Seminars and Colloquia by Series

An efficient numerical method for vesicle simulations

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 27, 2008 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
George BirosCSE, Georgia Tech
Fluid membranes are area-preserving interfaces that resist bending. They are models of cell membranes, intracellular organelles, and viral particles. We are interested in developing simulation tools for dilute suspensions of deformable vesicles. These tools should be computationally efficient, that is, they should scale well as the number of vesicles increases. For very low Reynolds numbers, as it is often the case in mesoscopic length scales, the Stokes approximation can be used for the background fluid. We use a boundary integral formulation for the fluid that results in a set of nonlinear integro-differential equations for the vesicle dynamics. The motion of the vesicles is determined by balancing the nonlocal hydrodynamic forces with the elastic forces due to bending and tension. Numerical simulations of such vesicle motions are quite challenging. On one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives and a milder constraint due to a transport-like stability condition. On the other hand, an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step. We present two semi-implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme. We discretize the equations by using a spectral method in space, and a multistep third-order accurate scheme in time. We use the fast multipole method to efficiently compute vesicle-vesicle interaction forces in a suspension with a large number of vesicles. We report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme. Joint work with: Shravan K. Veerapaneni, Denis Gueyffier, and Denis Zorin.

k-planes for classification

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, October 22, 2008 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Arthur SzlamUCLA

Please Note: SPECIAL TIME AND LOCATION FOR THIS WEEK ONLY

The k-planes method is the generalization of k-means where the representatives of each cluster are affine linear sets. In this talk I will describe some possible modifications of this method for discriminative learning problems.

The existence of three-dimensional generalized solitary waves

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 6, 2008 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Shengfu DengSchool of Mathematics, Georgia Tech
We consider the three-dimensional gravity-capillary waves on water of finite-depth which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The exact Euler equations are formulated as a spatial dynamical system in stead of using Hamiltonian formulation method. A center-manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic solution for the reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants.

Thin Elastic Materials Under Confinement

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 29, 2008 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Silas AlbenSchool of Mathematics, Georgia Tech
We discuss two problems. First: When a piece of paper is crumpled, sharp folds and creases form. These are distributed over the sheet in a complex yet fascinating pattern. We study experimentally a two-dimensional version of this problem using thin strips of paper confined within rings of shrinking radius. We find a distribution of curvatures which can be fit by a power law. We provide a physical argument for the power law using simple elasticity and geometry. The second problem considers confinement of charged polymers to the surface of a sphere. This is a generalization of the classical Thompson model of the atom and has applications in the confinement of RNA and DNA in viral shells. Using computational results and asymptotics we describe the sequence of configurations of a simple class of charged polymers.

Numerical Simulations with Uncertainty - Prediction and Estimation

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 22, 2008 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Dongbin XiuDivision of Applied Math, Purdue University
There has been growing interest in developing numerical methods for stochastic computations. This is motivated by the need to conduct uncertainty quantification in simulations, where uncertainty is ubiquitous and exists in parameter values, initial and boundary conditions, geometry, etc. In order to obtain simulation results with high fidelity, it is imperative to conduct stochastic computations to incorporate uncertainty from the beginning of the simulations. In this talk we review and discuss a class of fast numerical algorithms based on generalized polynomial chaos (gPC) expansion.The methods are highly efficient, compared to other traditional In addition to the forward stochastic problem solvers, we also discuss gPC-based methods for addressing "modeling uncertainty", i.e., deficiency in mathematical models, and solving inverse problems such as parameter estimation. ones, and suitable for stochastic simulations of complex systems.

Numerical Simulations of Global Approach for Photon Scanning Tunneling Microscopy - Coupling of Finite Element and Boundary Integral Methods

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 15, 2008 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Peijun LiDepartment of Mathematics, Purdue University
Near-field optics has developed dramatically in recent years due to the possibility of breaking the diffraction limit and obtaining subwavelength resolution. Broadly speaking, near-field optics concerns phenomena involving evanescent electromagnetic waves, to which the super-resolving capability of near-field optics may be attributed. In order to theoretically understand the physical mechanism of this capability, it is desirable to accurately solve the underlying scattering problem in near-field optics. We propose an accurate global model for one of the important experimental modes of near-field optics, photon scanning tunneling microscopy, and develop a coupling of finite element and boundary integral method for its numerical solution. Numerical experiments will be presented to illustrate the effectiveness of the proposed method and to show the features of wave propagation in photon scanning tunneling microscope.  The proposed model and developed method have no limitations on optical or geometrical parameters of probe and sample, they can be used for realistic simulations of various near-field microscope configurations.

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