## Seminars and Colloquia by Series

Monday, December 1, 2014 - 14:00 , Location: Skiles 005 , Raffaele D'Ambrosio , GA Tech , Organizer: Martin Short
It is the
purpose of this talk to analyze the behaviour of multi-value numerical
methods acting as structure-preserving integrators for the numerical
solution of ordinary and partial differential equations (PDEs), with
special emphasys to Hamiltonian problems and reaction-diffusion PDEs. As
regards Hamiltonian problems, we provide a rigorous long-term error
analyis obtained by means of backward error analysis arguments, leading
to sharp estimates for the parasitic solution components and for the
error in the Hamiltonian. As regards PDEs, we consider
structure-preservation properties in the numerical solution of
oscillatory problems based on reaction-diffusion equations, typically
modelling oscillatory biological systems, whose solutions oscillate both
in space and in time. Special purpose numerical methods able to
accurately retain the oscillatory behaviour are presented.
Monday, November 17, 2014 - 14:00 , Location: Skiles 005 , Dr. Mohammad Farazmand , GA Tech Physics , Organizer: Martin Short
Recent experimental and numerical observations have shown the significance
of the Basset--Boussinesq memory term on the dynamics of small spherical
rigid particles (or inertial particles) suspended in an ambient fluid flow.
These observations suggest an algebraic decay to an asymptotic state, as
opposed to the exponential convergence in the absence of the memory term.

I discuss the governing equations of motion for the inertial particles,
i.e. the Maxey-Riley equation, including a fractional order derivative in
time. Then I show that the observed algebraic decay is a universal property
of the Maxey--Riley equation. Specifically, the particle velocity decays
algebraically in time to a limit that is O(\epsilon)-close to the fluid
velocity, where 0<\epsilon<<1 is proportional to the square of the ratio of
the particle radius to the fluid characteristic length-scale. These results
follows from a sharp analytic upper bound that we derive for the particle
velocity.

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 , , 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.