### Computer Experiments

Session Chair: Max Morris, Iowa State University**Construction of Nested Orthogonal or Nearly Orthogonal Designs for Computer Experiments**

Jun Li and Peter Z. G. Qian

University of Wisconsin

We construct a new type of space-filling design, called a nested orthogonal or nearly orthogonal design, intended for computer experiments. These designs are constructed by exploiting nesting in a variety of discrete structures such as factorial designs, Hadamard matrices, orthogonal arrays and permutation matrices. They are useful for several important applications in computer experiments. The first application is multi-fidelity computer modeling. In this application, a pair of nested orthogonal or nearly orthogonal designs is used for conducting a slow but accurate computer experiment and a fast but less accurate computer experiment, which are respectively referred to as a high-accuracy experiment (HE) a low-accuracy experiment (LE), where the nesting between the two designs makes it easier to fit an adjustment model to link the HE data to the LE data. Motivated by the problem of estimating the expected values of the outputs of the HE and LE given a distribution of inputs, Qian (2009) introduced a method for using nested random permutations to construct a class of nested Latin hypercube designs. This work is motivated by a related but different problem that constructs suitable designs for the aim of building accurate prediction models based on HE and LE. It is also intuitively appealing to use a pair of nested orthogonal of nearly orthogonal designs for calibration and validation of a computer model, where the small design is useful conducting the physical experiment and the large design is useful for the computer experiments. Furthermore, a pair of nested orthogonal of nearly orthogonal designs can be used for sequential evaluation of a computer model, where the small design is first used for evaluating the model and more design points are added if necessary. The nested relationship facilitates sequential evaluation and makes it easy to combine data obtained from different stages.

**Multi-Layer Designs for Computer Experiments**

Shan Ba and V. Roshan Joseph

Georgia Institute of Technology

Abstract: Space-filling designs such as Latin hypercube designs (LHDs) are widely used in computer experiments. However, finding an optimal LHD with good space-filling properties is computationally cumbersome. On the other hand, the well-established factorial designs in physical experiments are unsuitable for computer experiments owing to the redundancy of design points when projected onto a subset of factor space. In this work, we present a new class of space-filling designs developed by splitting two-level factorial designs into multiple layers. The method takes advantages of many available results in factorial design theory and therefore, the proposed Multi-layer designs (MLDs) are easy to generate. Moreover, our numerical study shows that MLDs can have better space-filling properties than optimal LHDs.

**A Simple Approach to Emulation for Computer Models With Qualitative and Quantitative Factors**

Qiang Zhou, Peter Z.G. Qian, Shiyu Zhou

University of Wisconsin - Madison

Abstract: We propose a flexible yet computationally efficient approach for building Gaussian process models for computer experiments with both qualitative and quantitative factors. This approach uses the hypersphere parameterization to model the correlations of the qualitative factors, thus avoiding the need of directly solving optimization problems with positive definite constraints. This method is easy to compute and can be implemented straightforwardly in standard computational environments like R and Matlab. The effectiveness of the proposed method is successfully illustrated by several examples.

**Sequential Experiment Design for Emulator and Predictive Maturity**

Brian Williams

^{1}, Jason Loeppky,

^{2}and Leslie Moore

^{1}

^{1}Los Alamos National Laboratory and

^{2}University of British Columbia - Okanagan

We introduce batch sequential experiment design strategies for improving global prediction in the context of emulating a computer code or predicting discrepancy when calibrating a computer code to experimental data. Criteria used within a batch sequential design algorithm for efficiently achieving these objectives are discussed. A new class of experiment designs - projection array based designs - is introduced. This class generalizes orthogonal array based Latin hypercube design and facilitates space-filling augmentation of space-filling designs.