Lattices with minimal normalized second moments are designed using a new numerical optimization algorithm. Starting from a random lower-triangular generator matrix and applying stochastic gradient descent, all elements are updated towards the negative gradient, which makes it the most efficient algorithm proposed so far for this purpose. A graphical illustration of the theta series, called theta image, is introduced and shown to be a powerful tool for converting numerical lattice representations into their underlying exact forms. As a proof of concept, optimized lattices are designed in dimensions up to 16. In all dimensions, the algorithm converges to either the previously best known lattice or a better one. The dual of the 15-dimensional laminated lattice is conjectured to be optimal in its dimension and its exact normalized second moment is computed.

40 years ago, Conway and Sloane proposed using the highly symmetrical Coxeter–Todd latticeK12for quantization, and estimated its second moment. Since then, all published lists identifyK12as the best 12-dimensional lattice quantizer. Surprisingly,K12is not optimal: we construct two new 12-dimensional lattices with lower normalized second moments. The new lattices are obtained by gluing together products of two 6-dimensional lattices.

A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996 paper, Zamir and Feder show that the globally optimal lattice quantizer (which minimizes the mean square error) has white quantization error: for a uniformly distributed source, the covariance of the error is the identity matrix, multiplied by a positive real factor. We generalize the theorem, showing that the same property holds (i) for any lattice whose mean square error cannot be decreased by a small perturbation of the generator matrix, and (ii) for an optimal product of lattices that are themselves locally optimal in the sense of (i). We derive an upper bound on the normalized second moment (NSM) of the optimal lattice in any dimension, by proving that any lower- or upper-triangular modification to the generator matrix of a product lattice reduces the NSM. Using these tools and employing the best currently known lattice quantizers to build product lattices, we construct improved lattice quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions, these are the first reported lattices with normalized second moments below the best known upper bound.

Volunteer computing is a form of distributed computing in which the general public volunteers processing and storage to scientific research projects. BOINC, a middleware system for volunteer computing, is currently used by about 20 projects, to which 300,000 volunteers and 450,000 computers supply 350 TeraFLOPS of processing power. A BOINC client program runs on the volunteered hosts and manages the execution of applications. Together with a library linked to applications, it implements a runtime system providing process management, graphics control, checkpointing, file access, and other functions. This runtime system must handle widely varying applications, must provide features and properties desired by volunteers, and must work on many platforms. This paper describes the problems in designing a runtime system having these properties, and how these problems are solved in BOINC.