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Friday, November 14, 2014 - 11:00 ,
Location: Skiles 005 ,
Professor Andre Martinez-Finkelshtein ,
Universidad de Almería ,
Organizer: Martin Short

The medical imaging benefits from the advances in constructiveapproximation, orthogonal polynomials, Fourier and numerical analysis,statistics and other branches of mathematics. At the same time, the needs of the medical diagnostic technology pose new mathematical challenges. This talk surveys a few problems, some of them related to approximation theory, that have appeared in my collaboration with specialists studying some pathologies of the human eye, in particular, of the cornea, such as:- reconstruction of the shape of the cornea from the data collected bykeratoscopes- implementation of simple indices of corneal irregularity- fast and reliable computation of the through-focus characteristics of a human eye.

Monday, November 3, 2014 - 14:00 ,
Location: Skiles 005 ,
Dr. Matthew Calef ,
Los Alamos National Lab ,
Organizer: Martin Short

Observations of high energy density environments, from supernovae
implosions/explosions to inertial confinement fusion, are determined by
many different physical effects acting concurrently. For example, one
set of equations will describe material motion, while another set will
describe the spatial flow of energy. The relevant spatial and temporal
scales can vary substantially. Since direct measurement is difficult if
not impossible, and the relevant physics happen concurrently, computer
simulation becomes an important tool to understand how emergent behavior
depends on the constituent laws governing the evolution of the system.
Further, computer simulation can provide a means to use observation to
constrain underlying physical models.
This talk shall examine the challenges associated with developing
computational multiphysics simulation. In particular this talk will
outline some of the physics, the relevant mathematical models, the
associated algorithmic challenges, some of which are driven by emerging
compute architectures. The problem as a whole can be formidable and an
effective solution couples many disciplines together.

Monday, October 20, 2014 - 14:00 ,
Location: Skiles 005 ,
Dr. Maria D'Orsogna ,
Cal State University Northridge ,
Organizer: Martin Short

Given their ubiquity in physics, chemistry and materialsciences, cluster nucleation and growth have been extensively studied,often assuming infinitely large numbers of buildingblocks and unbounded cluster sizes. These assumptions lead to theuse of mass-action, mean field descriptions such as the well knownBecker Doering equations. In cellular biology, however, nucleationevents often take place in confined spaces, with a finite number ofcomponents, so that discrete and stochastic effects must be takeninto account. In this talk we examine finite sized homogeneousnucleation by considering a fully stochastic master equation, solvedvia Monte-Carlo simulations and via analytical insight. We findstriking differences between the mean cluster sizes obtained from ourdiscrete, stochastic treatment and those predicted by mean fieldones. We also study first assembly times and compare results obtained from processes where only monomer attachment anddetachment are allowed to those obtained from general coagulation-fragmentationevents between clusters of any size.

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.