A Note on Lower Bound of Centered L_2 -discrepancy on Combined Designs

This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered Lu-discrepancy values of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Our numerical results show that this lower bound is sharper than existing results when more factors reverse the signs in the initial design. Keywords Centered L2-discrepancy, optimal foldover plan, uniformity, uniformity pattern

A generalized L_2-discrepancy for cubature and uncertainty quantification of nonlinear structures

The numerical method for multi-dimensional integrals is of great importance, particularly in the uncertainty quantification of engineering structures. The key is to generate representative points as few as possible but of acceptable accuracy. A generalized L2（GL2）-discrepancy is studied by taking unequal weights for the point set. The extended Koksma-Hlawka inequality is discussed. Thereby, a worst-case error estimate is provided by such defined GL2-discrepancy, whose dosed-form expression is available. The characteristic values of GL2-discrepancy are investigated. An optimal strategy for the selection of the representative point sets with a prescribed cardinal number is proposed by minimizing the GL2-discrepancy. The three typical examples of the multi-dimensional integrals are investigated. The stochastic dynamic response analysis of a nonlinear structure is then studied by incorporating the proposed method into the probability density evolution method. It is shown that the proposed method is advantageous in achieving tradeoffs between the efficiency and accuracy of the exemplified problems. Problems to be further studied are discussed.

New Lower Bounds to Wrap-around L2-discrepancy for U-type Designs with Three-level

The objective of this paper is to study the issue of uniformity on three-level U-type designs in terms of the wrap-around L2-discrepancy.Based on the known formula,we present a new lower bound of wrap-around L2-discrepancy for three-level U-type designs and compare it with those existing ones through figures,numerical simulation and illustrative examples.

Efficient Asymmetrical Extended Designs Under Wrap-Around L2-Discrepancy

The purpose of the present article is to introduce a class of mixed two- and three-level extended designs obtained by adding some new runs to an existing mixed two- and three-level design. A formulation of wrap-around L2-discrepancy for the extended designs is developed. As a benchmark of obtaining （nearly） uniform asymmetrical extended designs, a lower bound to the wrap-around L2- discrepancy for our proposed designs is established. Thorough numerical results are displayed, which provide further corroboration to the derived theoretical results.

Effective Lower Bounds of Wrap-Around L_2-Discrepancy on Three-Level Combined Designs

How to obtain an effective design is a major concern of scientific research. This topic always involves high-dimensional inputs with limited resources. The foldover is a quick and useful technique in construction of fractional designs, which typically releases aliased factors or interactions.This paper takes the wrap-around L_2-discrepancy as the optimality measure to assess the optimal three-level combined designs. New and efficient analytical expressions and lower bounds of the wraparound L_2-discrepancy for three-level combined designs are obtained. The new lower bound is useful and sharper than the existing lower bound. Using the new analytical expression and lower bound as the benchmarks, the authors may implement an effective algorithm for constructing optimal three-level combined designs.

Optimal foldover plans of three-level designs with minimum wrap-around L-_2 discrepancy

Abstract The objective of this paper is to study the issue of employing the uniformity criterion measured by the wrap-around L2-discrepancy to assess the optimal foldover plans for three-level designs.For three-level fractional factorials as the original designs,the general foldover plan and combined design under a foldover plan are defined,some theoretical properties of the defined foldover plans are obtained,a tight lower bound of the wrap-around L2-discrepancy of combined designs under a general foldover plan is also obtained,which can be used as a benchmark for searching optimal foldover plans.For illustration of the usage of our theoretical results,a catalog of optimal foldover plans for uniform initial designs with s three-level factors is tabulated,where 2≤ s ≤11.

A Note on Optimal Foldover Four-level Factorials

The foldover is a quick and useful technique in construction of fractional factorial designs, which typically releases aliased factors or interactions. The issue of employing the uniformity criterion measured by the centered L2-discrepancy to assess the optimal foldover plans was studied for four-level design. A new analytical expression and a new lower bound of the centered L2-discrepancy for fourlevel combined design under a general foldover plan are respectively obtained. A necessary condition for the existence of an optimal foldover plan meeting this lower bound was described. An algorithm for searching the optimal four-level foldover plans is also developed. Illustrative examples are provided, where numerical studies lend further support to our theoretical results. These results may help to provide some powerful and efficient Mgorithms for searching the optimal four-level foldover plans.

Construction of Multi-level Space-filling Designs via Code Mappings

Space-filling designs are widely used in various fields because of their nice space-filling properties.Uniform designs are one of space-filling designs,which desires the experimental points to scatter uniformly over the experimental area.For practical need,the construction and their properties of nine-level uniform designs are discussed via two code mappings in this paper.Firstly,the algorithm of constructing nine-level uniform designs is presented from an initial three-level design by the Type-I code mapping and tripling technique.Secondly,the algorithm of constructing nine-level uniform designs is presented from a three-level base design by the Type-II code mapping and generalized orthogonal arrays.Moreover,relative properties are discussed based on the two code mappings.Finally,some numerical examples are given out for supporting our theoretical results.

Uniformity of Asymmetric Augmented Designs

Follow-up experimental designs are widely applied to explore the relationship between factors and responses step by step in various fields such as science and engineering.When some additional resources or information become available after the initial design of experiment is carried out,some additional runs and/or factors may be added in the follow-up stage.In this paper,the issue of the uniform row augmented designs and column augmented designs with mixed two-,three-and four-level is investigated.The uniformity of augmented designs is discussed under the wrap-around L_(2)-discrepancy.Some lower bounds of wrap-around L_(2)-discrepancy for the augmented designs are obtained,which can be used to assess uniformity of augmented design.Numerical results show that augmented designs have high efficiency,which have low discrepancy and close to the proposed lower bounds.