Speakers are named first.
2:15-2:30 Alexander Kurganov, Tulane University
Scheme Adaption for Hyperbolic Conservation Laws
Designing numerical methods for nonlinear hyperbolic systems of conservation laws is a challenging problem due to a complicated wave structure of their solutions, especially in the multidimensional case. Linear methods typically fail to achieve high resolution due to large numerical dissipation or dispersion that cannot be balanced without introducing a nonlinear mechanism -- limiters. There is a library of second- and higher-order limiters available. However, they are computationally expensive and, in some cases, problem-oriented.
The main idea of scheme adaption is to apply different numerical methods to different parts of the computed solution. In the context of hyperbolic conservation laws, one may take advantage of this approach by applying expensive limiters near nonlinear shock waves only, while treating the rest of the computational domain by a high-order linearly stable method. Obviously, scheme adaption requires a certain mechanism for distinguishing between the smooth and nonsmooth regions. We have used an indicator based on a weak local residual, which not only measures the smoothness of the computed solution, but also verifies its quality.
2:35-2:50 Anhua Lin, Department of Mathematical Sciences, Middle Tennessee State University
Using Ice-cream cone programs as subproblems to solve general nonlinear r programming problems
The classical SQP method for solving general nonlinear programs is not very effective to handle highly nonlinear constraints since the subproblems only use hyperplanes to approximate those constraints. Recently, methods of using subproblems which use quadratic surfaces to approximate the constraints are receiving serious attention due to the development of efficient interior point techniques for solving such subproblems via ice-cream cone programs. Several methods based on this idea have been proposed for the convex (feasible region) case. There are also some nonlocal anlysis for the general (maybe nonconvex) case. But a global method for the general nonlinear programming is still missing. We present a method and the corresponding global analysis which can fill this gap, to some extent. One important feature of this analysis is that it gives precise criteria that how well each subproblem should be approximately solved. In previous analysis of those algorithms, it is always assumed that the subproblems are solved to exact optimality which is impossible in practice, particularly when using interior-point methods.
2:55-3:10 Alina Chertock, North Carolina State University
Hybrid Finite-Volume Particle Method for Stiff Detonation Waves
We propose a hybrid finite-volume-particle method for hyperbolic conservation laws with stiff source terms. Such problems arise, among many other applications, in modeling chemical reactive flows, in which the reaction is fast and the time scale associated with the stiff reaction term is much smaller than that associated with the fluid advection. The main difficulty in numerically solving such equations is obtaining the correct propagation speed of detonation waves in the case of underresolved computations. Since the numerical dissipation in the computation of the mass fraction of the burnt/unburnt species may lead to an incorrect speed, a natural idea is to evolve the mass fraction field with a nonviscous numerical method. Our approach consists of two steps: solving the gas dynamics part of the system by a finite-volume method and applying a particle method to the mass fraction transport equation with the stiff reaction term. By doing so, we take an advantage of low dissipation and mesh-free feature of particle methods and demonstrate that applying particle methods to such models helps to accurately capture detonation waves even when a problem with a complicated nonlinear wave interactions is being considered.
3:15-3:30 Yingjie Liu, Georgia Tech and Todd F. Dupont
Back and Forth Error Compensation and Correction Methods for Semi-Lagrangian Schemes with Application to Interface Computation Using Level Set Method
Level set method uses a level set function, usually an approximate signed distance function, Φ, to represent the interface as the zero set of Φ. When Φ is advanced to the next time level by a transportation equation, its new zero level set will represent the new interface position. We update the level set function Φ forward in time and then backward to get another copy of the level set function, say Φ1. Φ1 and Φ should have been equal if there were no numerical error. Therefore Φ-Φ1 provides us the information of error and this information can be used to compensate Φ before updating Φ forward again in time. One nice property is that it has the convenience of possibly improving the temporal and spatial order of an odd order scheme simultaneously. We found that when applying this idea to semi-Lagrangian schemes, e.g., the CIR scheme(which has no CFL restriction, a nice feature for local refinement), the property is still valid (the MacCormack scheme has similar property but it may not be easily applied here). This technique coupled with a simple yet less diffusive redistancing technique produces a very efficient algorithm even for unstructured triangle meshes. Local constant velocity for the first two steps of this algorithm can be used to overcome the singularuities in the velocity field wherever they appear. Numerical results will be presented in the talk. Also we would like show some interesting theoretical results for applying this idea to a general linear scheme. For example, the center difference scheme (with forward Euler time stepping) is linearly unstable for hyperbolic equations. Coupled with this algorithm, it becomes stable and the order improves to two.
3:35-3:50 Robert E. Funderlic, North Carolina State University
The Mathematics of k-Modes, a Qualitative k-Means Like Clustering Method
Spherical k-means takes the normalized mean of a normalized cluster of vectors as its representative vector, whereas k-modes takes its mode vector. Spherical k-means yields a rich matrix algebra context as well as its set context. There are several analogies for non-numerical categorical vectors to normalized numerical vectors. The comparison of spherical k-means and k-modes is made in the environment of a word-document matrix and a matrix that comes from requests for the manufacturing of cabinets of electronic boards.
3:55-4:10 Tieling Chen, Univeristy of South Carolina Aiken
Wavelet Transformation in Edge Detection
Edge detection through wavelet transformation is equivalent to CannyÕs edge detector, which is currently the most popular edge detection algorithm. A wavelet transformation of a given image is proportional to the gradient of a corresponding smoothed image. The places where local extreme values of the wavelet transformation are reached give the edges of the image. A smoothing function derives a wavelet, and the support set and the shape of a smoothing function plays an important role in edge detection. Edges have their own properties in the evolution of the wavelet transform at different scale levels. A wavelet transformation may alter the positions of the edges but this alteration can be reduced by a proper chosen smoothing function. In this talk, we discuss how the evolution of the wavelet transformation characterizes different kinds of edges, and how the shape of a smoothing function affects the positions of extracted edges. An example of edge detection using wavelet transformation is also given.