Sequential Exploration of Complex Surfaces Using Minimum Energy Designs

A new space-filling design, called minimum energy design (MED), is proposed to explore unknown regions of the design space of particular interest to an experimenter. The key ideas involved in constructing the MED are the visualization of each design point as a charged particle inside a box, and minimization of the total potential energy of these particles. It is shown through theoretical arguments and simulations that with a proper choice of the charge function, the MED can asymptotically generate any arbitrary probability density function. A version of the MED, which adaptively updates the design by “learning” about the unknown response surface sequentially, is proposed and implemented. Two potential applications of MED in simulation of complex probability densities and optimization of complex response surfaces are discussed and demonstrated with examples. This article has supplementary material online.