Georgia Institute of Technology College of Sciences

Geometric modeling builds computer models for industrial design and manufacture from basic units, called patches, such as, Bézier curves and surfaces. The control polygon of a Bézier curve is well-defined and has geometric significance—there is a sequence of weights under which the limiting position of the curve is the control polygon. For a Bezier surface patch, there are many possible polyhedral control structures, and none are canonical. In this talk, I will present a not necessarily polyhedral control structure for surface patches, regular control surfaces, which are certain C^0 spline surfaces. While not unique, regular control surfaces are exactly the possible limiting positions of a Bezier patch when the weights are allowed to vary. While our primary interest is to explain the meaning of control nets for the classical rational Bezier patches, we work in the generality of Krasauskas’ toric Bezier patches. Toric Bezier patches are multi-sided parametric patches based on the geometry of toric varieties and depend on a polytope and some weights. Our results rely upon a construction in computational algebraic geometry called a toric degeneration.

The domain decomposition methods considered are preconditioned conjugate gradient methods designed for the very large algebraic systems of equations which often arise in finite element practice. They are designed for massively parallel computer systems and the preconditioners are built from solvers on the substructures into whichthe domain of the given problem is partitioned. In addition, to obtain scalability, there must be a coarse problem, with a small number of degrees of freedom for each substructure. The design of this coarse problem is crucial for obtaining rapidly convergent iterations and poses the most interesting challenge in the analysis.Our work will be illustrated by overlapping Schwarz methods for almost incompressible elasticity approximated by mixed finite element and mixed spectral element methods. These algorithms is now used extensively at the SANDIA, Albuquerque laboratories and were developed in close collaboration with Dr. Clark R. Dohrmann. These results illustrate two roles of the coarse component of the preconditioner.Currently, these algorithms are being actively developed for problems posed in H(curl) and H(div). This work requires the development of new coarse spaces. We will also comment on recent work on extending domain decomposition theory to subdomains with quite irregular boundaries.  This work is relevant because of the use of mesh partitioners in the decomposition of large finite element matrices. 

This is the second in the series of two talks aimed to discuss a recent work of Ontaneda which gives a poweful method of producing negatively curved manifolds. Ontaneda's work adds a lot of weight to the often quoted Gromov's prediction that in a sense most manifolds (of any dimension) are negatively curved. In the second talk I shall discuss some ideas of the proof.

The marginal distribution of identically distributed random variables having a constant sum is called a completely mixable distribution. In this talk, the concept, history and present research of the complete mixability will be introduced. I will discuss its relevance to existing problems in the Frechet class, i.e. problems with known marginal distributions but unknown joint distribution and its applications in quantitative risk management.