ALBERT

Description: Multigrid methods are very efficient iterative solvers for system of algebraic equations arising from finite element and finite difference discretization of elliptic boundary value problems. The main principle of multigrid methods is to complement the local exchange of information in point-wise iterative methods by a global one utilizing several related systems, called coarse levels, with a smaller number of variables. The coarse levels are often obtained as a hierarchy of discretizations with different characteristic meshsizes, but this requires that the discretization is controlled by the iterative method. To solve linear systems produced by existing finite element software, one needs to create an artificial hierarchy of coarse problems. The principal issue is then to obtain computational complexity and approximation properties similar to those for nested meshes, using only information in the matrix of the system and as little extra information as possible. Such algebraic multigrid method that uses the system matrix only was developed by Ruge. The prolongations were based on the matrix of the system by partial solution from given values at selected coarse points. The coarse grid points were selected so that each point would be interpolated to via so-called strong connections. Our approach is based on smoothed aggregation introduced recently by Vanek. First the set of nodes is decomposed into small mutually disjoint subsets. A tentative piecewise constant interpolation (in the discrete sense) is then defined on those subsets as piecewise constant for second order problems, and piecewise linear for fourth order problems. The prolongation operator is then obtained by smoothing the output of the tentative prolongation and coarse level operators are defined variationally.

Description: We analyze the convergence of a substructuring iterative method with Lagrange multipliers, proposed recently by Farhat and Roux. The method decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains and a coarse problem for the subdomain nullspace components. For linear conforming elements and preconditioning by the Dirichlet problems on the subdomains, we prove the asymptotic bound on the condition number C(1 + log(H/h))(sup gamma), gamma = 2 or 3, where h is the characteristic element size and H is the subdomain size.

Description: In our "Future Directions for Astronomical Image Display" project, the Smithsonian Astrophysical Observatory (SAO) and the National Optical Astronomy Observatories (NOAO) will evolve our existing image display software into a fully extensible, cross-platform image display server that can run stand-alone or be integrated seamlessly into astronomical analysis systems. We will build a Plug-in Image Extension (PIE) server for astronomy, consisting of a modular image display engine that can be customized using "plug-in" technology. We will create plug-ins that reproduce all the current functionality of SAOtng. We also will devise a messaging system and a set of distributed, shared data objects to support integrating the PIE server into astronomical analysis systems. Finally, we will migrate our PIE server, plug-ins, and messaging software from Unix and the X Window System to a platform-independent architecture that utilizes cross-platform technology such as Tcl/Tk or Java.