Max D. Gunzburger - US grants

9806358 Gunzburger In the last few years, the engineering and mathematical communities have shown increasing interest in least-squares finite element methods for solving a variety of problems in fluids, electromagnetics, elasticity, and other applications. The great promise of least-squares methods arises from the fact that, when compared to other discretization schemes, they lead to discrete problems that are much easier to solve on a computer. In the past the PI has studied numerous facets of least-squares finite element methods. These include: the use of mesh-dependent weights in least-squares functionals in order to achieve optimally accuracy, the solution of practical implementation issues that needed to be addressed in order to make these methods practical and competitive, and the application of these methods to problems with discontinuous coefficients that arise, e.g., from inhomogeneous media properties. The PI plans to apply least-squares finite element methodologies to optimization and control problems and to develop, analyze, and implement domain decomposition algorithms in the least-squares setting. Domain decomposition methods have attracted even more attention due to their usefulness in a parallel processing environment. The PI has developed novel non-overlapping domain decomposition methods based on optimization or optimal control ideas that posses numerous desirable features, the most important perhaps being that they are easily extended to nonlinear problems. The PI plans to introduce preconditioners to speed-up the performance of the methods, to look at different functionals and optimization parameters on which to base the decomposition into subdomains so that again more efficient algorithms are obtained, to apply and analyze algorithms to the solution of optimization problems for partial differential equations, to develop algorithms for time-dependent problems, and to implement algorithms on parallel computers consisting of clusters of Pentium processors.

9804748 Gunzburger This award provides support for a conference in "Recent Trends and Advances in PDEs and Numerical PDEs" to be held at Iowa State University between August 2-5, 1998 (dates to be confirmed). Partial differential equations (PDEs) is the language of much of science and engineering so that advances in PDEs and their approximate solution are of continuing importance to applications. The general themes of the conference are recent advances and trends in the theory and computation of PDEs, focusing on nonlinear models, algorithms, and theoretical and computational results that are of importance in applications. Interdisciplinary cross-fertilizations will be featured. This aspect of the conference will especially benefit junior mathematicians who will be exposed to the great variety of approaches and models that the general PDE community has developed. A distinguished list of speakers will be invited to the conference; there will also be ample time reserved for young researchers to present their results. The featured speaker at the conference will be Professor Olga Ladyzhenskaya of the Steklov Institute in St. Petersburg, Russia. A discussion, led by notable researchers, about the important trends that have been recently established and problems that the PDE community should be addressing will take place. The conference should be especially attractive to minorities and women. Speakers from these groups, including the featured speaker, will be prominent at the conference and help will be provided to junior participants representing these groups for their expenses.

A centroidal Voronoi tessellation (CVT) is a Voronoi tessellation of a
given set such that the associated generating points are centers of
mass of the corresponding Voronoi regions. Applications of CVT's
range from problems in image compression, vector quantization,
quadrature rules, grid generation and optimization, finite difference
schemes, distribution of resources, cellular biology, cluster analysis,
and the territorial behavior of animals. The goals are to develop highly
efficient parallel algorithms for the computation of CVT's, to gain
further understanding of interesting features related to the these
tessellations, to implement and test these algorithms, and to produce
a useful software suite that can serve as a design tool for many
problems in applications.

Among the theoretical questions to be studied are the complexity of
algorithms for the construction of CVT's, the effects of nonuniform
densities, CVT's in general metrics, constrained CVT's, and generalized
CVT's based on lines, curves and surfaces.

Parallel deterministic and probabilistic algorithms will be developed
and tested. In the former case, different averaging and communication
strategies will be examined; for the latter,domain decomposition
ideas will be exploited. This effort can lead to parallel grid generation
algorithms and other software useful in applications such as
clustering analysis.

Applications of CVT's will also be studied, with particular emphasis
on grid generation and optimization. The questions addressed in this
connection include the role of the density functions used to generate
the CVT's on the optimal placement of grid points, the construction of
grids with special properties and the the use of various generalized
CVT's. Another application that will be examined are the use of CVT's
in the clustering analysis of large data sets.