Audze-egla¯ js criterion for orthogonal and regular triangular grids

Abstract

Computer experiments have become a powerful tool for the investigation of problems encompassing the randomness of observed phenomena. To use this tool, it is necessary to prepare a plan of the simulations that should be performed, i.e. what is known as Design of Experiments (DoE) should be carried out. Such a design has to fulfill certain requirements in order to provide applicable results. As designs are often stochastic, the fulfillment of the requirements is not assured. Therefore, the originally random design is optimized with respect to a selected criterion which should guarantee that the demanded properties of the design will be achieved. One of the criteria that can be used during optimization is the Audze-Egla¯js (AE) criterion. It is a criterion that accentuates the space-filling property of the final design. Depending on the chosen method of optimization, the knowledge of the lower bound of the optimization criterion might be necessary for efficient control over the optimization algorithm. As the lower bound is not known for the AE criterion, this article describes two types of deterministic designs that are virtually ideal with regard to the uniform filling of space, and therefore should be close to optimal from the point of view of the AE criterion. Consequently, they are supposed to provide a lower bound of the criterion, or its estimate where its value cannot be evaluated due to the limits of these deterministic designs regarding the number of design points. The AE criterion values of these deterministic designs are compared to the values of various stochastic random and optimized designs. Furthermore, the description of a closed-form exact formula that enables faster evaluation of the AE criterion exploiting the regularity of deterministic design is presented for one of the two deterministic designs described in the article.