MSI Weekly Bulletin - Week starting Monday 1 May, 2006

Scientific simulation is progressing from the traditional approach of simulating individual processes and systems towards multiphysics simulation of complex systems comprising many mutually interacting subsystems. A classic example of this situation is a coupled climate model. Additionally, scientists are beginning to tackle simulation of multiscale systems that involve differing physics over multiple interacting length scales. The key technology enabling these bold enterprises is parallel computing, and in particular message-passing parallelism. This technological boon creates a new obstacle--the parallel coupling problem (PCP)--in the conveyance and transformation of data between multiple, distributed-memory applications. This presentation will begin with a brief survey multiphysics and multiscale applications, and how the computational complexity of their constituents give rise to the PCP. This will be followed by a discussion of the PCP and the challenges it presents in both coupled model architecture and parallel data processing. A brief overview of relevant available and emerging software technologies will be presented. Finally, a set of open problems will be stated.

This talk is based on a joint work with Bo Guan at Ohio-State Univ and Richard Schoen at Stanford Univ. An entire spacelike hypersurface with prescribe Gauss curvature in Minkowski space is reduced to an entire solution to the Monge-Ampere equation with gradient constraint in $R^n$. We will show a few existence results for the equation with different gradient constraints and discuss the completeness of the spacelike hypersurfaces with constant Gauss curvature.

Abstract: In the theory of partial differential operators one of the key issues to be resolved is to provide precise definition of the differential operators and solutions of partial differential equations. A natural and fruitful solution of the problem is to define differential operators as operators acting on some functional spaces. Lebesgue spaces $L^p$ seem to be the best and most common choice in this context. In my talk I discuss the boundedness of spectral multipliers of self adjoint differential operators on $L^p$ spaces. To avoid technicalities I consider mainly the standard Laplace operator and discuss the classical examples of restrictions theorems and optimal Bochner-Riesz summability.