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Monday, April 5, 2010 - 13:00 ,
Location: Skiles 255 ,
Jianfeng Cai ,
Dep. of Math. UCLA ,
Organizer: Haomin Zhou

Tight frame is a generalization of orthonormal basis. It inherits most good properties of orthonormal basis but gains more robustness to represent signals of intrests due to the redundancy. One can construct tight frame systems under which signals of interests have sparse representations. Such tight frames include translation invariant wavelet, framelet, curvelet, and etc. The sparsity of a signal under tight frame systems has three different formulations, namely, the analysis-based sparsity, the synthesis-based one, and the balanced one between them. In this talk, we discuss Bregman algorithms for finding signals that are sparse under tight frame systems with the above three different formulations. Applications of our algorithms include image inpainting, deblurring, blind deconvolution, and cartoon-texture decomposition. Finally, we apply the linearized Bregman, one of the Bregman algorithms, to solve the problem of matrix completion, where we want to find a low-rank matrix from its incomplete entries. We view the low-rank matrix as a sparse vector under an adaptive linear transformation which depends on its singular vectors. It leads to a singular value thresholding (SVT) algorithm.

Friday, April 2, 2010 - 13:00 ,
Location: Skiles 269 ,
Sookkyung Lim ,
Department of Mathematical Sciences, University of Cincinnati ,
Organizer: Sung Ha Kang

We investigate the effects of electrostatic and steric repulsion on thedynamics of pre-twisted charged elastic rod, representing a DNA molecule,immersed in a viscous incompressible fluid. Equations of motion of the rod, whichinclude the fluid-structure interaction, rod elasticity, and electrostatic interaction, are solved by the generalized immersed boundary method. Electrostatic interaction is treated using a modified Debye-Huckel repulsive force in which the electrostatic force depends on the salt concentration and the distance between base pairs, and a close range steric repulsion force to prevent self-penetration. After perturbation a pretwisted DNA circle collapses into a compact supercoiled configuration. The collapse proceeds along a complex trajectory that may pass near several equilibrium configurations of saddle type, before it settles in a locally stable equilibrium. We find that both the final configuration and the transition path are sensitive to the initial excess link, ionic stregth of the solvent, and the initial perturbation.

Monday, March 29, 2010 - 13:00 ,
Location: Skiles 255 ,
Luca Gerardo Giorda ,
Dep. of Mathematics and Computer Science, Emory University ,
Organizer: Sung Ha Kang

Schwarz algorithms have experienced a second youth over the lastdecades, when distributed computers became more and more powerful andavailable. In the classical Schwarz algorithm the computational domain is divided into subdomains and Dirichlet continuity is enforced on the interfaces between subdomains. Fundamental convergence results for theclassical Schwarzmethods have been derived for many partial differential equations. Withinthis frameworkthe overlap between subdomains is essential for convergence. More recently, Optimized Schwarz Methods have been developed: based on moreeffective transmission conditions than the classical Dirichlet conditions at theinterfaces between subdomains, such algorithms can be used both with and without overlap. On the other hand, such algorithms show greatly enhanced performance compared to the classical Schwarz method. I will present a survey of Optimized Schwarz Methods for the numerical approximation of partial differential equation, focusing mainly on heterogeneous convection-diffusion and electromagnetic problems.

Monday, March 15, 2010 - 13:00 ,
Location: Skiles 255 ,
Maria Cameron ,
Courant Institute, NYU ,
Organizer:

The overdamped Langevin equation is often used as a model in molecular dynamics. At low temperatures, a system evolving according to such an SDE spends most of the time near the potential minima and performs rare transitions between them. A number of methods have been developed to study the most likely transition paths. I will focus on one of them: the MaxFlux functional.The MaxFlux functional has been around for almost thirty years but not widely used because it is challenging to minimize. Its minimizer provides a path along which the reactive flux is maximal at a given finite temperature. I will show two ways to derive it in the framework of transition path theory: the lower bound approach and the geometrical approach. I will present an efficient way to minimize the MaxFlux functional numerically. I will demonstrate its application to the problem of finding the most likely transition paths in the Lennard-Jones-38 cluster between the face-centered-cubic and icosahedral structures.

Monday, March 8, 2010 - 13:00 ,
Location: Skiles 255 ,
Chun Liu ,
Penn State/IMA ,
Organizer:

Almost all models for complex fluids can be fitted into the energetic variational framework. The advantage of the approach is the revealing/focus of the competition between the kinetic energy and the internal "elastic" energies. In this talk, I will discuss two very different engineering problems: free interface motion in Newtonian fluids and viscoelastic materials. We will illustrate the underlying connections between the problems and their distinct properties. Moreover, I will present the analytical results concerning the existence of near equilibrium solutions of these problems.

Monday, March 1, 2010 - 13:00 ,
Location: Skiles 255 ,
James G. Nagy ,
Mathematics and Computer Science, Emory University ,
Organizer: Sung Ha Kang

Large-scale inverse problems arise in a variety of importantapplications in image processing, and efficient regularization methodsare needed to compute meaningful solutions. Much progress has beenmade in the field of large-scale inverse problems, but many challengesstill remain for future research. In this talk we describe threecommon mathematical models including a linear, a separable nonlinear,and a general nonlinear model. Techniques for regularization andlarge-scale implementations are considered, with particular focusgiven to algorithms and computations that can exploit structure in theproblem. Examples will illustrate the properties of these algorithms.

Monday, February 22, 2010 - 13:00 ,
Location: Skiles 255 ,
Heasoon Park ,
CSE, Georgia Institute of Technology ,
Organizer: Sung Ha Kang

Nonnegative Matrix
Factorization (NMF) has attracted much attention during the past
decade as a dimension reduction method in machine learning and data
analysis. NMF provides a lower rank approximation of a nonnegative
high dimensional matrix by factors whose elements are also
nonnegative. Numerous success stories were reported in application
areas including text clustering, computer vision, and cancer class
discovery.
In
this talk, we present novel algorithms for NMF and NTF (nonnegative
tensor factorization) based on the alternating non-negativity
constrained least squares (ANLS) framework. Our new algorithm for NMF
is built upon the block principal pivoting method for the
non-negativity constrained least squares problem that overcomes some
limitations of the classical active set method. The proposed NMF
algorithm can naturally be extended to obtain highly efficient NTF
algorithm for PARAFAC (PARAllel FACtor) model. Our algorithms
inherit the convergence theory of the ANLS framework and can easily
be extended to other NMF formulations such as sparse NMF and NTF with
L1 norm constraints. Comparisons of algorithms using various data
sets show that the proposed new algorithms outperform existing ones
in computational speed as well as the solution quality.
This
is a joint work with Jingu Kim and Krishnakumar Balabusramanian.

Monday, February 15, 2010 - 13:00 ,
Location: Skiles 255 ,
Lek-Heng Lim ,
UC Berkeley ,
Organizer: Haomin Zhou

Numerical linear algebra is often regarded as a workhorse of scientific and
engineering computing. Computational problems arising from optimization,
partial differential equation, statistical estimation, etc, are usually reduced
to one or more standard problems involving matrices: linear systems, least
squares, eigenvectors/singular vectors, low-rank approximation, matrix
nearness, etc. The idea of developing numerical algorithms for multilinear
algebra is naturally appealing -- if similar problems for tensors of higher
order (represented as hypermatrices) may be solved effectively, then one would
have substantially enlarged the arsenal of fundamental tools in numerical
computations.
We will see that higher order tensors are indeed ubiquitous in applications;
for multivariate or non-Gaussian phenomena, they are usually inevitable.
However the path from linear to multilinear is not straightforward. We will
discuss the theoretical and computational difficulties as well as ways to avoid
these, drawing insights from a variety of subjects ranging from algebraic
geometry to compressed sensing. We will illustrate the utility of such
techniques with our work in cancer metabolomics, EEG and fMRI neuroimaging,
financial modeling, and multiarray signal processing.

Monday, February 1, 2010 - 13:00 ,
Location: Skiles 255 ,
Manu O. Platt ,
Biomedical Engineering (BME), Georgia Tech ,
Organizer:

Tissue remodeling
involves the activation of proteases, enzymes capable of degrading
the structural proteins of tissue and organs. The implications of
the activation of these enzymes span all organ systems and therefore,
many different disease pathologies, including cancer metastasis.
This occurs when local proteolysis of the structural extracellular
matrix allows for malignant cells to break free from the primary
tumor and spread to other tissues. Mathematical models add value to
this experimental system by explaining phenomena difficult to test at
the wet lab bench and to make sense of complex interactions among the
proteases or the intracellular signaling changes leading to their
expression. The papain family of cysteine proteases, the cathepsins,
is an understudied class of powerful collagenases and elastases
implicated in extracellular matrix degradation that are secreted by
macrophages and cancer cells and shown to be active in the slightly
acidic tumor microenvironment. Due to the tight regulatory
mechanisms of cathepsin activity and their instability outside of
those defined spaces, detection of the active enzyme is difficult to
precisely quantify, and therefore challenging to target
therapeutically. Using valid assumptions that consider these complex
interactions we are developing and validating a system of ordinary
differential equations to calculate the concentrations of mature,
active cathepsins in biological spaces. The system of reactions
considers four enzymes (cathepsins B, K, L, and S, the most studied
cathepsins with reaction rates available), three substrates (collagen
IV, collagen I, and elastin) and one inhibitor (cystatin C) and
comprise more than 30 differential equations with over 50 specified
rate constants. Along with the mathematical model development, we
have been developing new ways to quantify proteolytic activity to
provide further inputs. This predictive model will be a useful tool
in identifying the time scale and culprits of proteolytic breakdown
leading to cancer metastasis and angiogenesis in malignant tumors.

Monday, January 11, 2010 - 13:00 ,
Location: Skiles 255 ,
Peter Blomgren ,
San Diego State University ,
Organizer: Sung Ha Kang

We describe two computational frameworks for the assessment of contractileresponses of enzymatically dissociated adult and neonatal cardiac myocytes.The proposed methodologies are variants of mathematically sound andcomputationally robust algorithms very well established in the imageprocessing community. The physiologic applications of the methodologies areevaluated by assessing the contraction in enzymatically dissociated adultand neonatal rat cardiocytes. Our results demonstrate the effectiveness ofthe proposed approaches in characterizing the true 'shortening' in thecontraction process of the cardiocytes. The proposed method not onlyprovides a more comprehensive assessment of the myocyte contraction process,but can potentially eliminate historical concerns and sources of errorscaused by myocyte rotation or translation during contraction. Furthermore,the versatility of the image processing techniques makes the methodssuitable for determining myocyte shortening in cells that usually bend ormove during contraction. The proposed method can be utilized to evaluatechanges in contractile behavior resulting from drug intervention, diseasemodeling, transgeneity, or other common applications to mammaliancardiocytes.This is research is in collaboration with Carlos Bazan, David Torres, andPaul Paolini.