2007 Barrett Memorial Lectures University of Tennessee , 28 - 30 April 2007

Gregory Beylkin, University of Colorado, Boulder Fast Algorithms for Adaptive Application of Integral Operators in High Dimensions In physics, chemistry and other applied fields, many important problems may be formulated using integral equations, typically involving Green’s functions as their kernels. Often such formulations are preferable to those via partial differential equations (PDEs). For example, evaluating the integral expressing solution of the Poisson equation in free space, i.e., the convolution of the Green’s function with the mass or charge density, avoids issues associated with the high condition number of a PDE formulation. Since many physically significant operators depend only on the distance between interacting objects, the operators such as (−∆ + μ2I)−α, where μ ≥ 0 and 0 < α < 3/2, and certain singular operators, such as the projector on divergence-free functions, fall into the class of operators for which our approach yields fast adaptive algorithms. This talk will describe adaptive multi resolution algorithms for applying integral operators with radially symmetric kernels in dimensions one, two and three. In dimensions two and higher kernels of operators are represented in separated form by approximating radial kernels by a sum of Gaussians, for any finite but arbitrary accuracy. We will briefly discuss an extension to dimensions higher than three. Russel Caflisch, University of California, Los Angeles Growth, Structure and Pattern Formation for Thin Films An epitaxial thin film consists of layers of atoms whose lattice properties are determined by those of the underlying substrate. This talk will describe mathematical modeling, analysis and simulation of growth, structure and pattern formation for epitaxial systems. Epitaxial growth involves physics on both atomistic and continuum length scales. For example, diffusion of adatoms can be coarse-grained, but nucleation of new islands and breakup for existing islands are best described atomistically. Our growth simulations use an island dynamics model with a level set simulation method. The level set velocity comes from a detailed model for a step edge or island boundary on an epitaxial surface. Through asymptotic analysis of this model, we derive the Gibbs-Thomson formula for anisotropic step stiffness. In heteroepitaxial growth, e.g., Germanium on Silicon, mismatch between the lattice spacing of the Silicon substrate and the Germanium film will introduce a strain into the film, which can significantly influence the material structure, for example leading to formation of quantum dots. Strain computations can be computationally intensive, so that effective simulation of atomistic strain effects relies on an accelerated method that incorporates algebraic multigrid and an artificial boundary condition. Technological applications of epitaxial structures, such as quantum dot arrays, require a degree of geometric uniformity that has been difficult to achieve. Modeling and simulation