Workshop on Group Theory and Numerical Analysis

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

The funds in this grant will be used to support U.S. researchers, postdocs and graduate students to attend and present lectures and posters in a Workshop on Group Theory and Numerical Analysis at the Centre de Recherches Mathematiques, Universite de Montreal, during the week May 26-31, 2003. The workshop features one aspect of the emerging theory of "geometric integration," that advocates incorporating known qualitative features of systems of differential equations, such as invariants, symmetries, conservation laws, phase-space dynamics, and symplectic structure, in their numerical solution algorithms. In particular, the symmetry properties of a system of differential equations have proven to be a very powerful means of understanding the qualitative properties of its solutions as well as facilitating the determination of exact analytical solutions. Such special solutions often serve as prototypes and govern the asymptotics for more complicated situations, and also as useful testing grounds for developing numerical integration schemes. The methods of symmetry analysis and integrability for differential equations have, more recently, been adapted with considerable success to analyze discrete equations. On the other hand, the incorporation of symmetry in the design of invariant geometric numerical schemes for a variety of differential equations arising in applications has recently been the subject of increasing research activity and advances. The key objective of the workshop is to bring experts on applications of group theory to discrete equations together with practitioners of geometric numerical methods, thereby stimulating a mutually beneficial exchange of ideas and recent developments and fostering a cross-fertilization between such closely related subjects. Group theory is the mathematical formalization of the concept of symmetry. Most interesting systems that arise in applications in physics, engineering, chemistry, computer vision, biology, etc., possess a rich variety of symmetries. While exploiting the symmetries of continuous systems has a long and distinguished record of achievement, only recently have significant developments occurred within the discrete regime. Indeed, standard numerical solution methods used by computers typically fail to properly respect the underlying symmetry properties. This workshop will build on recent developments in symmetry analysis of discrete systems and the development of symmetry-preserving numerical solution techniques. The participants are experts in symmetry methods and in geometric numerical integration. The workshop will promote interactions and significant progress in both subjects, as well as their applications in mathematics and the sciences. The results of the workshop, as presented in the talks and discussions, will be summed up in a book to be so focused as to be of permanent value. In addition, an assessment of the current state of the art in the subjects and new scientific opportunities stemming from this workshop will be the focus of a panel discussion and white paper to be distributed and posted on the web.