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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.

Monday, January 14, 2013 - 14:00 ,
Location: Skiles 005 ,
Xue-Cheng Tai ,
University of Bergen, Department of Mathematics, Norway ,
Organizer: Sung Ha Kang

In this talk, we present a new global optimization based approach to contour evolution, with or without the novel variational shape constraint that imposes a generic star shape using a continuous max-flow framework. In theory, the proposed continuous max-flow model provides a dual perspective to the reduced continuous min-cut formulation of the contour evolution at each discrete time frame, which proves the global optimality of the discrete time contour propagation. The variational analysis of the flow conservation condition of the continuous max-flow model shows that the proposed approach does provide a fully time implicit solver to the contour convection PDE, which allows a large time-step size to significantly speed up the contour evolution. For the contour evolution with a star shape prior, a novel variational representation of the star shape is integrated to the continuous max-flow based scheme by introducing an additional spatial flow. In numerics, the proposed continuous max-flow formulations lead to efficient duality-based algorithms using modern convex optimization theories. Our approach is implemented in a GPU, which significantly improves computing efficiency. We show the high performance of our approach in terms of speed and reliability to both poor initialization and large evolution step-size, using numerous experiments on synthetic, real-world and 2D/3D medical images.This talk is based in a joint work by: J. Yuan, E. Ukwatta, X.C. Tai, A. Fenster, and C. Schnorr.

Monday, November 26, 2012 - 14:00 ,
Location: Skiles 005 ,
Prashant Athavale ,
Fields Institute, Dep. of Math, University of Toronto, ,
prashant@math.utoronto.ca ,
Organizer: Sung Ha Kang

Images consist of features of varying scales. Thus, multiscale image processing techniques are extremely valuable, especially for medical images. We will discuss multiscale image processing techniques based onvariational methods, specifically (BV, L^2) and (BV, L^1) decompositions. We will discuss the applications to real time denoising, deblurring and image registration.

Monday, November 19, 2012 - 14:00 ,
Location: Skiles 005 ,
Hao Gao ,
Dep of Math and CS/ Dep of Radiology and Imaging Sciences, Emory University ,
haog@mathcs.emory.edu ,
Organizer: Sung Ha Kang

I will talk about (1) a few sparsity models for 4DCBCT; (2) the split Bregman method as an efficient algorithm for solving L1-type minimization problem; (3) an efficient implementation through fast and highly parallelizable algorithms for computing the x-ray transform and its adjoint.

Monday, November 12, 2012 - 14:00 ,
Location: Skiles 005 ,
Antonio Cicone ,
GT Math ,
cicone@math.gatech.edu ,
Organizer: Sung Ha Kang

Given F, finite set of square matrices of dimension n, it is possible to define the Joint Spectral Radius or simply JSR as a generalization of the well known spectral radius of a matrix. The JSR evaluation proves to be useful for instance in the analysis of the asymptotic behavior of solutions of linear difference equations with variable coefficients, in the construction of compactly supported wavelets of and many others contexts. This quantity proves, however, to be hard to compute in general. Gripenberg in 1996 proposed an algorithm for the computation of lower and upper bounds to the JSR based on a four member inequality and a branch and bound technique. In this talk we describe a generalization of Gripenberg's method based on ellipsoidal norms that achieve a tighter upper bound, speeding up the approximation of the JSR. We show the performance of this new algorithm compared with Gripenberg's one. This talk is based on joint work with V.Y.Protasov.