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Friday, April 17, 2015 - 14:05 ,
Location: Skiles 005 ,
Stephen Sprigle ,
Schools of Industrial Design and Applied Physiology, Georgia Tech ,
Organizer: Guillermo Goldsztein

The Rehabilitation Engineering and Applied Research Lab (REARLab) performs
both experimental research and product development activities focused on
persons with disabilities. The REARLab seeks collaboration from the School
of Mathematics on 2 current projects. This session will introduce
wheelchair seating with respect to pressure ulcer formation and present two
projects whose data analysis would benefit from applied mathematics.
3D Tissue Deformation- Sitting induces deformation of the
buttocks tissues. Tissue deformation has been identified as the underlying
cause of tissue damage resulting from external loading. The REARLab has
been collecting multi-planar images of the seated buttocks using MRI. This
data clearly shows marked differences between persons, as expected. We are
interested in characterizing tissue deformation as a combination of
displacement and distortion. Some tissues- such as muscle- displace
(translate within the sagittal, coronal and transverse planes) and distort
(change shape). Other tissue such as skin and subcutaneous fat, simple
distorts. We seek a mathematical means to characterize tissue deformation
that reflects its multi-planar nature.
Categorizing Weight-shifting behaviors - many wheelchair users have
limitations to their motor and/or sensory systems resulting in a risk of
pressure ulcers. Pressure ulcers occur when localized loading on the skin
causes ischemia and necrosis. In an attempt to reduce risk of pressure
ulcer occurrence, wheelchair users are taught to perform weight-shifts.
Weight shifts are movements that re-distribute loads off the buttocks for
short periods of time. The REARLab is measuring weight shifting behaviors
of wheelchair users during their everyday lives. We seek a means to
classify patterns of behavior and relate certain patterns to healthy
outcomes versus other patterns that result in unhealthy outcomes.

Monday, April 13, 2015 - 14:00 ,
Location: Skiles 005 ,
Seong Jun Kim ,
Georgia Tech ,
skim396@math.gatech.edu ,
Organizer:

We introduce a new parallel in time (parareal) algorithm which couples multiscale integrators with fully resolved fine scale integration and computes highly oscillatory solutions for a class of ordinary differential equations in parallel.
The algorithm computes a low-cost approximation of all slow variables in the system. Then, fast phase-like variables are obtained using the parareal iterative methodology and an alignment algorithm. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances.

Monday, April 6, 2015 - 14:00 ,
Location: Skiles 005 ,
Prof. Molei Tao ,
Georgia Tech School of Math. ,
mtao@gatech.edu ,
Organizer: Molei Tao

We show how to control an oscillator by periodically perturbing its stiffness, such that its amplitude follows an arbitrary positive smooth function. This also motivates the design of circuits that harvest energies contained in infinitesimal oscillations of ambient electromagnetic fields. To overcome a key obstacle, which is to compensate the dissipative effects due to finite resistances, we propose a theory that quantifies how small/fast periodic perturbations affect multidimensional systems. This results in the discovery of a mechanism that reduces the resistance threshold needed for energy extraction, based on coupling a large number of RLC circuits.

Monday, March 30, 2015 - 14:05 ,
Location: Skiles 005 ,
Professor Andrei Martinez-Finkelshtein ,
University of Almería ,
Organizer: Martin Short

The importance of the 2D Fourier transform in mathematical
imaging and vision is difficult to overestimate. For instance, the impulse
response of an optical system can be defined in terms of diffraction
integrals, that are in turn Fourier transforms of a function on a disk.
There are several popular competing approaches used to calculate
diffraction integrals, such as the extended Nijboer-Zernike (ENZ) theory.
In this talk, an alternative efficient method of computation of two
dimensional Fourier-type integrals based on approximation of the integrand
by Gaussian radial basis functions is discussed. Its outcome is a rapidly
converging series expansion for the integrals, allowing for their accurate
calculation. The proposed method yields a reliable and fast scheme for
simultaneous evaluation of such kind of integrals for several values of the
defocus parameter, as required in the characterization of the through-focus
optics.

Tuesday, March 24, 2015 - 11:00 ,
Location: Skiles 005 ,
Prof. Yifei Lou ,
UT Dallas ,
Organizer: Sung Ha Kang

A fundamental problem in compressed sensing (CS) is to reconstruct a sparsesignal under a few linear measurements far less than the physical dimensionof the signal. Currently, CS favors incoherent systems, in which any twomeasurements are as little correlated as possible. In reality, however, manyproblems are coherent, in which case conventional methods, such as L1minimization, do not work well. In this talk, I will present a novelnon-convex approach, which is to minimize the difference of L1 and L2 norms(L1-L2) in order to promote sparsity. Efficient minimization algorithms areconstructed and analyzed based on the difference of convex functionmethodology. The resulting DC algorithms (DCA) can be viewed as convergentand stable iterations on top of L1 minimization, hence improving L1 consistently. Through experiments, we discover that both L1 and L1-L2 obtain betterrecovery results from more coherent matrices, which appears unknown intheoretical analysis of exact sparse recovery. In addition, numericalstudies motivate us to consider a weighted difference model L1-aL2 (a>1) todeal with ill-conditioned matrices when L1-L2 fails to obtain a goodsolution. An extension of this model to image processing will be alsodiscussed, which turns out to be a weighted difference of anisotropic andisotropic total variation (TV), based on the well-known TV model and naturalimage statistics. Numerical experiments on image denoising, imagedeblurring, and magnetic resonance imaging (MRI) reconstruction demonstratethat our method improves on the classical TV model consistently, and is onpar with representative start-of-the-art methods.

Monday, March 23, 2015 - 14:05 ,
Location: Skiles 005 ,
Yoonsang Lee ,
Courant Institute of Mathematical Sciences ,
ylee@cims.nyu.edu ,
Organizer:

Backscatter is the process of energy transfer from small to large scales in turbulence; it is crucially important in the inverse energy cascades of two-dimensional and quasi-geostrophic turbulence, where the net transfer of energy is from small to large scales. A numerical scheme for stochastic backscatter in the two-dimensional and quasi-geostrophic inverse kinetic energy cascades is developed and analyzed. Its essential properties include a local formulation amenable to implementation in finite difference codes and non-periodic domains, smooth behavior at the coarse grid scale, and realistic temporal correlations, which allows detailed numerical analysis, focusing on the spatial and temporal correlation structure of the modeled backscatter. The method is demonstrated in an idealized setting of quasi-geostrophic turbulence using a low-order finite difference code, where it produces a good approximation to the results of a spectral code with more than 5 times higher nominal resolution. This is joint work with I. Grooms and A. J. Majda

Friday, March 13, 2015 - 14:00 ,
Location: Skiles 168 ,
Richard Tsai ,
University of Texas at Austin ,
Organizer:

I will present a new approach for computing boundary integrals that are defined on implicit interfaces, without
the need of explicit parameterization. A key component of this approach is a volume integral which is identical to the integral over the interface. I will show results applying this approach to simulate interfaces that evolve according to Mullins-Sekerka dynamics used in certain phase transition problems. I will also discuss our latest results in generalization of this approach to summation of unstructured point clouds and regularization of hyper-singular integrals.

Monday, March 9, 2015 - 14:00 ,
Location: Skiles 005 ,
Prof. Jianfeng Lu ,
Duke University ,
jianfeng@math.duke.edu ,
Organizer: Molei Tao

Understanding rare events like transitions of chemical system
from reactant to product states is a challenging problem due to the time
scale separation. In this talk, we will discuss some recent progress in
mathematical theory of transition paths. In particular, we identify and
characterize the stochastic process corresponds to transition paths. The
study of transition path process helps to understand the transition
mechanism and provides a framework to design and analyze numerical
approaches for rare event sampling and simulation.

Monday, March 2, 2015 - 14:00 ,
Location: Skiles 005 ,
Professor Scott McCalla ,
Montana State University ,
Organizer: Martin Short

The existence, stability, and bifurcation structure of localized
radially symmetric solutions to the Swift--Hohenberg equation is
explored both numerically through continuation and analytically through
the use of geometric blow-up techniques. The bifurcation structure for
these solutions is elucidated by formally treating the dimension as a
continuous parameter in the equations. This reveals a family of
solutions with an anomalous amplitude scaling that is far larger than
expected from a formal scaling in the far field. One key advantage of
the geometric blow-up techniques is that a priori knowledge of this
scaling is unnecessary as it naturally emerges from the construction.
The stability of these patterned states will also be discussed.

Monday, February 16, 2015 - 14:00 ,
Location: Skiles 005 ,
Prof. Matthew Lin ,
National Chung Cheng University, Georgia Tech ,
mhlin@ccu.edu.tw ,
Organizer: Chi-Jen Wang

Reference[1] Moody T. Chu

, Nonnegative Inverse Eigenvalue and Singular Value Problems, SIAM J. Numer. Anal (1992).[2] Wei Ma and Zheng-J. Bai, A regularized directional derivative-based Newton method for inverse singular value problems, Inverse Problems (2012).

Nonnegative inverse eigenvalue and singular value problems have been a research focus for decades. It is true that an inverse problem is trivial if the desired matrix is not restricted to any structure. This talk is to present two numerical procedures, based on a conquering procedure and an alternating projection process, to solve inverse eigenvalue and singular value problems for nonnegative matrices, respectively. In theory, we also discuss the existence of nonnegative matrices subject to prescribed eigenvalues and singular values. Though the focus of this talk is on inverse eigenvalue and singular value problems with nonnegative entries, the entire procedure can be straightforwardly applied to other types of structure with no difficulty.