Integration of Normative Decision-Making and Batch Sampling for Global Metamodeling

The cost of adaptive sampling for global metamodeling depends on the total number of costly function evaluations and to which degree these evaluations are performed in parallel. Conventionally, samples are taken through a greedy sampling strategy that is optimal for either a single sample or a handful of samples. The limitation of such an approach is that they compromise optimality when more samples are taken. In this paper, we propose a thrifty adaptive batch sampling (TABS) approach that maximizes a multistage reward function to find an optimal sampling policy containing the total number of sampling stages, the number of samples per stage, and the spatial location of each sample. Consequently, the first batch identified by TABS is optimal with respect to all potential future samples, the available resources, and is consistent with a modeler’s preference and risk attitude. Moreover, we propose two heuristic-based strategies that reduce numerical complexity with a minimal reduction in optimality. Through numerical examples, we show that TABS outperforms or is comparable with greedy sampling strategies. In short, TABS provides modelers with a flexible adaptive sampling tool for global metamodeling that effectively reduces sampling costs while maintaining prediction accuracy.

A multi-response adaptive sampling approach for global metamodeling

Some adaptive sampling approaches have been developed to efficiently and accurately build global metamodels for the deterministic single-response problems. Most complex engineering problems, however, yield multiple responses during one simulation. This article adjusts the framework of the CV-Voronoi adaptive sampling approach for a multi-response system. In the proposed multi-response CV-Voronoi (mCV-Voronoi) sampling approach, a new strategy that combines a weighted-sum term and an extreme term is presented to properly estimate the cell errors by simultaneously considering the characteristics of multiple responses. The performance of this approach is investigated on 57 multi-response systems and two engineering design problems. The results show that mCV-Voronoi is very promising for global metamodeling of deterministic multi-response systems.

An Adaptive Bayesian Sequential Sampling Approach for Global Metamodeling

Computer simulations have been increasingly used to study physical problems in various fields. To relieve computational budgets, the cheap-to-run metamodels, constructed from finite experiment points in the design space using the design of computer experiments (DOE), are employed to replace the costly simulation models. A key issue related to DOE is designing sequential computer experiments to achieve an accurate metamodel with as few points as possible. This article investigates the performance of current Bayesian sampling approaches and proposes an adaptive maximum entropy (AME) approach. In the proposed approach, the leave-one-out (LOO) cross-validation error estimates the error information in an easy way, the local space-filling exploration strategy avoids the clustering problem, and the search pattern from global to local improves the sampling efficiency. A comparison study of six examples with different types of initial points demonstrated that the AME approach is very promising for global metamodeling.

A Robust Error-Pursuing Sequential Sampling Approach for Global Metamodeling Based on Voronoi Diagram and Cross Validation

Surrogate models are widely used in simulation-based engineering design and optimization to save the computing cost. The choice of sampling approach has a great impact on the metamodel accuracy. This article presents a robust error-pursuing sequential sampling approach called cross-validation (CV)-Voronoi for global metamodeling. During the sampling process, CV-Voronoi uses Voronoi diagram to partition the design space into a set of Voronoi cells according to existing points. The error behavior of each cell is estimated by leave-one-out (LOO) cross-validation approach. Large prediction error indicates that the constructed metamodel in this Voronoi cell has not been fitted well and, thus, new points should be sampled in this cell. In order to rapidly improve the metamodel accuracy, the proposed approach samples a Voronoi cell with the largest error value, which is marked as a sensitive region. The sampling approach exploits locally by the identification of sensitive region and explores globally with the shift of sensitive region. Comparative results with several sequential sampling approaches have demonstrated that the proposed approach is simple, robust, and achieves the desired metamodel accuracy with fewer samples, that is needed in simulation-based engineering design problems.

Approximation Assisted Multiobjective Optimization With Combined Global and Local Metamodeling

Approximation Assisted Optimization (AAO) is widely used in engineering design problems to replace computationally intensive simulations with metamodeling. Traditional AAO approaches employ global metamodeling for exploring an entire design space. Recent research works in AAO report on using local metamodeling to focus on promising regions of the design space. However, very limited works have been reported that combine local and global metamodeling within AAO. In this paper, a new approximation assisted multiobjective optimization approach is developed. In the proposed approach, both global and local metamodels for objective and constraint functions are used. The approach starts with global metamodels for objective and constraint functions and using them it selects the most promising points from a large number of randomly generated points. These selected points are then “observed”, which means their actual objective/constraint function values are computed. Based on these values, the “best” points are grouped in multiple clustered regions in the design space and then local metamodels of objective/constraint functions are constructed in each region. All observed points are also used to iteratively update the metamodels. In this way, the predictive capabilities of the metamodels are progressively improved as the optimizer approaches the Pareto optimum frontier. An advantage of the proposed approach is that the most promising points are observed and that there is no need to verify the final solutions separately. Several numerical examples are used to compare the proposed approach with previous approaches in the literature. Additionally, the proposed approach is applied to a CFD-based engineering design example. It is found that the proposed approach is able to estimate Pareto optimum points reasonably well while significantly reducing the number of function evaluations.

On Sequential Sampling for Global Metamodeling in Engineering Design

Approximation models (also known as metamodels) have been widely used in engineering design to facilitate analysis and optimization of complex systems that involve computationally expensive simulation programs. The accuracy of metamodels is directly related to the sampling strategies used. Our goal in this paper is to investigate the general applicability of sequential sampling for creating global metamodels. Various sequential sampling approaches are reviewed and new approaches are proposed. The performances of these approaches are investigated against that of the one-stage approach using a set of test problems with a variety of features. The potential usages of sequential sampling strategies are also discussed.