Problems and Techniques

Abstract

This issue features two papers involving optimization problems, one targeted at minimizing temperature differences across a plate, and the other at multidimensional integration. 1. Want to turn the graphics card in your laptop or desktop into a personal supercomputer? Eddie Wadbro and Martin Berggren show you how, in their paper “Megapixel Topology Optimization on a Graphics Processing Unit.” The advantage of graphics cards is that they are cheap, relatively speaking, and that they allocate more resources to data processing than a CPU would. Graphics cards are also becoming easier to program (especially for those of us who grew up with Connection Machines and FPS-164s). Graphics processing units are natural hardware platforms for problems that are highly data parallel. This includes the topology optimization problem considered here, where a limited amount of high-conductivity material is to be distributed across a heated plate so that its temperature field is as even as possible. The authors express this as an area-to-point flow optimization problem. A subsequent finite element discretization gives a symmetric positive-definite linear system that is solved by a diagonally preconditioned conjugate gradient method. As it turns out, the optimal distribution of the high-conductivity material emanates like a root from the heat sink, with increasing girth and finer branches as the discretization is refined. 2. In their paper “Approximate Volume and Integration for Basic Semialgebraic Sets,” Didier Henrion, Jean Bernard Lasserre, and Carlo Savorgnan are concerned with deterministic techniques for difficult multidimensional integration, of the type where only brute force Monte Carlo methods have a chance at producing acceptable approximations. The bodies can be disconnected or nonconvex, and are described by sets of polynomial inequalities (i.e., semialgebraic sets). The foundation for this work was laid more than a hundred years ago, when Chebyshev, Markov, and Stieltjes showed how to approximate one-dimensional integrals by sequences of moments. In this paper, the authors formulate the multidimensional integration as an infinite-dimensional linear programming problem, and approximate the required moments by a hierarchy of semidefinite programming problems. Numerical examples illustrate that the approach produces accurate approximations in two and three dimensions.