Space-filling Criterion Research Articles

Some Improvements on Good Lattice Point Sets

Good lattice point (GLP) sets are a type of number-theoretic method widely utilized across various fields. Their space-filling property can be further improved, especially with large numbers of runs and factors. In this paper, Kullback-Leibler (KL) divergence is used to measure GLP sets. The generalized good lattice point (GGLP) sets obtained from linear-level permutations of GLP sets have demonstrated that the permutation does not reduce the criterion maximin distance. This paper confirms that linear-level permutation may lead to greater mixture discrepancy. Nevertheless, GGLP sets can still enhance the space-filling property of GLP sets under various criteria. For small-sized cases, the KL divergence from the uniform distribution of GGLP sets is lower than that of the initial GLP sets, and there is nearly no difference for large-sized points, indicating the similarity of their distributions. This paper incorporates a threshold-accepting algorithm in the construction of GGLP sets and adopts Frobenius distance as the space-filling criterion for large-sized cases. The initial GLP sets have been included in many monographs and are widely utilized. The corresponding GGLP sets are partially included in this paper and will be further calculated and posted online in the future. The performance of GGLP sets is evaluated in two applications: computer experiments and representative points, compared to the initial GLP sets. It shows that GGLP sets perform better in many cases.

Maxpro Designs for Experiments with Multiple Types of Branching and Nested Factors.

Contemporary experiments often involve special factors known as branching factors. The levels of such factors determine the presence of some certain factors, referred to as nested factors. The design criteria for investigating the goodness of such designs are rarely developed. Furthermore, the existing criteria for such designs pay less attention to the space-filling property of low-dimensional projections of the design. The efficiencies of designs yielded by such criteria can markedly decrease when only a few factors are significant. To address this issue, this paper proposes a novel space-filling criterion based on the maximum projection criterion to evaluate the performance of the designs with branching and nested factors. A framework to construct optimal designs under the proposed criterion is also provided. Compared with the existing works, the resulting designs have better space-filling properties in all possible low-dimensional projections. Moreover, our strategy imposes no constraints on run size, level, and type of any factor, demonstrating its broad applicability.

Stratification pattern enumerator and its applications

Abstract Space-filling designs are widely used in computer experiments. A minimum aberration-type space-filling criterion was recently proposed to rank and assess a family of space-filling designs including orthogonal array-based Latin hypercubes and strong orthogonal arrays. However, it is difficult to apply the criterion in practice because it requires intensive computation for determining the space-filling pattern, which measures the stratification properties of designs on various subregions. In this article, we propose a stratification pattern enumerator to characterise the stratification properties. The enumerator is easy to compute and can efficiently rank space-filling designs. We show that the stratification pattern enumerator is a linear combination of the space-filling pattern. Based on the connection, we develop efficient algorithms for calculating the space-filling pattern. In addition, we establish a lower bound for the stratification pattern enumerator and present construction methods for designs that achieve the lower bound using multiplication tables over Galois fields. The constructed designs have good space-filling properties in low-dimensional projections and are robust under various criteria.

Ensemble learning of multi-kernel Kriging surrogate models using regional discrepancy and space-filling criteria-based hybrid sampling method

Kriging surrogate model has been widely used to simulate expensive models in engineering application. Ensemble of multi-kernel Kriging surrogate models can integrate the information of different kernel functions and enhance the predictive robustness. The performance of the ensemble Kriging model is dependent on the collection of samples. However, the traditional sampling methods primarily concentrate on the accuracy of individual-kernel Kriging, disregarding the difference among different kernel functions. To deal with the issue of sample selection in the context of ensemble Kriging, a regional discrepancy and space-filling criteria-based hybrid sampling (RSHS) method is proposed in this paper. According to the bias-variance decomposition, the expected leave-one-out cross validation error is introduced to compute the weight of each Kriging component. Considering the disagreement of different kernel-based Kriging models, the combination of regional correlation and regional predictive error is derived to measure the regional discrepancy for pursuing local accuracy. The space-filling criterion is also employed to ensure the even coverage across the entire space for global exploration. Therefore, the hybrid sampling criterion is developed to select sample point sequentially by maximizing the regional discrepancy with subject to the constraint of space-filling criterion. The performance of the proposed method is validated by six benchmark examples and an engineering application of airfoil shape model. The results demonstrate the RSHS sampling method can provide promising accuracy and robustness for ensemble Kriging metamodeling.

Searching for optimal Latin hypercube designs by a local greedy strategy

The Latin hypercube design (LHD), because of its one-dimensional projection uniformity, is commonly used in computer experiment. The randomly generated LHD may have too many concentrated design points, and factors may be highly correlated. In this article, we suggested a local greedy strategy for searching optimal LHDs. Our strategy consists of two parts. One is a swap process for doing a local greedy search in a polynomial time. The other is a simulated annealing process for jumping out of the possible local optima. Our strategy is flexible and adapts to various space-filling criteria of LHDs. The simulated experiments illustrated that our proposed algorithm can produce LHDs with well space-filling property and orthogonality. Compared to other classical design algorithms, our algorithm performed better on the criteria related to the point distance and the column correlation. Moreover, for the response surface approximation, the Kriging model using our produced optimal LHD performed more robust on the surface prediction.

Design and Analysis of Computer Experiments with both Numeral and Distributional Inputs

Nowadays stochastic computer simulations with both numeral and distributional inputs are widely used to mimic complex systems which contain a great deal of uncertainty. This article studies the design and analysis issues of such computer experiments. First, we provide preliminary results concerning the Wasserstein distance in probability measure spaces. To handle the product space of the Euclidean space and the probability measure space, we prove that, through the mapping from a point in the Euclidean space to the mass probability measure at this point, the Euclidean space can be isomorphic to the subset of the probability measure space, which consists of all the mass measures, with respect to the Wasserstein distance. Therefore, the product space can be viewed as a product probability measure space. We derive formulas of the Wasserstein distance between two components of this product probability measure space. Second, we use the above results to construct Wasserstein distance-based space-filling criteria in the product space of the Euclidean space and the probability measure space. A class of optimal Latin hypercube-type designs in this product space are proposed. Third, we present a Wasserstein distance-based Gaussian process model to analyze data from computer experiments with both numeral and distributional inputs. Numerical examples and real applications to a metro simulation are presented to show the effectiveness of our methods.

Deterministic construction methods for uniform designs

Space-filling designs are useful for exploring the relationship between the response and factors, especially when the true model is unknown. The wrap-around L2-discrepancy is an important measure of the uniformity, and has often been used as a type of space-filling criterion. However, most obtained designs are generated through stochastic optimization algorithms, and cannot achieve the lower bound of the discrepancies and are only nearly uniform. Then deterministic construction methods for uniform designs are desired. This paper constructs uniform designs under the wrap-around L2-discrepancy by generator matrices of linear codes. Several requirements on the generator matrices, such as a necessary and sufficient condition for generating uniform designs, are derived. Based on these, two simple deterministic constructions for uniform designs are given. Some examples illustrate the effectiveness of them. Moreover, the resulting designs can be regarded as a generalization of good lattice point sets, and also enjoy good orthogonality.

Optimization and experimental research on treelike joints based on generative design and powder bed fusion

Optimization and intelligent construction are of particular interest and important to improve the traditional design and manufacturing methods of cast steel joints. The optimal design, intelligent construction, and experimental research of treelike joints are conducted is this study based on generative design and laser power bed fusion (L-PBF) technology. First, the principle of generative design is comprehensively discussed in accordance of the Audze space-filling criteria. This method is applied to the optimal design of the treelike cast steel joint for Jinming Campus Museum of Henan University. Second, static behaviors of the generative design joints are compared with those of other joints obtained via parametric design through ANSYS version 2022R1 (64-bit) software. Finally, instead of traditional casting, all joints are produced using the L-PBF technology, and the manufactured 316L stainless steel joints are subjected to experimental research and finite element analysis. Results showed the minimum mass, optimal static behaviors, and evenly distributed stress of the generative design joint, thereby indicating that the generative design can effectively optimize treelike cast steel joints. The failure mode of 316Lstainless steel joints obtained via the L-PBF technology is similar to that of traditional cast steel joints but with higher yield load and ultimate bearing capacity. Therefore, employing the generative design and L-PBF technology in the optimization and intelligent construction of cast steel joints is effective and feasible.

A study of orthogonal array-based designs under a broad class of space-filling criteria

Space-filling designs based on orthogonal arrays are attractive for computer experiments for they can be easily generated with desirable low-dimensional stratification properties. Nonetheless, it is not very clear how they behave and how to construct good such designs under other space-filling criteria. In this paper, we justify orthogonal array-based designs under a broad class of space-filling criteria, which include commonly used distance-, orthogonality- and discrepancy-based measures. To identify designs with even better space-filling properties, we partition orthogonal array-based designs into classes by allowable level permutations and show that the average performance of each class of designs is determined by two types of stratifications, with one of them being achieved by strong orthogonal arrays of strength 2+. Based on these results, we investigate various new and existing constructions of space-filling orthogonal array-based designs, including some strong orthogonal arrays of strength 2+ and mappable nearly orthogonal arrays.

Isovolumetric adaptations to space-filling design of experiments

A brief review of methods in design of experiments and criteria to determine space-filling properties of a set of samples is given. Subsequently, the so-called curse of dimensionality in sampling is reviewed and used as motivation for the proposal of an adaptation to the strata creation process in Latin hypercube sampling based on the idea of nested same-sized hypervolumes. The proposed approach places samples closer to design space boundaries, where in higher dimensions the majority of the design space volume is located. The same idea is introduced for Monte Carlo considering an affordable number of samples as an a-posteriori transformation. Both ideas are studied on different algorithms and compared using different distance-based space-filling criteria. The proposed new sampling approach then enables more efficient sampling for optimization especially for high-dimensional problems, i.e. for problems with a high number of design variables.

A minimum aberration-type criterion for selecting space-filling designs

Summary Space-filling designs are widely used in computer experiments. Inspired by the stratified orthogonality of strong orthogonal arrays, we propose a criterion of minimum aberration-type for assessing the space-filling properties of designs based on design stratification properties on various grids. A space-filling hierarchy principle is proposed as a basic assumption of the criterion. The new criterion provides a systematic way of classifying and ranking space-filling designs, including various types of strong orthogonal arrays and Latin hypercube designs. Theoretical results and examples are presented to show that strong orthogonal arrays of maximum strength are favourable under the proposed criterion. For strong orthogonal arrays of the same strength, the space-filling criterion can further rank them based on their space-filling patterns.

Gaussian Process Assisted Active Learning of Physical Laws

ABSTRACT In many areas of science and engineering, discovering the governing differential equations from the noisy experimental data is an essential challenge. It is also a critical step in understanding the physical phenomena and prediction of the future behaviors of the systems. However, in many cases, it is expensive or time-consuming to collect experimental data. This article provides an active learning approach to estimate the unknown differential equations accurately with reduced experimental data size. We propose an adaptive design criterion combining the D-optimality and the maximin space-filling criterion. In contrast to active learning for other regression models, the D-optimality here requires the unknown solution of the differential equations and derivatives of the solution. We estimate the Gaussian process (GP) regression models from the available experimental data and use them as the surrogates of these unknown solution functions. The derivatives of the estimated GP models are derived and used to substitute the derivatives of the solution. Variable selection-based regression methods are used to learn the differential equations from the experimental data. Through multiple case studies, we demonstrate the proposed approach outperforms the D-optimality and the maximin space-filling design alone in terms of model accuracy and data economy.

On Design Orthogonality, Maximin Distance, and Projection Uniformity for Computer Experiments

Space-filling designs are widely used in both computer and physical experiments. Column-orthogonality, maximin distance, and projection uniformity are three basic and popular space-filling criteria proposed from different perspectives, but their relationships have been rarely investigated. We show that the average squared correlation metric is a function of the pairwise L 2-distances between the rows only. We further explore the connection between uniform projection designs and maximin L 1-distance designs. Based on these connections, we develop new lower and upper bounds for column-orthogonality and projection uniformity from the perspective of distance between design points. These results not only provide new theoretical justifications for each criterion but also help in finding better space-filling designs under multiple criteria. Supplementary materials for this article are available online.

Versatile sequential sampling algorithm using Kernel Density Estimation

Understanding the physical mechanisms governing scientific and engineering systems requires performing experiments. Therefore, the construction of the Design of Experiments (DoE) is paramount for the successful inference of the intrinsic behavior of such systems. There is a vast literature on one-shot designs such as low discrepancy sequences and Latin Hypercube Sampling (LHS). However, in a sensitivity analysis context, an important property is the stochasticity of the DoE which is partially addressed by these methods. This work proposes a new stochastic, iterative DoE – named KDOE – based on a modified Kernel Density Estimation (KDE). It is a two-step process: (i) candidate samples are generated using Markov Chain Monte Carlo (MCMC) based on KDE, and (ii) one of them is selected based on some metric. The performance of the method is assessed by means of the C2-discrepancy space-filling criterion. KDOE appears to be as performant as classical one-shot methods in low dimensions, while it presents increased performance for high-dimensional parameter spaces. It is a versatile method which offers an alternative to classical methods and, at the same time, is easy to implement and offers customization based on the objective of the DoE.

Metamodel-based design optimization employing a novel sequential sampling strategy

PurposeMetamodels are widely used to replace simulation models in engineering design optimization to reduce the computational cost. The purpose of this paper is to develop a novel sequential sampling strategy (weighted accumulative error sampling, WAES) to obtain accurate metamodels and apply it to improve the quality of global optimization.Design/methodology/approachA sequential single objective formulation is constructed to adaptively select new sample points. In this formulation, the optimization objective is to select a sample point with the maximum weighted accumulative predicted error obtained by analyzing data from previous iterations, and a space-filling criterion is introduced and treated as a constraint to avoid generating clustered sample points. Based on the proposed sequential sampling strategy, a two-step global optimization approach is developed.FindingsThe proposed WAES approach and the global optimization approach are tested in several cases. A comparison has been made between the proposed approach and other existing approaches. Results illustrate that WAES approach performs the best in improving metamodel accuracy and the two-step global optimization approach has a great ability to avoid local optimum.Originality/valueThe proposed WAES approach overcomes the shortcomings of some existing approaches. Besides, the two-step global optimization approach can be used for improving the optimization results.

Flexible Nested Latin Hypercube Designs for Computer Experiments

Multi-level and sequential computer experiments are commonly used to study complex systems in engineering and science. Suitable designs for such experiments require nested structures. Because a nested Latin hypercube design (NLHD) (Qian (2009)) can be constructed only when the run size in each larger design is a multiple of that in a smaller one, the use of NLHDs faces some limitations in practice. In this paper, we propose a random sampling procedure for constructing flexible NLHDs, which have no such a restriction. We also develop an efficient sequential algorithm to search for the optimal NLHDs with respect to some space-filling criteria. Simulation results show that our methods can yield flexible NLHDs with good space-filling properties.

Sequential forward-inverse design for genetic network modeling

This paper proposes methods for forward and inverse system modeling using Bayesian and least squares regression. These methods are based on both space-filling design criteria for multiple response problems and linear optimality criteria focusing on D-optimality. Modeling with and without the constant term is considered motivated by the case study application of genetic network modeling. We propose extended one-factor-at-a-time experimentation followed by augmentation of next stage design which offers biologists simplicity. Results are illustrated both numerical examples, a test problem from the literature, and a case study motivated by an real world biological research related to genetic network modeling.

A novel extension algorithm for optimized Latin hypercube sampling

ABSTRACTLatin hypercube sampling (LHS), as an efficient sampling method, has been widely used in computer experiments. But it is difficult to choice the sample size while applying LHS, especially for expensive simulations. The effective way is to add sample points sequentially. Nevertheless, the oversampling problem may be countered while extending the sample size with the existing extension algorithms of LHS. To alleviate this problem and obtain extension sample with good space-filling properties, a novel extension algorithm of optimized LHS (OLHS) is proposed. According to the extending rule, a new LHS is constructed by adding sample points of size n each time firstly. Then each additional sample points are optimized by the enhanced stochastic evolutionary algorithm based on the space-filling criterion. The extension algorithm is illustrated by two test functions and appears to perform well in both efficiency and convergence compared with the traditional extension algorithm.

Optimal design of computer experiments for surrogate models with dimensionless variables

This paper presents a method for constructing optimal design of experiments (DoE) intended for building surrogate models using dimensionless (or non-dimensional) variables. In order to increase the fidelity of the model obtained by regression, the DoE needs to optimally cover the dimensionless space. However, in order to generate the data for the regression, one still needs a DoE for the physical variables, in order to carry out the simulations. Thus, there exist two spaces, each one needing a DoE. Since the dimensionless space is always smaller than the physical one, the challenge for building a DoE is that the relation between the two spaces is not bijective. Moreover, each space usually has its own domain constraints, which renders them not-surjective. This means that it is impossible to design the DoE in one space and then automatically generate the corresponding DoE in the other space while satisfying the constraints from both spaces. The solution proposed in the paper transforms the computation of the DoE into an optimization problem formulated in terms of a space-filling criterion (maximizing the minimum distance between neighboring points). An approach is proposed for efficiently solving this optimization problem in a two steps procedure. The method is particularly well suited for building surrogates in terms of dimensionless variables spanning several orders of magnitude (e.g. power laws). The paper also proposes some variations of the method; one when more control is needed on the number of levels on each non-dimensional variable and another one when a good distribution of the DoE is desired in the logarithmic scale. The DoE construction method is illustrated on three case studies. A purely numerical case illustrates each step of the method and two other, mechanical and thermal, case studies illustrate the results in different configurations and different practical aspects.

Revealing the surface deformation induced by deep CO2 injection in vegetated/agricultural areas: The combination of corner-reflectors, reservoir simulations and spatio-temporal statistics

The performance of Permanent Scatterers PS Interferometry (PSI) analysis is highly limited where the presence of large vegetated cover (agricultural terrains/forests) reduces signal coherence. A possible solution relies on the installation on ground of artificial devices (Corner Reflectors CR) to complement the existing PS network. Yet, the number of such CRs (spatial density typically ~1/km2) can be limited when the deformation pattern affects a large area (tens of km2) especially for CO2 geological storage in open aquifers. In order to support the surveillance of such sites, we address here the question of how to estimate the spatio-temporal distribution of the ground displacements over the whole area using only the sparse CR+PS network. We propose to test the feasibility of the Geostatistics Output Perturbation (GOP) method, enabling to combine: 1. The results of the reservoir model calibrated on the limited number of ground displacements' time series and, 2. The spatial correlation between such calibrated results and the observations. A test case was constructed using the signal measured during CO2 injection (from 2004–2009) in the KB501 well of the In-Salah site, Algeria at only a few tens of spatial measurement points either corresponding to: 1. a “realistic” PS network selected in a highly vegetated region in western France or 2. a series of CR selected through a space-filling criterion. The comparison with the observations over the whole area confirmed that 80% of the temporal observations were fitted by the GOP method over the region defined by the CR network with a density of ~1CR/km2. Comparisons with the calibrated model results and with the direct application of a spatio-temporal kriging confirmed the better performance of GOP as well.