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Monday, October 21, 2013 - 14:00 ,
Location: Skiles 005 ,
Jeff Geronimo ,
GT Math ,
Organizer: Sung Ha Kang

The Alpert multiwavelets are an extension of the Haar wavelet to higher degree piecewise polynomials thereby giving higher approximation order. This system has uses in numerical analysis in problems where shocks develop. An orthogonal basis of scaling functions for this system are the Legendre polynomials and we will examine the consequence of this. In particular we will show that the coefficients in the refinement equation can be written in terms of Jacobi polynomials with varying parameters. Difference equationssatisfied by these coefficients will be exhibited that give rise to generalized eigenvalue problems. Furthermore an orthogonal basis of wavelet functions will be discussed that have explicit formulas as hypergeometric polynomials.

Monday, September 9, 2013 - 14:00 ,
Location: Skiles 005 ,
Seong Jun Kim ,
GT Math ,
Organizer: Sung Ha Kang

The main aim of this talk is to design efficient and novel numerical algorithms for highly oscillatory dynamical systems with multiple time scales. Classical numerical methods for such problems need temporal resolution to resolve the finest scale and become very inefficient when the longer time intervals are of interest. In order to accelerate computations and improve the long time accuracy of numerical schemes, we take advantage of various multiscale structures established from a separation of time scales. The framework of the heterogeneous multiscale method (HMM) will be considered as a general strategy both for the design and for the analysis of multiscale methods.(Keywords: Multiscale oscillatory dynamical systems, numerical averaging methods.)

Monday, April 22, 2013 - 14:00 ,
Location: Skiles 005 ,
Prof. Seyed Moghadas ,
York University ,
Organizer: Haomin Zhou

Modelling and computational approaches provide powerful tools in the study of disease dynamics at both the micro- and macro-levels. Recent advances in information and communications technologies have opened up novel vistas and presented new challenges in mathematical epidemiology. These challenges are central to the understanding of the collective dynamics of heterogeneous ensembles of individuals, and analyzing pertinent data that are less coarse and more complex. The evolution of dynamic modelling is typified by the agent-based modelling (ABM) as a shifting paradigm, a lattice-distributed collection of autonomous decision-making entities (i.e., agents), the interactions of which unveil the dynamics and emergent properties of a real-life problem, such as an infectious disease outbreak. In this talk, we show a general framework for developing an ABM that can be used to computationally optimize intervention strategies for novel influenza viruses with pandemic potential. Our findings contrast previous results !

Monday, April 15, 2013 - 14:05 ,
Location: Skiles 005 ,
Alexander Kurganov ,
Tulane University ,
Organizer: Yingjie Liu

I will first give a brief review on simple and robust central-upwind schemes
for hyperbolic conservation laws. I will then discuss their application to
the Saint-Venant system of shallow water equations. This can be done in a
straightforward manner, but then the resulting scheme may suffer from the
lack of balance between the fluxes and (possibly singular) geometric source
term, which may lead to a so-called numerical storm, and from appearance of
negative values of the water height, which may destroy the entire computed
solution. To circumvent these difficulties, we have developed a special
technique, which guarantees that the designed second-order central-upwind
scheme is both well-balanced and positivity preserving.
Finally, I will show how the scheme can be extended to the two-layer shallow
water equations and to the Savage-Hutter type model of submarine landslides
and generated tsunami waves, which, in addition to the geometric source
term, contain nonconservative interlayer exchange terms. It is well-known
that such terms, which arise in many different multiphase models, are
extremely sensitive to a particular choice their numerical discretization.
To circumvent this difficulty, we rewrite the studied systems in a different
way so that the nonconservative terms are multiplied by a quantity, which
is, in all practically meaningful cases, very small. We then apply the
central-upwind scheme to the rewritten system and demonstrate robustness and
superb performance of the proposed method on a number numerical examples.

Monday, March 25, 2013 - 14:00 ,
Location: Skiles 005 ,
Mario Arioli ,
Rutherford Appleton Laboratory, United Kingdom ,
mario.arioli@stfc.ac.uk ,
Organizer: Sung Ha Kang

We derive discrete norm representations associated with projections of interpolation spaces onto finite dimensional subspaces. These norms are products of integer and non integer powers of the Gramian matrices associated with the generating pair of spaces for the interpolation space. We include a brief description of some of the algorithms which allow the efficient computation of matrix powers. We consider in some detail the case of fractional Sobolev spaces both for positive and negative indices together with applications arising in preconditioning techniques. Several other applications are described.

Monday, March 11, 2013 - 14:00 ,
Location: Skiles 005 ,
Francesco G. Fedele ,
Georgia Tech, Civil & Environmental Engineering ,
fedele@gatech.edu ,
Organizer: Sung Ha Kang

I will present some results on the space-time stereo reconstruction of nonlinear sea waves off the Venice coast using a Variational Wave Acquisition Stereo System (VWASS). Energy wave spectra, wave dispersion and nonlinearities are then discussed. The delicate balance of dispersion and nonlinearities may yield the formation of solitons or traveling waves. These are introduced in the context of the Euler equations and the associatedthird order compact Zakharov equation. Traveling waves exist also in the axisymmetric Navier-Stokes equations. Indeed, it will be shown that the NS equations can be reduced to generalized Camassa-Holm equations that support smooth solitons and peakons.

Monday, March 4, 2013 - 14:00 ,
Location: Skiles 005 ,
Xiaojing Ye ,
Georgia Tech, School of Math ,
xye33@math.gatech.edu ,
Organizer: Sung Ha Kang

We consider the modeling and computations of random dynamical processes of viral signals propagating over time in social networks. The viral signals of interests can be popular tweets on trendy topics in social media, or computer malware on the Internet, or infectious diseases spreading between human or animal hosts. The viral signal propagations can be modeled as diffusion processes with various dynamical properties on graphs or networks, which are essentially different from the classical diffusions carried out in continuous spaces. We address a critical computational problem in predicting influences of such signal propagations, and develop a discrete Fokker-Planck equation method to solve this problem in an efficient and effective manner. We show that the solution can be integrated to search for the optimal source node set that maximizes the influences in any prescribed time period. This is a joint work with Profs. Shui-Nee Chow (GT-MATH), Hongyuan Zha (GT-CSE), and Haomin Zhou (GT-MATH).

Monday, February 4, 2013 - 14:05 ,
Location: Skiles 005 ,
Robert Lipton ,
LSU ,
Organizer: Guillermo Goldsztein

Metamaterials are a new form of structured materials used to control
electromagnetic waves through localized resonances. In this talk we
introduce a rigorous mathematical framework for controlling localized
resonances and predicting exotic behavior inside optical metamaterials.
The theory is multiscale in nature and provides a rational basis for
designing microstructure using multiphase nonmagnetic materials to create
backward wave behavior across prescribed frequency ranges.

Friday, January 25, 2013 - 14:00 ,
Location: Skiles 005 ,
Sangwoon Yun ,
Sung Kyun Kwan Univ. (Korea) ,
yswmathedu@skku.edu ,
Organizer: Sung Ha Kang

In this talk, we introduce coordinate gradient descent methods for nonsmooth separable minimization whose objective function is the sum of a smooth function and a convex separable function and for linearly constrained smooth minimization. We also introduce incremental gradient methods for nonsmooth minimization whose objective function is the sum of smooth functions and a convex function.

Friday, January 18, 2013 - 14:00 ,
Location: Skiles 005 ,
Michael Klibanov ,
University of North Carolina, Charlotte ,
Organizer: Haomin Zhou

Coefficient Inverse Problems (CIPs) are the hardest ones to work with in the field of Inverse Problems. Indeed, they are both nonlinear and ill-posed. Conventional numerical methods for CIPs are based on the least squares minimization. Therefore, these methods suffer from the phenomenon of multiple local minima and ravines. This means in turn that those methods are locally convergent ones. In other words, their convergence is guaranteed only of their starting points of iterations are located in small neighborhoods of true solutions. In the past five years we have developed a new numerical method for CIPs for an important hyperbolic Partial Differential Equation, see, e.g. [1,2] and references cited there. This is a globally convergent method. In other words, there is a rigorous guarantee that this method delivers a good approximation for the exact solution without any advanced knowledge of a small neighborhood of this solution. In simple words, a good first guess is not necessary. This method is verified on many examples of computationally simulated data. In addition, it is verified on experimental data. In this talk we will outline this method and present many numerical examples with the focus on experimental data.REFERENCES [1] L. Beilina and M.V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. [2] A.V. Kuzhuget, L. Beilina and M.V. Klibanov, A. Sullivan, L. Nguyen and M.A. Fiddy, Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm, Inverse Problems, 28, 095007, 2012.