Applied and Computational Mathematics (ACM) Seminar

Department of Mathematics
Faculty of Science
National University of Singapore

Schedule for   Semester II,   2005/2006
Regular meeting time:    Tuesday   4pm -- 5pm
Venue:    Colloquium Room A   (S14-03-10)

 Date Speaker Title   /   Abstract 18 April Reading week No meeting 11 April Michael   NG Hong Kong Baptist University Iterative Methods for Saddle Point Systems In this talk, we review iterative methods for saddle point systems.  We present new preconditioners based on Hermitian and skew-Hermitian splitting for saddle point systems.  Both theoretical and numerical results are reported. 4 April ZHAO   GongYun Representing the space of linear programs as a Grassmannian We represent the space of linear programs as the space of projection matrices.  Projection matrices of the same dimension and rank comprise a Grassmannian, which has rich geometric and algebraic structures.  An ordinary differential equation on the space of projection matrices defines a path for each projection matrix associated with a linear programming instance and the path leads to a projection matrix associated with an optimal basis of the instance.  In this way, any point (projection matrix) in the Grassmannian is connected to a stationary point of the differential equation.  We will present some basic properties of the stationary points, in particular, the characteristics of eigenvalues and eigenvectors.  We will show that there are only a finite number of stable points.  Thus, the Grassmannian can be partitioned into a finite number of attraction regions, each associated with a stable point.  The structures of the attraction regions will be important for applications which will be discussed at the end of this paper. 28 March LIU   Tie-Gang Institute of High Performance Computing This talk is at S16 #05-98/99 The Modified Ghost Fluid Method and its Application to Compressible Fluid-Deformable Structure Coupling Numerical oscillations are inevitably encountered when a well-established numerical method for single medium flow is directly applied to the multi-medium flow.  The Ghost Fluid Method (GFM) developed a few years ago has provided a simple and flexible way to treat the multi-medium flow.  However, we have shown that the original GFM tends to give wrong solution when directly applied to the problems with a strong wave interaction at the material interface.   In fact, we have shown that in order for a GFM-based algorithm to work correctly, the influence of wave interaction at the material interface and the effect of material properties on the interfacial status have to be faithfully taken into account in the definition of the ghost fluid status.  As such, we have developed a modified GFM (MGFM).  In this talk, we shall show my recent work on analyzing the accuracy of a GFM-based algorithm, and the latest development of the MGFM applied to compressible fluid coupled to deformable structure, where the pressure in the structure or flow can vary from an extremely high value such that the solid medium can be treated as a fluid, to a moderate level such that the structure is in linear elastic deformation, or to a very lower level so that cavitation can occur in the fluid and negative stress can appear next to the structure interface.  I shall also discuss on the Eulerian-Lagrangian coupling using the MGFM. 21 March DAI   Min Finite-horizon optimal investment with transaction costs: a parabolic double obstacle problem This talk concerns optimal investment problem of a CRRA investor who faces proportional transaction costs and finite time horizon.  Using a partial differential equation approach, we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies.  This enables us to make use of the well developed theory of variational inequality to study the problem.  The \$C^{2,1}\$ regularity of the value function is proven and the optimal investment policies are completely characterized.  Relying on the double obstacle problem, we extend the binomial method widely used in option pricing to determine the optimal investment policies.  Numerical examples are presented as well 14 March LAI   Ying-Cheng Arizona State University Synchronization in complex networks In their seminal work on small-world networks, Watts and Strogatz showed that the addition of a small number of shortcut links to a regular, locally connected network can reduce the average network distance greatly while keeping the network locally clustered.   Many complex networks have also been found to have the scale-free property in that their connectivity (or degree) distributions are algebraic.   Barabasi and Albert suggested a model of growing networks, in which preferential attachment of new links to nodes having high degrees results in the scale-free property.   In the past few years there has been intense research on complex networks.   The structural properties of a complex network can affect the dynamical processes taking place on it, such as synchronization.  The aims of this talk are to introduce the theory of synchronization in complex networks and to present some recent results from the nonlinear dynamics group at Arizona State University.  Specific topics include stability analysis, effect of network topology on synchronizability, synchronization in social networks, and desynchronization wave-pattern formation. 7 March WANG   Li-Lian Nanyang Technological University Efficient and stable Jacobi spectral methods and their applications In recent years, spectral methods have become increasingly popular among computational scientists and engineers because of their superior accuracy and efficiency when properly implemented.   In this talk, we will implement and analyze spectral-Gakerlin algorithms using Jacobi and generalized Jacobi polynomials as basis functions for constructing efficient solvers for partial differential equations (PDEs).  As examples of applications, we will consider the Helmholtz equation (in wave scattering) and KdV-type equations.   We will also present a set of optimal Jacobi approximation results in non-uniformly weighted Sobolev spaces, which play essential roles in the analysis of spectral methods for PDEs. 28 February FU   Libin BIAPCM Self-trapping of two weakly coupled BECs and its periodic modulation With phase space analysis approach, we investigate thoroughly the self-trapping phenomenon for two weakly coupled Bose-Einstein condensates (BEC) in a symmetric double-well potential.   We identify two kinds of self-trapping by their different relative phase behavior.  With applying a periodic modulation on the energy bias of the system we find the occurrence of the self-trapping can be controlled, saying, the transition parameters can be adjusted effectively by the periodic modulation.  Analytic expressions for the dependence of the transition parameters on the modulation parameters are derived for high and low frequency modulations.  For an intermediate frequency modulation, we find the resonance between the periodic modulation and nonlinear Rabi oscillation dramatically affects the tunnelling dynamics and demonstrate many novel phenomena.   Finally, we study the effects of many-body quantum fluctuation on self-trapping and discuss the possible experimental realization of the model. 22 February Mid-term break No meeting 14 February YAN   Jie Statistics of loop formation along double helix DNAs and its applications Recent experiments (T.E. Cloutier, J. Widom, Mol. Cell 14, 355-62 2004) indicate that double-stranded DNA molecules of approximately 100 base pairs in length have a probability of cyclization which is up to 100,000 times larger than that expected based on the known bending modulus of the double helix.  We show that for short molecules, `flexible hinges' made of small (few base pair) regions of single-stranded DNA, which are generated by thermal excitation, can explain the experimental data.   Our calculation is based on a novel statistical-mechanical calculation using a transfer-matrix approach.  It allows very general calculations for different types of local disorders, various looping boundary conditions, and nonlinear polymer elasticity.  It also allows calculations for sequence-dependent properties.  Applications of this type of calculation to these situations, which may play important roles in DNA higher-order structure, will also be discussed. 7 February KONG   Yong Some recent results on monomer-dimer problem The monomer-dimer problem is one of the classical lattice models in statistical physics whose origin goes back at least to 1935.  A special case of the model, namely the dimer model where the lattice is fully covered by dimers, was solved exactly in 1961 by Kasteleyn, Temperley, and Fisher independently for planar lattices.  The general monomer-dimer model, however, remains unsolved and is classified as a "#P-complete" problem.  In this talk I will give some recent results on the monomer-dimer problem.  By using exact computational method, it was found that for an arbitrary number of monomers, there is a logarithmic correction term in the finite-size correction of the free energy.  The coefficient of this logarithmic correction term depends not only on the number of monomers, but also on the parity of the lattice strip.  The results will be discussed in the context of universality for two-dimensional systems at criticality as predicted by conformal field theory and finite-size scaling.  I'll also discuss some remarkable number-theoretical properties found in packing dimers on odd-by-odd lattices (where there is always a vacancy) in parallel to those found in close-packed lattices. 3 February PANG   Zhan The Chinese University of Hong Kong Design and evaluation of long-term supply contracts in the presence of spot market We consider a make-to-stock (MTS) manufacturing system that produces a single product that can be sold not only in a spot market but also through a long-term supply contractual channel, such as the OEM contract.  A typical example of the spot market is a B2B online commodity market. The price in the spot market is random, and evolves as a continuous-time Markov chain.  The demand in the spot market comes in the arrivals of a Markov Modulated Poisson Process (MMPP), and can be rejected or accepted by instant or delayed fulfillment. However, price in the contractual channel is pre-specified and demand in the contractual channel comes at a homogeneous Poisson process with a committed constant arrival rate.  The system is obligate to fulfill contract orders. In this setting, the coordination of production and spot sales arises to be a core of decisions.  The long-run average profit of the system is provably optimized by a simple and intuitively structured threshold policy, which consists of spot-dependent BASE-STOCK and SALE ADMISSION thresholds.  We then show how a contract should be evaluated and designed when bargaining with the contractual partner.  Finally, we unveil that the manufacturer and the partner can reach an economic Nash equilibrium point to strike a deal.  An effective algorithm is proposed for computing optimal threshold control and Nash equilibrium point. 24 January Bruce   KELLOGG University of South Carolina Corner singularities for the steady-state compressible Navier-Stokes system The solution of an elliptic boundary value problem in a polygonal domain has singularities at the corners of the polygon.  There is a well-developed theory that deals with this; the theory includes the Stokes system and can even be extended to the incompressible Navier-Stokes system in a plane polygonal domain.  The question arises: is there a similar theory for the the compressible Navier-Stokes system?  In this lecture we discuss recent work on this problem.  The lecture will not assume a knowledge of corner singularities.  It will focus on the essential difficulties in the problem, and will include some unsolved questions.