Optimal Centered L₂-discrepancy Designs Based On Good Lattice Point

Computer experiments are increasingly being used to investigate complex systems,space-filling design can reduce trials and ensure the robustness of the model,so it is crucial for building a high quality computer model.According to different criteria,several class of optimal design are proposed,such as Latin hypercube designs,maximin distance designs and uniform designs.Uniform designs have made many contributions in the military and agriculture,but the construction of optimal uniform design is challenged.Uniform designs spread the points evenly throughout the experimental area,and it mainly include the good lattice point design?GLPD?.Recently,Zhou and Xu?2015?pointed that linear permutation will not decrease the 1-distance of the GLPD.Wang,Xiao and Xu?2018?performed the Williams transform on the basis of the linear permutation of a GLPD and found the conditions for achieving the optimal design under the maximin distance criteria.All the studies above show that the GLPD has many excellent properties,but how to choose the optimal design is still a meaningful and urgent research topic.We use the centered L₂-discrepancy to measure the uniformity of the design and construct two types of space-filling designs based on the GLPD: one is linear permutation of GLPD,another is Williams transformation based on linear permutation of GLPD.These two designs can shorten the experimental time and improve the estimation efficiency when constructing multi-factor models.We give some suggestions for constructing optimal space-filling designs.Finally,we select two functions with multiple independent variables for simulation.Compared with the existing design methods,the proposed designs have smaller prediction error.