Math Sciences Colloquia - Spring 2004

All Math Sciences colloquia take place at 3 p.m. in 280 or 657 Rhodes Hall.

• February 2 - Bharath Rangarajan, ORIE, "Polynomial Convergence of Interior-Point Algorithms on Symmetric Cones"
  Abstract: Conic programs are optimization problems where a linear function is minimized over the intersection of an affine space and a closed convex cone. An interior-point algorithm for conic programs over symmetric cones derived from associative algebras is presented. This setting naturally generalises semidefinite programming. In the process, a Lyapunov-type lemma is established for this framework. The algorithms start at an initial point that is in the interior of the cone but not necessarily in the affine space. The difficulty of analysing these methods over those that start at a feasible point is highlighted. The iterates are restricted to a wide-neighborhood of the central path, inside of the cone. A polynomial convergence result for the algorithm is presented.
  Key Words: Infeasible-Interior-Point Methods, Symmetric Cones, Euclidean Jordan Algebras, Polynomial Convergence.
• March 3 - Will Noid, Chemistry, "Classical and Semiclassical Dynamics in Nonlinear Response Theory"
  Abstract: Nonlinear optical response theory describes the interaction of light and matter relevant to modern ultrafast higher order spectroscopies. Rigorous modeling of this interaction requires a solution to the time dependent Schrodinger equation - a prohibitively difficult problem for most molecular systems. A standard approach to theoretically interpreting the spectroscopic results is to approximate quantum dyamics with classical dynamics, which only involves numerically integrating ordinary differential equations (Newton's equations).
  We illustrate the successful application of classical mechanics to theoretically predict nonlinear response functions and interpret vibrational echo experiments performed by colleagues at Stanford. We also demonstrate that for a model system without dissipation, classical mechanics does not even qualitatively reproduce quantum dynamics. We then introduce semiclassical approximations to quantum mechanics in the context of optical response theory. This semiclassical theory reproduces the "quantum effects," while using only information from classical mechanics.
• March 15 - Suzanne Shontz, CAM, "Algorithms for Mesh Warping with Applications to Cardiology"
  Abstract: Moving meshes arise in cardiology, computer graphics, animation, and crash simulation, among other applications in science and engineering. With moving meshes, the mesh is updated at each step in time due to a moving domain boundary, thus resulting in potentially drastically varying mesh quality from step to step. One problem that can occur at each timestep is element inversion. Our focus is on the development of mesh warping algorithms that maintain good mesh quality at each timestep.
  We have developed several different algorithms for the warping of tetrahedral meshes. These methods are based upon weighted Laplacian smoothing, the finite element method, and quadratic programming. We start with a 3D domain which is bounded by a triangulated surface mesh and has a tetrahedral volume mesh as its interior. We then suppose that a movement of the surface mesh is prescribed and use any of our mesh warping algorithms to update the nodes of the volume mesh. Each method determines a set of local weights for each interior node which describe the relative positions of the node to each of its neighbors. After a boundary transformation is applied, the method solves a system of linear equations based upon the weights and the new boundary positions to determine the final positions of the interior nodes.
  We study mesh invertibility and prove a theorem which gives sufficient conditions for a mesh to resist inversion by a transformation. Our theory shows that two of our methods yield exact results for affine mappings, and we state a conjecture for more general mappings. In addition, our theory ensures that the same methods yield the mesh to which both the local weighted laplacian smoothing algorithm and the Gauss-Seidel algorithm for linear systems converge. We test the robustness of our methods and present some numerical results. Finally, we use our algorithm to study the movement of the beating canine heart.
  Part of this talk represents joint work with S. Vavasis.
• April 19 - Jay Henniger, CAM, "Tracking an index with a small portfolio of stock"
  Abstract: I will discuss the problem of forming a portfolio consisting of a small number of stocks to track the return on an index (like the S&P 500). Different measures of tracking error will be described and a gradual non-convexation algorithm will be presented to give an approximate solution to the problem. Other current methods of solving this problem will be discussed and numerical results will be presented.
• May 3 - Yannet Interian-Fernandez, CAM, "Approximation algorithm for Maximum random k-SAT"
  Abstract: k-Satisfiability (k-SAT) problem is an NP-Complete problem. We consider the most studied random model for k-SAT in which all the clauses are taken uniformly at random. We consider the problem finding an assingment that satisfies the maximum number of clauses. We propose a polynomial approximation algorithm that gives a 10/9.5 -approximation. The previous result for this problem was a 9/8-approximation. The paper can be found in: http://www.cam.cornell.edu/~interian/sat04.pdf
• May 11 - Joe Tien, CAM, "A Michaelis-Menten style model for the autocatalytic enzyme prostaglandin H synthase"
  Abstract: The arachidonic acid metabolic pathway is involved in a wide variety of biological processes, including inflammation, blood clotting, renal function, and tumorigenesis. A key player in this pathway is the autocatalytic enzyme prostaglandin H synthase (PGHS). Here we present a Michaelis-Menten style model for PGHS. A stability analysis determines when the reaction becomes self-sustaining, and can help explain the regulation of PGHS activity {\it in vivo}. We also consider a quasi-steady-state approximation (QSSA) for the model, and present conditions under which the QSSA is expected to be a good approximation. Applying the QSSA for this model can be useful in computationally intensive modeling endeavors involving PGHS.