Applied and Computational Mathematics (ACM) Seminar

Department of Mathematics
Faculty of Science
National University of Singapore

Schedule for   Semester 1,   2006/2007
Regular meeting time:    Thursday   3:00 -- 4:00pm
Venue:    Colloquium Room A   (S14-03-10)

Date Speaker Title   /   Abstract
  30 November Han   Bin Interpolating and Approximating Hermite Subdivision Schemes and Wavelets
In this talk, we shall discuss interpolating and approximating Hermite subdivision schemes, which are of interest in CAGD and consist of an important family of refinable functions in wavelet analysis.  First, we shall discuss the analysis and construction of interpolating scalar subdivision schemes by the projection method and the CBC algorithm. Next, we shall introduce interpolating Hermite subdivision schemes and investigate their properties such as convergence, smoothness, construction and optimality.  Finally, we shall talk about approximating Hermite subdivision schemes and their connections to the spline theory.
23 November Mark   Ainsworth
Strathclyde University
Diagonal Scaling of Discrete Differential Forms
The use of discrete differential forms in the construction of finite element discretisations of the Sobolev spaces H^s, H(div) and H(curl) is now routinely applied by numerical analysts and engineers alike. However, little attention has been paid to the conditioning of the resulting stiffness matrices, particularly in the case of the non-uniform meshes that arise when adaptive refinement algorithms are used. We study this issue and show that the matrices are generally rather poorly conditioned. Typically, diagonal scaling is applied (often unwittingly) as a preconditioner. However, whereas diagonal scaling removes the effect of the mesh non-uniformity in the case of Sobolev spaces H^s, we show this is not so in the case of the spaces H(curl) and H(div). We trace the reason behind this difference, and give a simple remedy for curing the problem.
16 November
Jean-Claude   Saut
Universite de Paris-Sud
Recent results on the KP equations
The Kadomtsev-Petviashvili equations (KP) are "universal" models for the description of long, wealkly nonlinear waves propagating mainly in one direction with weak transverse effects. There are two versions, the (focusing) KP I equation and the (defocusing) KP II equation. The talk will review recent progress on those equations with focus on the Cauchy problem for the KP I equation. Modelling aspects and the KP I limit of the Gross-Pitaevskii equation will be also considered if time allows.
  9 November Zhang   Yanzhi Quantized Vortex Dynamics in Rotating Bose-Einstein Condensate
One manifestation of superfluidity in rotating Bose-Einstein condensate (BEC) is the appearance of quantized vortices. In this talk, I will show numerically and analytically the rich phenomena of quantized vortices in rotating BEC. From our numerical results, we found that central vortex states with winding number |m|=1 are dynamically stable, and this fact makes our following investigations on single vortex (|m|=1) dynamics meaningful. Firstly, we study the dynamics of the single vortex by shifting its center from the origin, and find that the motion of the vortex center is dominated by the angular speed of the rotating BEC and further its trajectory can be given by an ODE system. Secondly, we investigate the interaction of two single vortices with same or opposite winding numbers, which is a quite complicated problem and so far there has almost no report about it in the literature. Based on our interesting numerical results, we make some conjectures about it. Finally, I end my talk by showing some interesting dynamics of vortex lattices.
  2 November Sun   Defeng Understanding Augmented Lagrangian Methods
Penalty methods are a popular choice among most practitioners in selecting numerical methods for solving constrained optimization problems. The dominant reason for this is the simplicity of these methods, but not the efficiency. On the contrary, penalty methods suffer from several deficiencies both numerically and theoretically. Nevertheless, penalty methods are the precursors of much more efficient, and yet not too complicated, augmented Lagrangian methods. Augmented Lagrangian functions were proposed by economists Arrow and Solow in 1958 and the research on augmented Lagrangian methods was initiated by optimization people like Hestenes, Powell, and Rockafellar from 1969. Since augmented Lagrangian methods can be regarded as gradient ascent direction methods and gradient ascent direction methods are known to be very slow in convergence, it is quite natural to think that augmented Lagrangian methods may converge slowly too. This is, however, not true at all. In this talk, we aim to explain why augmented Lagrangian methods for constrained optimization problems possess fast convergence and enjoy numerical stability. Some recently developed tools like semismooth analysis and variational analysis on metric projectors over closed convex sets play key roles in our explanations and will be introduced briefly.
  26 October Lin   Ping A quasi-continuum approximation for the large-scale energy minimization in material sciences
In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacting with all others through a pair potential energy. The equillibrium configuration of the material is the minimizer of the total energy of the system. The computational cost is high since the number of atoms is huge. Recently much attention has been paid to a so-called quasicontinuum (QC) approximation. It is a mixed atomistic/continuum model and a representative of various multi-scale models for material simulation. The QC method solves a fully atomic scale problem in regions where the material contains defects, but uses continuum finite elements to integrate out the majority of the atomistic degrees of freedom in no-defect regions. However, numerical analysis is still at its infancy. In this talk we will introduce this topic and conduct a convengence analysis of the QC method in the case that there is no serious defect or that the defect region is small. The difference of our analysis from conventional finite element analysis is that our exact solution is not a solution of a well-established continuous partial differential equation but a discrete atomic scale solution which is not simply related to any conventional partial differential equation. We will consider both one dimensional and two dimensional cases. Some thoughts about the dynamics may be mentioned as well. The QC method may be related to many other fields such as model reduction, pre-conditioning and multi-pole method.
  19 October Ji   Hui A computational study on human perception of 3D geometry
Optical illusions are fascinating to most people and some of them have been known since antiquity. Since the nineteenth century, the scientific study of visual illusions has been an important strategy in the study of human perception. If you search on the net for the explanations for these illusions, you probably form the impression that every illusion has its own cause. In this talk, I will first review a framework proposed by Prof. C. Fermuller, which study the optical illusions from its computational nature. Many well-known optical illusions related to the human perception on low-level visual features are successfully predicted by this general framework. Then, I will present our research on optical illusions related to human's high-level visual process on 3D spatial geometry, and some important implications to computer vision scientists from this study will also be discussed.
  12 October Lee   Seng-Luan Geometric Modeling of Free-Form Curves and Surfaces
The present technology for computer aided design (CAD) and computer graphics and animation (CGA) is based largely on NURBS, which is the acronym for NonUniform Rational B-Splines. NURBS are extensions of Bezier curves and surfaces for digital representation of free-form geometric objects while allowing users to manipulate their shapes interactively. A NURBS surface is rendered by subdividing its control polyhedron, which is a rectangular mesh in space in which every vertex has 4 edges. However, in most practical situations the meshes may contain singular vertices, which, in the case of a rectangular mesh, are vertices with number of edges other than 4. Subdivision surface methods are extensions of NURBS to model surfaces from meshes with singular points, which NURBS cannot handle. Subdivision surface methods are well-known in CAD and CGA because of its simplicity and efficiency. Another method for modeling surfaces with singular points, which is not much known and not developed, was proposed by Goodman. Goodman's idea is to define blending functions with domain on a polyhedral mesh in space instead of a two dimensional region, and to use these blending functions as a basis for the representation of surfaces. In this talk we will give a quick overview of Bezier curves and surfaces, NURBS, subdivision surfaces, Goodman method and some joint work with Hwee Huat and Majid.
5 October Qi   Li-Qun
City University of
Hong Kong
Eigenvalues of Tensors and Their Applications
Recently, I defined eigenvalues and eigenvectors of a real supersymmetric tensor, and explored their practical applications in determining positive definiteness of an even degree multivariate form, and finding the best rank-one approximation to a supersymmetric tensor. This work extended the classical concept of eigenvalues of square matrices, and has potential applications in mechanics and physics as well as the classification of hypersurfaces and the study of hypergraphs.
Independently, Lek-Heng Lim, a Ph.D. student of Gene Golub at Stanford University,  who was originally from Singapore, also defined eigenvalues for tensors. Notably, he proposed a multilinear generalization of the Perron-Frobenius theorem based upon the notion of eigenvalues  of tensors.
  28 September
Song   Lianfa Modeling and Numerical Studies of the Nonlinear Behaviors of a Long Narrow Reverse Osmosis Filtration Channel
Reverse osmosis membrane process is playing the key role in the reclamation of high quality water from non-conventional sources, such as wastewater, brackish water and seawater. The common configuration of reverse osmosis filtration process is characterized by a long narrow channel (typically several meters in length but a fraction of millimeter in height). The feed water is fed in one end and the concentrate exits from the other end of the membrane channel. Because of the substantial variations in the operating parameters and water properties along the channel, the membrane filtration system presents strong nonlinear behaviors that cannot be adequately explained within the framework of the classic membrane filtration theories. Mathematical model for the heterogeneous membrane system is first developed. The model is then solved numerically and analytically for a better understanding of the long narrow reverse osmosis filtration channel. Interesting results of paramount importance to the reverse osmosis processes have been obtained and will be reported in this talk, such as the role of concentration polarization in the spiral wound membrane modules, controlling mechanisms of the performance of the long narrow reverse osmosis channel, and characteristics of membrane fouling.
  21 September Karthik B. Natarajan   Persistency Model and Its Applications
Given a discrete optimization problem, with random objective coefficients, we are interested in evaluating the expected optimal objective value and the probability that the decision variable takes a particular value. This is called the persistency problem for a discrete optimization problem under uncertain objective. In this talk, we show that a subclass of this problem can be solved in polynomial time. Our results extends the results of Meilijson and Nadas (JAP 1979) to general integer programs (with more than two values) using marginal distributions or marginal moment information. We discuss applications of the model in discrete choice models and integer knapsack.
  14 September Yip   M.-H., Andy Super-resolution Image Reconstruction Using Fast Inpainting Algorithms
In this talk,  I will present a total variation based model for multi-frame super-resolution imaging problems.  The model allows an arbitrary pattern of missing pixels and missing frames. Moreover, the use of total variation is of two-fold: it induces regularization in regions where the pixels are observed and induces inpainting in regions where the pixels are missing. We also investigate a fast algorithm which uses an inverse preconditioner in solving the linear system arisen from each lagged diffusivity fixed point iteration.
7 September Tony   Ware
University of Calgary
Swing options with continuous exercise
Swing options with continuous exercise can be used to model gas-storage contacts; options with discrete exercise opportunities (common in power delivery contracts) can be viewed as limiting cases. The value of such an option may be found by solving a semilinear PDE, and I will discuss the numerical solution of such equations.
  31 August Bao   Weizhu The dynamics and interaction of quantized vortices in Ginzburg-Landau-Schrodinger equation
In this talk, we investigate the dynamical laws of quantized vortex interaction in the Ginzburg-Lanau-Schrodinger equation (GLSE) analytically and numerically. We begin with a review of the reduced dynamic laws governing the motion of vortex centers in GLSE and solve the nonlinear ordinary differential equations (ODEs) of the reduced dynaqmical laws analytically with a few types of initial data. By directly simulating the GLSE with an efficient and accurate numerical method proposed recently by us, we can compare quantized vortex interaction patterns of GLSE with those from the reduced dynamic laws qualitatively and quantitatively. Some conclusive experimental findings are obtained, and discussions on numerical and theoretical results are made to provide further understanding of vortex interactions in GLSE. Finally, the vortex motion under an inhomogeneous potential is also studied.

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