Measures of scalability

Scalable frames are frames with the property that the frame vectors can be rescaled resulting in tight frames. However, if a frame is not scalable, one has to aim for an approximate procedure. For this, in this paper we introduce three novel quantitative measures of the closeness to scalability for frames in finite dimensional real Euclidean spaces. Besides the natural measure of scalability given by the distance of a frame to the set of scalable frames, another measure is obtained by optimizing a quadratic functional, while the third is given by the volume of the ellipsoid of minimal volume containing the symmetrized frame. After proving that these measures are equivalent in a certain sense, we establish bounds on the probability of a randomly selected frame to be scalable. In the process, we also derive new necessary and sufficient conditions for a frame to be scalable.Comment: 27 pages, 5 figure

On Optimal Frame Conditioners

A (unit norm) frame is scalable if its vectors can be rescaled so as to result into a tight frame. Tight frames can be considered optimally conditioned because the condition number of their frame operators is unity. In this paper we reformulate the scalability problem as a convex optimization question. In particular, we present examples of various formulations of the problem along with numerical results obtained by using our methods on randomly generated frames.Comment: 11 page

From Cluster to Grid: A Case Study in Scaling-Up a Molecular Electronics Simulation Code

This paper describes an ongoing project whose goal is to significantly improve the performance and applicability of a molecular electronics simulation code. The specific goals are to (1) increase computational performance on the simulation problems currently being solved by our physics collaborators; (2) allow much larger problems to be solved in reasonable time; and (3) expand the set of resources available to the code, from a single homogeneous cluster to a campus-wide computational grid, while maintaining acceptable performance across this larger set of resources. We describe the sequential performance of the code, the performance of two parallel versions, and the benefits of problem-specific load balancing strategies. The grid context motivates the need for runtime algorithm selection; we present a component-based software framework that makes this possible