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The paper introduces a practical method for the construction of large-scale point sets for analysis of computer models. The constructed experimental design is useful for (probabilistic) integration, construction of approximation or a screening. The essence of the presented approach is the stratification of the design domain into an orthogonal grid of substrata and a subsequent tiling with tiles of points. If optimized, such tiles experience a major reduction of the number of degrees of freedom in the optimization process. That way, optimal or near-optimal point patterns can be feasibly identified and are further utilized for construction of larger point sets, thanks to the idea of self-similarity and structured space stratification The space-filling properties of the resulting point sets may be further enhanced by various "scrambling" strategies, which may remove the undesired sample collapsibility achieved via regular tiling. The performance of the constructed point sets is compared to Quasi Monte Carlo (QMC), Randomized Quasi Monte Carlo (RQMC) sequences, which are still today considered by engineers and even scientists as choices for variance reduction of numerical integration Further, the mentioned sampling strategies are compared in the terms of robustness when integrating a multivariate function with a localized feature. It is concluded that the proposed sampling approach reaches a superior performance in numerical integration and identification of function extremes as compared to sampling methods used by practicing researchers and engineers. Additionally, the reader is supplied with the open-access, ready-to-use implementation of the presented algorithm named SampleTiler. • Construction of large-scale point samples by balancing the bias–variance tradeoff. • Stratification of the design domain and subsequent tiling with tiles of points. • Construction of large optimized experimental designs due to self-similarity. • Superior performance of samples as compared to commonly used methods. • Open-access Python code provided on a GitHub repository. [ABSTRACT FROM AUTHOR]

Highlights • Rasterized approximation of hyper-dimensional Voronoi diagram. • Exctraction of selected scalar topological attributes of cells. • Efficient massively parallel solution providing significant speedup in comparison with full analytic solution. • Explicit estimation of rasterization error in general dimension. Abstract This paper concerns an algorithm for fast parallel approximation of selected attributes of hyper-dimensional Voronoï cells in a unit hypercube. The presented algorithm does not require the construction of a corresponding Voronoï diagram (usually by employing the Quick Hull algorithm) which typically is a highly computationally demanding task, especially when performed in higher dimensions. The algorithm is suitable for both the clipped and periodic variants of Voronoï tessellation and provides a significant speedup in a convenient range of practical usage. For the purposes of approximation of selected scalar properties of Voronoï cells, only the distances of points in sample are evaluated using an adequately fine underlying orthogonal mesh. The algorithm estimates e.g. the volumes and centroids of Voronoï cells, radii and coordinates of centers of the corresponding Delaunay hyper-circles. The paper also provides quite accurate explicit error estimators for the extracted attributes due to rasterization and thus the user is given a control over the accuracy by selecting an appropriate discretization density. Among numerous fields in which Voronoï diagrams are being utilized, the authors are concerned with optimization of point samples for the Design of Experiments and also with weighting of integration points in Monte Carlo type integration. In these applications, selected scalar topological descriptors of Voronoï diagrams are being repeatedly computed. As the full Voronoï diagram is not of interest but the resulting cell volumes or shape descriptions have to be repeatedly computed, the presented parallel solution seems highly suitable for these applications. [ABSTRACT FROM AUTHOR]

Highlights • Efficient parallel implementation of multi-dimensional particle system using GPGPU. • Analogy between potential energy of particle system and point distribution evaluation in Design of Experiments. • Periodically repeated design domain. • Generalized utilization of on-chip memory. • Efficient serialization of thread writing requests. Abstract The presented paper deals with possible approaches to parallel implementation of solution of a hyper-dimensional dynamical particle system. The proposed implementation approaches are generally applicable for similar particle systems of interest in various research and engineering fields. The original motivation for the present work was a simulation of particles that represent a space-filling design to be optimized for further use in design of experiments. Due to the underlying purpose of this particle system, the dimension of the particle system of interest is considered to be entirely arbitrary. Such a hyper-dimensional space is further folded into a periodically repeated domain. The theoretical background of the proposed particle system is provided along with the derivation of equations of motion of the dynamical system. As the complexity of the system is not limited by the number of particles nor the number of dimensions, the possibilities of utilizing the GPGPU platform are more restricted in comparison with today's fast parallel implementations of common particle systems. Two distinct approaches to parallel implementation are presented, one aiming at a generalized usage of the fast on-chip resources, the other entirely relying on the GPU's on-board global memory. Despite unambiguous mutual differences in their performance, both parallel implementations deliver major speedup compared to the single-thread CPU solution as well as a better scaling of execution time when increasing both the number of particles and dimensions. [ABSTRACT FROM AUTHOR]

The Audze–Eglājs (AE) criterion was developed to achieve a uniform distribution of experimental points in a hypercube. However, the paper shows that the AE criterion provides strongly nonuniform designs due to the effect of the boundaries of the hypercube. We propose a simple remedy that lies in the assumption of periodic boundary conditions. The biased behavior of the original AE criterion and excellent performance of the modified criterion are demonstrated using simple numerical examples focused on (i) the uniformity of sampling density over the design space and, (ii) statistical sampling efficiency measured through the ability to correctly estimate the statistical parameters of functions of random variables. An engineering example of reliability calculation is presented, too. [ABSTRACT FROM AUTHOR]

Highlights: [•] FReET software for statistical and reliability analyses has been developed. [•] Small-sample simulation techniques have been implemented. [•] Degradation modeling enables degradation module FReET-D. [•] Applications of software for reliability analysis of concrete structures are listed. [ABSTRACT FROM AUTHOR]

• Major refinements of the distance-based criterion for uniform space-filling designs. • The lower bound on the exponent in ϕ p criterion for space-filling design is derived. • The improved criterion becomes useful for small-sample designs and high dimensions. • The correct shape of the interaction domain for a pair of points is proposed. • Energy and force contrast in high dimensions is derived for the Euclidean metric. In this paper, the family of ϕ p optimization criteria for space-filling designs is critically reviewed, with a focus on its behavior in moderate to large dimensions, especially for small sample sizes (low saturations of the design domain). Problems that arise during the standard use of the ϕ p criteria for the optimization of point sets in standard hypercubic design domains are identified and adequate remedies are proposed. It is shown how the distance exponent in the distance-based criteria should be dependent on the domain dimension. In cases of small sample sizes, we propose utilizing multiple repetitions of a periodic hyper-toroidal domain. We show that the naïve use of the ϕ p criterion for the construction of optimized designs can produce undesired orthogonal grid patterns (either complete or incomplete). We show how this behavior is related to the directional non-uniformity of hypercubical volume considered in the objective function, and we propose a simple remedy that involves limiting the interaction to a rotationally symmetrical neighborhood. Use of the recently proposed minimum image convention may provide too crude an approximation of the full periodic extension of the design space. We propose that a finite but sufficiently large interaction radius be considered for the evaluation of the pairwise potential. The upper bound on the interaction radius can be set to contain a sufficient number of points within the periodically repeated domain. These enhancements are embodied in the proposed ψ p criterion for space-filling designs. We show that the new criterion favors designs with better space-filling property, better projection properties and also with lower discrepancy. Euclidean distances among points within high-dimensional objects tend to concentrate and the resolution between distances decreases. We show that despite the decreasing contrast of distances, the desired resolution ability of the refined criterion is retained even when this isotropic metric is used. [ABSTRACT FROM AUTHOR]

• numerical integration with designs optimized via miniMax (mM) criterion is developed. • mM designs with standard metric are nonuniform which may cause biased integration. • the proposed periodic metric ensures point uniformity and unbiased robust integration. • efficient construction algorithm for mM designs (standard or periodic) is developed. The selection of points for numerical integration of the Monte Carlo type, largely used in analysis of engineering problems, is developed. It is achieved by modification of the metric in the minimax optimality criterion. The standard minimax criterion ensures the design exhibits good space-filling property and therefore reduces the variance of the estimator of the integral. We, however, show that the points are not selected with the same probability over the space of sampling probabilities: some regions are over- or under-sampled when designs are generated repetitively. This violation of statistical uniformity may lead to systematically biased integral estimators. We propose that periodic metric be considered for calculation of the minimax distance. Such periodic minimax criterion provides statistically uniform designs leading to unbiased integration results and also low estimator variance due to retained space-filling property. These conclusions are demonstrated by examples integrating analytical functions. The designs are constructed by two different algorithms: (i) a new time-stepping algorithm resembling a damped system of attracted particles developed here, and (ii) the heuristic swapping of coordinates. The designs constructed by the time-stepping algorithm are attached to the paper as a supplementary material. The computer code for construction of the designs is attached too. [ABSTRACT FROM AUTHOR]

• Uniform distribution of points in a unit hypercube. • Optimal Design of Experiments for computationally intensive models. • The analogy to a dynamical system of repulsive particles in periodic space is exploited. A method is proposed for the construction of uniformly distributed point sets within a design domain using an analogy to a dynamical system of interacting particles. The possibility of viewing various distance-based optimality criteria as formulas representing the potential energy of a system of charged particles is discussed. The potential energy is employed in deriving the equations of motion of the particles. The particles are either attracted or repelled and dissipative dynamical systems can be simulated to achieve optimal and near-optimal arrangements of points. The design domain is set up as an N var -dimensional unit hypercube, with N var being the number of variables (factors). The number of points is equal to the number of simulations (levels). The periodicity assumption of the design domain is shown to be an elegant way to obtain statistically uniform coverage of the design domain. The ϕ p criterion, which is a generalization of the Maximin criterion, is selected in order to demonstrate its analogy with an N-body system. This criterion guarantees that the points are spread uniformly within the design domain. The solution to such an N-body system is presented. The obtained designs are shown to outperform the existing optimal designs in various types of applications: multidimensional numerical integration, statistical exploration of computer models, reliability analyses of engineering systems, and screenings or exploratory designs for the global optimization/minimization of functions. [ABSTRACT FROM AUTHOR]