Kriging based on Gaussian random fields is widely used in reconstructing unknown functions. The kriging method has pointwise predictive distributions which are computationally simple. However, in many applications one would like to predict for a range of untried points simultaneously. In this work we obtain some error bounds for the simple and universal kriging predictor under the uniform metric. It works for a scattered set of input points in an arbitrary dimension, and also covers the case where the covariance function of the Gaussian process is misspecified. These results lead to a better understanding of the rate of convergence of kriging under the Gaussian or the Matérn correlation functions, the relationship between space-filling designs and kriging models, and the robustness of the Matérn correlation functions.

Computer experiments have become ubiquitous in science and engineering. Commonly, runs of these simulations demand considerable time and computing, making experimental design extremely important in gaining high quality information with limited time and resources. Broad principles of experimental design are proposed and justified which ensure high nominal, numeric, and parameter estimation accuracy for Gaussian process emulation of deterministic simulations. The space-filling properties "small fill distance" and "large separation distance" are only weakly conflicting and ensure well-controlled nominal, numeric, and parameter estimation error. Nonstationarity indicates a greater density of experimental inputs in regions of the input space with more quickly decaying correlation, while non-constant regression functions indicate a balancing of traditional design features with space-fillingness. This work provides robust, rigorously justified, and practically useful overarching sufficient principles for scientists and engineers selecting combinations of simulation inputs with high information content.

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Scientists and engineers commonly use simulation models to study real systems for which actual experimentation is costly, difficult, or impossible. Many simulations are stochastic in the sense that repeated runs with the same input configuration will result in different outputs. For expensive or time-consuming simulations, stochastic kriging (Ankenman et al., 2010) is commonly used to generate predictions for simulation model outputs subject to uncertainty due to both function approximation and stochastic variation. Here, we develop and justify a few guidelines for experimental design, which ensure accuracy of stochastic kriging emulators. We decompose error in stochastic kriging predictions into nominal, numeric, parameter estimation and parameter estimation numeric components and provide means to control each in terms of properties of the underlying experimental design. The design properties implied for each source of error are weakly conflicting and broad principles are proposed. In brief, space-filling properties "small fill distance" and "large separation distance" should balance with replication at distinct input configurations, with number of replications depending on the relative magnitudes of stochastic and process variability. Non-stationarity implies higher input density in more active regions, while regression functions imply a balance with traditional design properties. A few examples are presented to illustrate the results.

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