This chapter is concerned with the following experimental situation: a fixed number of experiments are to be used to learn as much as possible about a space of potential materials. The researcher is required to first commit him-/herself to the specific materials that are to be investigated before these are all evaluated simultaneously. We start from the assumption that the response of interest (such as activity, toxicity, durability, permeability, etc.) shows some spatial correlation over the experimental space to be explored. This means that the evaluation of a specific material tells the researcher something about the properties of similar materials, where similarity decreases with distance in material space. The lack of knowledge of the response surfaces—which motivates experimentation in the first place—is taken into account by using stochastic processes. We give quantitative expressions for the uncertainty in the response that persists after experiments will have been performed and discuss strategies to find designs that minimize this uncertainty. Furthermore, we introduce lattices that become optimal under certain limiting conditions and use these to illustrate the nature of the space-filling designs that are derived from our assumptions. Finally, we discuss both the case in which the specific materials to be investigated can be chosen arbitrarily as well as the case in which the samples must be selected from a finite set, e.g., an existing collection or library. Throughout the chapter, special emphasis is put on the problems arising in multidimensional spaces. All required tools (stochastic processes, the best linear unbiased estimator and lattices) are introduced to make the chapter self-
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