Monday, April 19, 2010 - 13:00 , Location: Skiles 255 , Jae-Hun Jung , Mathematics, SUNY Buffalo , Organizer: Sung Ha Kang
Solutions of differential equations with singular source terms easily becomenon-smooth or even discontinuous. High order approximations of suchsolutions yield the Gibbs phenomenon. This results in the deterioration ofhigh order accuracy. If the problem is nonlinear and time-dependent it mayalso destroy the stability. In this presentation, we focus on thedevelopment of high order methods to obtain high order accuracy rather thanregularization methods. Regularization yields a good stability condition,but may lose the desired accuracy. We explain how high order collocationmethods can be used to enhance accuracy, for which we will adopt severalmethods including the Green’s function approach and the polynomial chaosmethod. We also present numerical issues associated with the collocationmethods. Numerical results will be presented for some differential equationsincluding the nonlinear sine-Gordon equation and the Zerilli equation.
Monday, April 12, 2010 - 13:00 , Location: Skiles 255 , Samuel Isaacson , Boston University Mathematics Dept. , Organizer:
We will give an overview of our recent work investigating the influence of incorporating cellular substructure into stochastic reaction-diffusion models of gene regulation and expression. Extensions to the reaction-diffusion master equation that incorporate effects due to the chromatin fiber matrix are introduced. These new mathematical models are then used to study the role of nuclear substructure on the motion of individual proteins and mRNAs within nuclei. We show for certain distributions of binding sites that volume exclusion due to chromatin may reduce the time needed for a regulatory protein to locate a binding site.
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.
[Special day and location] Electrostatic effects on DNA dynamics in fluid by the generalized immersed boundary methodFriday, 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.