UNC Charlotte
DEPARTMENT OF MATHEMATICS AND STATISTICS

Applied and Computational Mathematics Seminar, Wed, 4:00-5:00, Math Conference Room

Spring 2008 Speakers


 


2/29 SPECIAL TIME Anne Gelb, Math, Arizona State, High Order Image Reconstruction from Fourier Data.

  Fourier spectral data is often the source of image information. For example, magnetic resonance imaging (MRI) and synthetic aperture radar (SAR) sensors measure the values of the Fourier coefficients of an image. The FFT provides an efficient way to reconstruct the image. Since the underlying image often contains edges, the reconstruction is polluted by the Gibbs ringing artifact. Although this is somewhat alleviated by filtering, the ``blurring'' at the internal boundaries of the image can lead to difficulties in segmentation and misdiagnosis. This talk discusses how to determine the internal boundaries and how reconstruct the image without blurring or the Gibbs ringing effects. As a result, image segmentation is much more accurate. Some examples from MRI are included.


3/5 Weizhu Bao, Quantized vortex stability and dynamics in superfluidity and superconductivity

  In this talk, I will review our recent work on quantized vortex stability and dynamics in Ginzburg-Landau-Schroedinger and nonlinear wave equation for modeling superfluidity and superconductivity as well as nonlinear optics. The reduced dynamic laws for quantized vortex interaction are reviewed and solved analytically in several cases. Direct numerical simulation results for Ginzburg-Landau-Schroedinger and nonlinear wave equations are reported for quantized vortex dynamics and they are compared with those from the reduced dynamics laws.


3/14 11:00 FRET 207, joint with MATH PHYSICS SEMINAR Ibrahim Fatkullin, Math, U Arizona, Nematic liquid crystals: Onsager model and beyond.

  In 1949 Lars Onsager introduced a variational model describing isotropic-nematic phase transition in liquid crystals. In this model equilibrium states of a liquid-crystalline system correspond to minimizers of a free energy functional. I will review the model and present a complete classification of all critical points of the Onsager functional with Maier-Saupe interaction. Then I will discuss extensions of Onsager's theory that take into account spatial variations of nematic ordering.


3/19 Di Liu, Math, Michigan State, Numerical methods for stochastic bio-chemical reacting networks with multiple time and concentration scales.

  Multiscale and stochastic approaches play a crucial role in faithfully capturing the dynamical features and making insightful predictions of cellular reacting systems involving gene expression. A Genetic Regulatory Networks (GRN), describing all the reacting channels and species involved in gene expression, consists of a set of genes, proteins, small molecules and their mutual regulatory interactions. From the point of view of modeling, Genetic Regulatory Networks, unlike metabolism networks, involve fewer species and lower concentrations of molecules in a small volume within a cell; therefore stochastic effects have a significant impact on the system. Despite their accuracy, the standard stochastic simulation algorithms are necessarily inefficient for most of the realistic problems with a multiscale nature characterized by 1.) Rare events arising from the metastability of the system, 2.) Multiple time scales induced by widely disparate reactions rates, and 3.) Multiple well separated concentration scales of the reacting species. In this talk, I will discuss some recent progress on using asymptotic techniques for probability theory, e.g. Random Homogenization and Large Deviation Theory, to simplify the complex networks and help to design efficient numerical schemes.


3/26 Shanker Balasubramaniam, EE, Michigan State, O(N) methods for rapidly computing pairwise potentials in large systems

  Abstract


4/9 George Yin, Math, Wayne State, Two-time-scale Markovian Systems and Applications

  Originated from applications in signal processing, random evolution, telecommunications, financial engineering, and manufacturing systems, two-time-scale Markovian systems have received much attention recently. This talk summaries some of our recent work. It includes asymptotic expansions of solutions to the forward equations, scaled and unscaled occupation measures, approximation error bounds, and associated switching diffusion processes. Controlled dynamic systems, Markov decision processes, and two-time-scale diffusions will also be mentioned.


4/16 Weiqing Ren, Courant, Coupled atomistic-continuum methods for fluids

  I will present a multiscale method for the study of fluid systems with unknown constitutive relations and/or unknown boundary conditions. The method captures the macroscale behavior of the fluid system by combining conservation laws (macro model) and molecular dynamics (micro model). The macro and micro models are coupled in a seamless way that does not require going back and forth between the macro and micro states of the system. I will discuss the details of the coupling method and its application to complex fluids, as well as the major difficulties in implementation.


4/23 Yoon Mo Jung, Math, Duke, Multiphase Image Segmentation via Modica-Mortola Phase transition

  We propose a novel multiphase segmentation model built upon the phase transition model of Modica and Mortola in material sciences and a properly synchronized fitting term. The proposed sine-sinc model outputs a single multiphase distribution from which each individual segment or phase can be easily extracted. It includes the gamma-convergence behavior of the proposed model and the existence of its minimizers. Since the model is not quadratic nor convex, for computation we adopted the convex-concave procedure (CCCP). Numerical details and experiments on both synthetic and natural images are presented.


4/30 Peter Mucha, Math, UNC Chapel Hill, Stochastic Dynamics in Near-Wall Velocimetry

  The tracking of small, colloidal particles is a common technique for measuring fluid velocities, highly successful at the micro-scale and recently extended to measurements at nano-scales. While the Brownian fluctuations of the colloidal tracers are typically isotropic in the bulk, such fluctuations in the near-wall region are strongly affected by the hydrodynamic interaction with the wall and by the no-flux condition imposed there. Such wall effects can, under appropriate conditions, bias particle image velocimetry (PIV) measurements based on colloidal tracers, potentially leading to significant overestimation of near-wall velocities and potentially skewing the measurement of slip. The quantification of the resulting bias is presented in terms of the size of the imaged region and the time interval between PIV images. The effect of the steady state particle distribution is additionally explored, and implications for near-wall velocimetry measurements are briefly discussed. This talk represents collaborative work with Christel Hohenegger, Minami Yoda, Reza Sadr, and Haifeng Li.


5/14 Stanislav Molchanov, Math, UNC Charlotte, Determinantal theory of random distribution of particles

 

Fall 2007 Speakers


 


09/24 Scott Kelly, Mech Engineering, UNCC , Geometric Mechanics and Biomorphic Aquatic Locomotion.

  The geometric formalism of analytical mechanics provides a natural setting in which to realize models for aquatic locomotion as nonlinear control systems. Methods of Lagrangian and Hamiltonian reduction provide, in particular, for the identification of low-order models for systems which exhibit Lie groups of symmetries. This approach is most direct for problems in the driftless extremes of irrotational inviscid flow and Stokes flow, but can be adapted to problems in macroscopic swimming hinging on the shedding of coherent vortex structures from deformable solid surfaces. This talk will present complementary theoretical, computational, and experimental research pertaining to both solitary and cooperative biomorphic swimming, linking research in Lagrangian and Hamiltonian mechanics to the development of robotic platforms with novel flow sensors for feedback control.


10/01 Kai Fan, Math, UNCC, A Generalized Discontinuous Galerkin (GDG) Method for PDE with Nonsmooth Solutions .

  To model optical wave propagations in inhomogenous waveguides under the paraxial approximation, we need to solve time dependent Schrödinger equations with nonsmooth solutions as a result of field discontinuities at material interfaces. We will present a new type of discontinuous Galerkin method based on split distributions and their incorporations into the PDEs to account for jumps in solutions and derivatives. Special integration by parts formula for the split distributions is developed. The resulting generalized discontinuous Galerkin (GDG) method will be flexible to handle various types of interface jump conditions (time dependent and nonlinear) with high accuracy and easy to extend to multi-dimensional and other type PDEs with nonsmooth solutions. A full vector GDG-BPM (beam propagation method) will be developed to study gain guided fiber laser for efficient generations of high energy power sources.


10/15 Greg Gbur, Physics, UNCC , Simulating Partially Coherent and Vortex Beams in Atmospheric Turbulence

  In recent years, there has been increased interest in the use of special beam classes to improve free space optical communications in atmospheric turbulence. Partially coherent beams, with 'built-in' random fluctuations at the source, potentially result in lower intensity fluctuations at the detector. Vortex beams, with inherent angular momentum and the related 'rips' in their wavefronts, potentially provide an alternative method for carrying information. In this talk we discuss techniques for simulating partially coherent beams and vortex beams in atmospheric turbulence, as well as techniques for assessing their performance.


11/12 Laurent Demanet, Department of Mathematics, Stanford , Time upscaling of the wave equation beyond the CFL restriction .

  The complexity of solving the time-dependent wave equation via traditional methods scales faster than linearly in the complexity of the initial data. This behavior is mostly due to the necessity of timestepping at the CFL level, and is starting to hamper the resolution of large-scale inverse scattering problems such as reflection seismology, where massive datasets need to be processed. In this talk I will report on two different ideas that can be used to solve the wave equation beyond CFL timestepping: 1) a solver based on a sparse, block-separated decomposition of the Green's function into multiscale wave atoms; and 2) discrete symbol calculus for pseudodifferential and Fourier integral operators. Joint work with Lexing Ying from UT Austin.


11/16 David Ambrose, Department of Mathematics, Clemson , Free surface problems in fluid dynamics.

  Fluid flows in the presence of free surfaces occur in a great many situations in nature; examples include waves on the ocean and the flow of groundwater. In this talk, I will discuss my contributions to the analysis of the systems of nonlinear partial differential equations which model such phenomena. In particular, I will discuss short-time well-posedness theorems for three particular free surface problems: vortex sheets with surface tension, water waves, and interfacial flows in porous media. Each of these problems is considered in both two and three space dimensions; some of these results are from joint work with Nader Masmoudi. The main ingredients in the proofs are a reformulation of the evolution equations using a convenient parameterization of the free surface, and approximations of singular integrals using Hilbert or Riesz transforms. With the well-posedness of the problems established for short times, many questions remain about the long-time behavior. To address one such question, I will present numerical evidence of formation of isolated curvature singularites for the two-dimensional vortex sheet with surface tension.


11/26 John Strain, Department of Mathematics, UC Berkeley, SLC methods for moving interfaces

  Many moving interface problems evolve complex material interfaces through topological changes, under velocities determined by elliptic systems of partial differential equations. Robust efficient methods for such problems are built with semi-Lagrangian contouring and fast elliptic boundary integral solvers. The interface motion is converted to a contouring problem with an explicit second-order semi-Lagrangian advection formula, and grid-free adaptive refinement resolves complex interface geometry. Elliptic systems are solved with a fast new locally-corrected boundary integral formulation derived by Ewald summation and accelerated by new geometric nonuniform fast Fourier transforms. High-resolution computations with the resulting methods reproduce complex features of geometric, Stokes and viscoelastic flows.


11/29 Lexing Ying, Department of Mathematics, UT Austin, Fast Directional Multilevel Algorithms for Oscillatory Kernels

  In this talk, we introduce a new directional multilevel algorithm for solving N-body or N-point problems with highly oscillatory kernels. These systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non-uniformity of the particle distribution. We address the problem by first proving that the interaction between a ball of radius r and a well-separated region has an approximate low rank representation, as long as the well-separated region belongs to a cone with a spanning angle of O(1/r) and is at a distance which is at least O(r2) away from from the ball. We then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated, low rank representation. Based on the resulting representations, our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency. The algorithm performs well on a large group of highly oscillatory kernels. Our algorithm is proved to have O(N\log N) computational complexity for any given accuracy when the points are sampled from a two dimensional surface.

 
 
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