann-Doering theory to predict the shape and travel velocity of the
jamitons. We furthermore demonstrate that the existence of jamiton
solutions can serve as an explanation for multi-valued parts that
fundamental diagrams of traffic flow are observed to exhibit.

Monday, March 24, 2014 - 14:00 ,
Location: Skiles 005 ,
Seth Marvel ,
University of Michigan ,
Organizer: Martin Short

In this talk, I will present work on two very different
problems, with the only common theme being a substantial departure from
standard approaches. In the first part, I will discuss how the spread of
many common contagions may be more accurately modeled with nonlocal
approaches than with the current standard of local approaches, and I will
provide a minimal mathematical foundation showing how this can be done. In
the second part, I will present a new computational method for ranking
items given only a set of pairwise preferences between them. (This is
known as the minimum feedback arc set problem in computer science.) For a
broad range of cases, this method appears to beat the current "world
record" in both run time and quality of solution.

Monday, March 10, 2014 - 14:00 ,
Location: Skiles 005 ,
Ray Treinen ,
Texas State, San Marcos ,
Organizer: John McCuan

The symmetric configurations for the equilibrium shape of a fluid interfaceare given by the geometric differential equation mean curvature isproportional to height. The equations are explored numerically tohighlight the differences in classically treated capillary tubes andsessile drops, and what has recently emerged as annular capillary surfaces. Asymptotic results are presented.