Space-filling Designs Research Articles

Optimization of the three-segment aerofoil arrangement based on wind tunnel tests and numerical analysis

This paper presents the optimization of multi-element aerofoil for the LAR-3 Puffin -- STOL light transport aircraft concept proposal. Based on the geometry and aerodynamic characteristics of the well-known and proven in flight three-segment NACA 63A416 aerofoil, the authors explore the possibility of enhancing its high-lift performance by the movement of slot and flap position in extended (deployed) aerodynamic configuration. In order to determine the optimum positions of aerofoil segments (elements), a multi-step optimization approach was developed. It combines computational fluid dynamics simulations that were used for design space screening and preliminary optimization together with low-turbulence wind tunnel tests which yielded certain results. To decrease the numerical cost of the computer simulation campaign, Design of experiment methods (optimal space-filling design among others) were employed instead of exhausting full factorial (parametric) design. Response surface models of major aerodynamic coefficients (lift, drag, pitching moment) at predicted maximum lift coefficient ( C L max) point allowed to narrow down search space and identify several candidates for optimal configuration to be checked experimentally. Wind tunnel tests campaign confirmed the major trends observed in computational fluid dynamics derived response surface contour plots. For the optimum aerodynamic configuration, chosen experimental C L max is over 3.9, which is a 10% increase over the baseline (initial slat and flap positions) case. In parallel, the maximum lift-to-drag ratio gain at that point was almost 19%. The research outlined in this paper was conducted on behalf of the aircraft production company and its results will be applied in a newly designed transport aircraft.

A budget allocation strategy minimizing the sample set quantile for initial experimental design

The increased complexity of manufacturing systems makes the acquisition of the system performance estimate a black-box procedure (e.g., simulation tools). The efficiency of most black-box optimization algorithms is affected significantly by initial designs (populations). In most population initializers, points are spread out to explore the entire domain, e.g., space-filling designs. Some population initializers also consider exploitation procedures to speed up the optimization process. However, they are either application-dependent or require an additional budget. This article proposes a generic method to generate, without an additional budget, several good solutions in the initial design. The aim of the method is to optimize the quantile of the objective function values in the generated sample set. The proposed method is based on a clustering of the solution space; feasible solutions are clustered into groups and the budget is allocated to each group dynamically based on the observed information. The asymptotic performance of the proposed method is analyzed theoretically. The numerical results show that, if proper clustering rules are applied, an unbalanced design is generated in which promising solutions have higher sampling probabilities than non-promising solutions. The numerical results also show that the method is robust to wrong clustering rules.

A comparison of experimental designs for calibration

The impact of experimental design choice on the performance of statistical calibration is largely unknown. Calibration is a technique that uses available experimental data to model the relationship between input and response variables to ultimately infer inputs based on newly observed response values. The purpose of this article is to investigate the performance of several experimental designs with regards to inverse prediction via a comprehensive simulation study. Specifically, we compare several design types including traditional response surface designs, algorithmically generated variance optimal designs, and space-filling designs. Results indicate that the choice of design has an impact on calibration performance and provides overall support for the use of I-optimal designs.

Optimal dose calibration in radiotherapy

In this paper, the tools provided by the theory of Optimal Experimental Design are applied to a nonlinear calibration model. This is motivated by the need of estimating radiation doses using radiochromic films for radiotherapy purposes. The calibration model is in this case nonlinear and the explanatory variable cannot be worked out explicitly from the model. In this case an experimental design has to be found on the dependent variable. For that, the inverse function theorem will be used to obtain an information matrix to be optimized. Optimal designs on the response variable are computed from two different perspectives, first for fitting the model and estimating each of the parameters and then for predicting the proper dose. While the first is a common point of view in a general context of the Optimal Experimental Design, the latter is actually the main objective of the calibration problem for the practitioners and algorithms for computing these optimal designs are also provided.The optimal designs obtained have just three different points in their support, but practitioners usually demand for more support points. Thus, a methodology for computing space-filling designs is also provided when the support points are forced to follow some mathematical rule, such as arithmetic or geometric sequences. Cross efficiencies of all these designs are computed in order to show their ability for different goals.

Support vector regression based metamodel by sequential adaptive sampling for reliability analysis of structures

Support vector regression (SVR) based metamodel is a powerful mean to alleviate computational challenge of Monte Carlo simulation (MCS) based reliability analysis of structure involving implicit limit state function. But, the sample size requirement is an important issue to achieve accuracy of estimated reliability. A two-stage iterative algorithm is explored to address this issue. The algorithm is hinged on the prediction accuracy of a metamodel near the failure surface region. In the first stage, an initial design of experiment is built by a space-filling design over the entire physical domain of the random variables. In the next stage, based on the prediction at MCS points using the previous SVR model, a subset of MCS samples are selected. These are now used to enrich existing design by adding more data points sequentially such that the new points are closer to the limit state and also as far as possible from the existing points. A comparative performance of reliability estimate by SVR with the proposed sequential adaptive approach and that of obtained by the relevance vector machines, Kriging and moving least square method based metamodels are performed to numerically demonstrate the improved reliability estimation capability of the proposed approach.

Weighted symmetrized centered discrepancy for uniform design

Uniformity and projection uniformity are very important criteria in space-filling design and studied by many authors. In this paper, a new criterion, called weighted symmetrized centered discrepancy (WSCD), is put forwarded for measuring the uniformity of design array. Firstly, a new kernel function is deduced by symmetrizing the reproducing kernel of centered L 2 discrepancy (CD). The new criterion WSCD is generated from the symmetrized the reproducing kernel, which make it avoid the well-known shortcomings of CD. Secondly, the customized weights are set to all the sub-dimensions to conform with the Effect Hierarchy principle. The WSCD has similar computation formula with the popularly used discrepancies, hence all the construction methods for uniform design still work. Thirdly, the weights choice is also discussed, and a practical suggestion is made. A naive empirical comparison is also conducted to show that the new criterion performs better than those existed discrepancies in the variable screening.

Non-uniform space filling (NUSF) designs

Space-filling designs are a convenient and effective approach for exploring the input space for experiments. However, standard choices for these designs strive to provide uniform density of points throughout the region of interest. There are numerous situations where flexibility to adapt the density of points to match specific design objectives would be advantageous to maximize the efficiency of the design. In this paper, we propose non-uniform space-filling (NUSF) designs to achieve a user-specified desired density distribution of design points across the input space and demonstrate how to implement NUSF designs in different ways to provide the experimenters with flexibility to match their goals. The approach is flexible for a variety of scenarios where the experimenter wishes to control the density of points throughout the region while still preserving the space-filling characteristic. Details are provided about how to translate a problem into an appropriate weight structure to generate several designs which can then be compared using graphical methods, including the Closest Distance by Weight plot, to determine if the desired characteristics have been achieved. The methods are demonstrated with two real examples with different requirements for design point placement.

Fully-sequential space-filling design algorithms for computer experiments

Fully-sequential (i.e., with design points added one-at-a-time) space-filling designs are useful for global surrogate modeling of expensive computer experiments when the number of design points required to achieve a suitable accuracy is unknown in advance. We develop and investigate three fully-sequential space-filling (FSSF) design algorithms that are conceptually simple and computationally efficient and that achieve much better space-filling properties than alternative methods such as Sobol sequences and more complex batch-sequential methods based on sliced or nested optimal Latin hypercube designs (LHDs). Remarkably, at each design size in the sequence, our FSSF algorithms even achieve much better space-filling properties than a one-shot LHD optimized for that specific size. The algorithms we propose also scale well to very large design sizes. We provide an R package to implement the approaches.

Construction of orthogonal marginally coupled designs

Marginally coupled designs (MCDs) were first introduced by Deng et al. (Stat Sin 25:1567–1581, 2015), as more economical designs than sliced space-filling designs which are the popular choices for computer experiments with both qualitative and quantitative factors. In an MCD, the design for qualitative factors is an orthogonal array, and the one for quantitative factors is a Latin hypercube design (LHD) with its rows corresponding to each level of any qualitative factor also forming a small LHD. As we know, orthogonality is a popular and important property for evaluating LHDs, but was not considered in existing results on MCDs. In this paper, we propose some approaches to constructing a new class of MCDs with orthogonality. In some cases, the designs for quantitative factors also satisfy the two dimensional space-filling property. Besides, the run sizes of the obtained designs are more flexible than the existing ones.

Bayesian Quadrature, Energy Minimization, and Space-Filling Design

A standard objective in computer experiments is to approximate the behavior of an unknown function on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, covering radius) or a discrepancy criterion measuring distance to uniformity. The paper investigates connections between design for integration (quadrature design), construction of the (continuous) BLUE for the location model, space-filling design, and minimization of energy (kernel discrepancy) for signed measures. Integrally strictly positive definite kernels define strictly convex energy functionals, with an equivalence between the notions of potential and directional derivative, showing the strong relation between discrepancy minimization and more traditional design of optimal experiments. In particular, kernel herding algorithms, which are special instances of vertex-direction methods used in optimal design, can be applied to the construction of point sequences with suitable space-filling properties.

On considering the output in space-filling test signal designs for the identification of dynamic Takagi-Sugeno models

The model-based design of test signals for the identification of dynamical Takagi-Sugeno (TS) fuzzy models is addressed. The multi-model structure is exploited to reduce computational cost. Space-filling designs usually only address the input but the nonlinear behavior of dynamic systems depends on the lagged output in general. This is considered as an additional constraint regarding the test signal design. This contribution investigates whether a control input can be calculated exploiting the local structure and whether considering the output in space-filling designs yields identified models of higher quality.

Parallel 3d shape optimization for cellular composites on large distributed-memory clusters

Skin modeling is an ongoing research area that highly benefits from modern parallel algorithms. This article aims at applying shape optimization to compute cell size and arrangement for elastic energy minimization of a cellular composite material model for the upper layer of the human skin. A gradient-penalized shape optimization algorithm is employed and tested on the distributed-memory cluster Hazel Hen, HLRS, Germany. The performance of the algorithm is studied in two benchmark tests. First, cell structures are optimized with respect to purely geometric aspects. The model is then extended such that the composite is optimized to withstand applied deformations. In both settings, the algorithm is investigated in terms of weak and strong scalability. The results for the geometric test reflect Kelvin's conjecture that the optimal space-filling design of cells with minimal surface is given by tetrakaidecahedrons. The PDE-constrained test case is chosen in order to demonstrate the influence of the deformation gradient penalization on fine inter-cellular channels in the composite and its influence on the multigrid convergence. A scaling study is presented for up to 12,288 cores and 3 billion DoFs.

Construction results for strong orthogonal arrays of strength three

Strong orthogonal arrays were recently introduced as a class of space-filling designs for computer experiments. The most attractive are those of strength three for their economical run sizes. Although the existence of strong orthogonal arrays of strength three has been completely characterized, the construction of these arrays has not been explored. In this paper, we provide a systematic and comprehensive study on the construction of these arrays, with the aim at better space-filling properties. Besides various characterizing results, three families of strength-three strong orthogonal arrays are presented. One of these families deserves special mention, as the arrays in this family enjoy almost all of the space-filling properties of strength-four strong orthogonal arrays, and do so with much more economical run sizes than the latter. The theory of maximal designs and their doubling constructions plays a crucial role in many of theoretical developments.

An application study of response surface method on axial gadolinium designs of BWR fuel assemblies

To solve the optimization problem of a BWR fuel assembly axial gadolinium (Gd) design, a response surface method was applied to approximate the objective function of the design to create a surrogate model for optimization search. The response surface function consists of a mean value and an error term that are modeled by using Kriging function. An initial response surface model is built based on space-filling design, and further improvement of the model is achieved by adaptively sampling the promising design space that have high probability of better solution. The objective function used for evaluating the axial Gd designs is the weighted sum of core design parameters such as MFLPD, MFLCPR and CSDM. Applications of this optimization approach showed satisfactory results that meet all the core design requirements. It was demonstrated that this optimization approach is applicable for axial Gd design of BWR fuel assembly.

Free energy calculation using space filled design and weighted reconstruction: a modified single sweep approach

ABSTRACTA modified single sweep approach is proposed for generating free energy landscapes. The approach replaces the use of temperature-accelerated molecular dynamics (TAMD) to generate centres in collective variable (CV) space at which mean forces are computed using restrained molecular dynamics (MD) simulations with a sequential space-filling design. This approach also modifies the radial basis function reconstruction step of the traditional single sweep approach and proposes a weighted reconstruction of the free energy surface using the previously generated mean forces. The modified approach is compared to the traditional single sweep (SS) approach on the (φ, ψ) dihedral free-energy map of solvated alanine dipeptide (AD). It is found that the new approach results in a more accurate reconstructed free energy than does the traditional approach when compared to the directly-computed reference free energy landscape. It is shown that the increased accuracy of the overall map stems from the improved 1-dimensional space filling (projective) property of the proposed design compared to that of the TAMD generated centres. A further enhancement in the accuracy of the crucial lower energy regions is enabled by the introduction of weights in the reconstruction step that give more importance to lower energy-valued regions.

Distance-Distributed Design for Gaussian Process Surrogates

A common challenge in computer experiments and related fields is to efficiently explore the input space using a small number of samples, that is, the experimental design problem. Much of the recent focus in the computer experiment literature, where modeling is often via Gaussian process (GP) surrogates, has been on space-filling designs, via maximin distance, Latin hypercube, etc. However, it is easy to demonstrate empirically that such designs disappoint when the model hyperparameterization is unknown, and must be estimated from data observed at the chosen design sites. This is true even when the performance metric is prediction-based, or when the target of interest is inherently or eventually sequential in nature, such as in blackbox (Bayesian) optimization. Here we expose such inefficiencies, showing that in many cases a purely random design is superior to higher-powered alternatives. We then propose a family of new schemes by reverse engineering the qualities of the random designs which give the best estimates of GP length scales. Specifically, we study the distribution of pairwise distances between design elements, and develop a numerical scheme to optimize those distances for a given sample size and dimension. We illustrate how our distance-based designs, and their hybrids with more conventional space-filling schemes, outperform in both static (one-shot design) and sequential settings.

Fast and simple dataset selection for machine learning

Abstract The task of data reduction is discussed and a novel selection approach which allows to control the optimal point distribution of the selected data subset is proposed. The proposed approach utilizes the estimation of probability density functions (pdfs). Due to its structure, the new method is capable of selecting a subset either by approximating the pdf of the original dataset or by approximating an arbitrary, desired target pdf. The new strategy evaluates the estimated pdfs solely on the selected data points, resulting in a simple and efficient algorithm with low computational and memory demand. The performance of the new approach is investigated for two different scenarios. For representative subset selection of a dataset, the new approach is compared to a recently proposed, more complex method and shows comparable results. For the demonstration of the capability of matching a target pdf, a uniform distribution is chosen as an example. Here the new method is compared to strategies for space-filling design of experiments and shows convincing results.

Evolving a Psycho-Physical Distance Metric for Generative Design Exploration of Diverse Shapes

AbstractIn this paper, a generative design approach is proposed that involves the users’ psychological aspect in the design space exploration stage to create distinct design alternatives. Users’ perceptual judgment about designs is extracted as a psycho-physical distance metric, which is then integrated into the design exploration step to generate design alternatives for the parametric computer-aided design (CAD) shapes. To do this, a CAD model is first parametrized by defining geometric parameters and determining ranges of these parameters. Initial design alternatives for the CAD model are generated using Euclidean distance-based sampling teaching–learning-based optimization (S-TLBO), which is recently proposed and can sample N space-filling design alternatives in the design space. Similar designs are then clustered, and a user study is conducted to capture the subjects’ perceptual response for the dissimilarities between the cluster pairs. In addition, a furthest-point-sorting technique is introduced to equalize the number of designs in the clusters, which are being compared by the subjects in the user study. Afterward, nonlinear regression analyses are carried out to construct a mathematical correlation between the subjects’ perceptual response and geometric parameters in the form of a psycho-physical distance metric. Finally, a psycho-physical distance metric obtained is utilized to explore distinct design alternatives for the CAD model. Another user study is designed to compare the diversification between the designs when the Euclidean and the suggested psycho-physical distance metrics are utilized. According to the user study, designs generated with the latter metric are more distinct.

Data-driven stochastic inversion via functional quantization

In this paper, we propose a new methodology for solving stochastic inversion problems through computer experiments, the stochasticity being driven by a functional random variables. This study is motivated by an automotive application. In this context, the simulator code takes a double set of simulation inputs: deterministic control variables and functional uncertain variables. This framework is characterized by two features. The first one is the high computational cost of simulations. The second is that the probability distribution of the functional input is only known through a finite set of realizations. In our context, the inversion problem is formulated by considering the expectation over the functional random variable. We aim at solving this problem by evaluating the model on a design, whose adaptive construction combines the so-called stepwise uncertainty reduction methodology with a strategy for an efficient expectation estimation. Two greedy strategies are introduced to sequentially estimate the expectation over the functional uncertain variable by adaptively selecting curves from the initial set of realizations. Both of these strategies consider functional principal component analysis as a dimensionality reduction technique assuming that the realizations of the functional input are independent realizations of the same continuous stochastic process. The first strategy is based on a greedy approach for functional data-driven quantization, while the second one is linked to the notion of space-filling design. Functional PCA is used as an intermediate step. For each point of the design built in the reduced space, we select the corresponding curve from the sample of available curves, thus guaranteeing the robustness of the procedure to dimension reduction. The whole methodology is illustrated and calibrated on an analytical example. It is then applied on the automotive industrial test case where we aim at identifying the set of control parameters leading to meet the pollutant emission standards of a vehicle.

Modification of the Maximin and ϕp (Phi) Criteria to Achieve Statistically Uniform Distribution of Sampling Points

This article proposes a sampling technique that delivers robust designs, that is, point sets selected from a design domain in the shape of a unit hypercube. The designs are guaranteed to provide a statistically uniform point distribution, meaning that every location has the same probability of being selected. Moreover, the designs are sample uniform, meaning that each individual design has its points spread evenly throughout the domain. The sample uniformity (often measured via a discrepancy criterion) is achieved using distance-based criteria ( or Maximin), that is, criteria normally used in space-filling designs. We show that the standard intersite metrics employed in distance-based criteria (Maximin and (phi)) do not deliver statistically uniform designs. Similarly, designs optimized via centered L2 discrepancy or support points are also not statistically uniform. When these designs (after optimization based on intersite distances) are used for Monte Carlo type of integration, their statistical nonuniformity is a serious problem as it may lead to a systematic bias. This article proposes using a periodic metric to guarantee the statistical uniformity of the family of distance-based designs. The presented designs used as benchmarks in the article are only taken from the class of Latin hypercube designs, which forces univariate projections to be uniform and improves accuracy in Monte Carlo integration of some functions. Supplementary materials for this article are available online.