Metamodeling Techniques Research Articles

Biogeography-based optimization of a variable camshaft timing system

Automotive simulations often prohibit the use of traditional optimization techniques because these simulations are complex and computationally expensive. These two qualities motivate the use of evolutionary algorithms and meta-modeling techniques respectively. In this work, we apply biogeography-based optimization (BBO) to optimize radial basis function (RBF)-based lookup table controls of a variable camshaft timing system for fuel economy. Also, we reduce computational search effort by finding an effective parameterization of the problem, optimizing the parameters of the BBO algorithm for the problem, and estimating the cost of a portion of the candidate solutions in BBO with design and analysis of computer experiments (DACE). We find that we can improve fuel economy by 1.7% over the original control parameters, and we find a tradeoff in population size, and an optimal value for mutation rate. Finally, we find that we can use a small number of samples to construct DACE models, and we can use these models to estimate a significant portion of the candidate solutions each generation to reduce computation effort and still obtain good BBO solutions.

Computer-aided definition of sizing procedures and optimization problems of mechatronic systems

This article is dedicated to the generation of sizing procedures used during the preliminary optimal design of physical parts of mechatronic systems. The knowledge necessary to such design is composed of two layers: mechatronic system layer and component layer. Such knowledge can be represented by algebraic equations directly or by the use of meta-modeling techniques. Constraint networks and graphs theory are used here to represent and order the constraints and the links between those algebraic models in order to obtain a well-posed optimization problem. The selected or developed algorithms are finally illustrated with typical problems of component selection.

An adaptive SVR-HDMR model for approximating high dimensional problems

Purpose – Popular regression methodologies are inapplicable to obtain accurate metamodels for high dimensional practical problems since the computational time increases exponentially as the number of dimensions rises. The purpose of this paper is to use support vector regression with high dimensional model representation (SVR-HDMR) model to obtain accurate metamodels for high dimensional problems with a few sampling points. Design/methodology/approach – High-dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input-output system behavior. Support vector regression (SVR) method can approximate the underlying functions with a small subset of sample points. Dividing Rectangles (DIRECT) algorithm is a deterministic sampling method. Findings – This paper proposes a new form of HDMR by integrating the SVR, termed as SVR-HDMR. And an intelligent sampling strategy, namely, DIRECT method, is adopted to improve the efficiency of SVR-HDMR. Originality/value – Compared to other metamodeling techniques, the accuracy and efficiency of SVR-HDMR were significantly improved. The SVR-HDMR helped engineers understand the essence of underlying problems visually.

Evaluation of convergence behavior of metamodeling techniques for bridging scales in multi-scale multimaterial simulation

The effectiveness of several metamodeling techniques, viz. the Polynomial Stochastic Collocation method, Adaptive Stochastic Collocation method, a Radial Basis Function Neural Network, a Kriging Method and a Dynamic Kriging Method is evaluated. This is done with the express purpose of using metamodels to bridge scales between micro- and macro-scale models in a multi-scale multimaterial simulation. The rate of convergence of the error when used to reconstruct hypersurfaces of known functions is studied. For sufficiently large number of training points, Stochastic Collocation methods generally converge faster than the other metamodeling techniques, while the DKG method converges faster when the number of input points is less than 100 in a two-dimensional parameter space. Because the input points correspond to computationally expensive micro/meso-scale computations, the DKG is favored for bridging scales in a multi-scale solver.

Sensitivity analysis and parameter identification of a time dependent constitutive model for rock salt

The tendency to shift from fossil and nuclear energy sources to renewable energy carriers has increased during the past couple of decades. Subsequently, development of effective energy storage systems has become more attractive. Nowadays, caverns excavated in rock salt formations are recognized as the appropriate places for underground storage of energy in the form of compressed air or hydrogen. Accurate design of these underground cavities requires suitable numerical simulations employing appropriate constitutive models to describe the material behavior of rock salt under various geological conditions. It is obvious that to have a realistic numerical simulation, it is essential to have a comprehensive knowledge concerning the unknown material parameters and their influence on the calculation results. In this paper, a time dependent model is selected to describe the mechanical response of the rock salt around the cavern. This model is implemented in a finite element code and its application in numerical modeling of salt caverns is illustrated. In addition, global sensitivity analysis is used to investigate the influence of material parameters on the mechanical behavior of the salt cavern. Finally, inverse analysis of the synthetic data is performed to identify the material parameters of the selected model. The applied global sensitivity and inverse analysis algorithms employ metamodeling technique in order to reduce the time which is needed for these computationally expensive calculations.

Optimization of traffic forecasting: Intelligent surrogate modeling

Transportation modelers are frequently faced with several optimization challenges related to model selection and parameter optimization for forecasting. The concept of surrogate modeling is discussed in order to tackle some limitations related to the practice of developing short-term forecasting algorithms. An automated meta-modeling technique is presented that uses heterogeneous information from multiple types of statistical and computationally intelligent models, along with multi-objective evolutionary strategies to optimize the model and parameter selection. A number of different models from the family of Support Vector Machines, Radial Base Functions and Neural Networks are jointly considered and optimized with the aim to improve the short-term predictability of travel speed. Results are presented and discussed in both a univariate and multivariate framework.

Multiple-optima search method based on a metamodel and mathematical morphology

This article investigates a non-population-based optimization method using mathematical morphology and the radial basis function (RBF) for multimodal computationally intensive functions. To obtain several feasible solutions, mathematical morphology is employed to search promising regions. Sequential quadratic programming is used to exploit the possible areas to determine the exact positions of the potential optima. To relieve the computational burden, metamodelling techniques are employed. The RBF metamodel in different iterations varies considerably so that the positions of potential optima are moving during optimization. To find the pair of correlative potential optima between the latest two iterations, a tolerance is presented. Furthermore, to ensure that all the output minima are the global or local optima, an optimality judgement criterion is introduced.

Effective strategies of metamodelling of industrial metallurgical processes

The main objective of the metamodelling is replacing the model of analysed process by its simple (with respect to the computation time) approximation. Metamodel gives a significant reduction of computation time of considered process simulation, as well as its further analysis (sensitivity analysis, optimization, etc.). The paper discusses the idea of metamodelling and compares the effectiveness of three techniques: Response Surface Methodology (RSM), Kriging method and Artificial Neural Network (ANN) applied to the benchmark functions. An example of the use of the considered metamodelling techniques in optimization of the problem of laminar cooling of rolled Dual Phase (DP) steel strips is presented. Metamodelling and optimization of a real industrial metal forming problems seems a novel approach in the field of research on Artificial Intelligence and Optimization practical applications.

Resource planning in the emergency departments: A simulation-based metamodeling approach

Patient’s congestion and their long waiting times in Emergency Departments (EDs) are the most common problems in hospitals. This paper extends application domain of metamodels into decision-making in the EDs by developing a discrete event simulation (DES) model combined with suitable metamodels. This is used as a novel decision support system to improve the patients flow and relieve congestion by changing the number of ED resources (i.e., the number of receptionists, nurses, residents, and beds). This new tool could be used for decision-making in operational, tactical, and strategic levels. In the first step, we develop a simulation routine of the ED in order to evaluate the system performance measure (total average waiting times of patients) for each configuration of resources. In the next step, we use different metamodel techniques and choose one with the maximum efficiency through a cross validation technique to replace the computationally expensive DES model with an accurate and efficient metamodel. Then the proposed model is used to minimize the total average waiting times of patients subject to both budget and capacity constraints. We implement our proposed model in an emergency department in Iran. Experimental results with the current ED budget verify that after using the resource allocation obtained from the proposed model, the total waiting time of patients is reduced by about 48%. Furthermore, to evaluate the efficiency of the selected metamodel, we compare the respective results with those obtained through OptQuest in terms of both the accuracy and the time needed to perform the optimization process.

POLYNOMIAL-CHAOS-BASED KRIGING

Computer simulation has become the standard tool in many engineering fields for designing and optimizing systems, as well as for assessing their reliability. To cope with demanding analysis such as optimization and reliability, surrogate models (a.k.a meta-models) have been increasingly investigated in the last decade. Polynomial Chaos Expansions (PCE) and Kriging are two popular non-intrusive meta-modelling techniques. PCE surrogates the computational model with a series of orthonormal polynomials in the input variables where polynomials are chosen in coherency with the probability distributions of those input variables. On the other hand, Kriging assumes that the computer model behaves as a realization of a Gaussian random process whose parameters are estimated from the available computer runs, i.e. input vectors and response values. These two techniques have been developed more or less in parallel so far with little interaction between the researchers in the two fields. In this paper, PC-Kriging is derived as a new non-intrusive meta-modeling approach combining PCE and Kriging. A sparse set of orthonormal polynomials (PCE) approximates the global behavior of the computational model whereas Kriging manages the local variability of the model output. An adaptive algorithm similar to the least angle regression algorithm determines the optimal sparse set of polynomials. PC-Kriging is validated on various benchmark analytical functions which are easy to sample for reference results. From the numerical investigations it is concluded that PC-Kriging performs better than or at least as good as the two distinct meta-modeling techniques. A larger gain in accuracy is obtained when the experimental design has a limited size, which is an asset when dealing with demanding computational models.

Multi-objective optimisation of hybrid S-shaped rails under oblique impact loading

This paper aims at optimising the crashworthiness capabilities of hybrid S-shaped rails of an automotive body under oblique impact loading. In the optimisation process, geometrical parameters, material types and the length ratios of aluminium part to the whole length are taken as design variables. Kriging metamodel techniques are performed to predict the crashworthiness criteria of specific energy absorption (SEA) and peak crushing force (PCF). In the following multi-objective optimisation design (MOD), non-dominated sorting genetic algorithm (NSGA-II) is adopted to obtain maximum SEA capacity and minimum PCF of hybrid S-shaped rails. It is found that the optimum design for a specific load angle does not necessarily guarantee the best solution when the load angle changes. More importantly, the change of weighting factor for each load angle affects the Pareto front of such MOD considerably, and thus selecting proper weights is of great importance in the crashworthiness design of hybrid S-shaped rails.

A Structured Approach for Rapidly Mapping Multilevel System Measures via Simulation Metamodeling

ABSTRACTWith increasing complexity of real‐world systems, especially for continuously evolving scenarios, systems analysts encounter a major challenge with the modeling techniques that capture detailed system characteristics defining input–output relationships. The models become very complex and require long time of execution. In this situation, techniques to construct approximations of the simulation model by metamodeling alleviate long run times and the need for large computational resources; it also provides a means to aggregate a simulation's multiple outputs of interest and derives a single decision‐making metric. The method described here leverages simulation metamodeling to map the three basic SE metrics, namely, measures of performance to measures of effectiveness to a single figure of merit. This enables using metamodels to map multilevel system measures supports rapid decision making. The results from a case study demonstrate the merit of the method. Several metamodeling techniques are compared and bootstrap error analysis and predicted residual sums of squares statistic are discussed to evaluate the standard error and error due to bias.

Designing scalable product families by the radial basis function–high-dimensional model representation metamodelling technique

Product family design is cost-efficient for achieving the best trade-off between commonalization and diversification. However, for computationally intensive design functions which are viewed as black boxes, the family design would be challenging. A two-stage platform configuration method with generalized commonality is proposed for a scale-based family with unknown platform configuration. Unconventional sensitivity analysis and information on variation in the individual variants’ optimal design are used for platform configuration design. Metamodelling is employed to provide the sensitivity and variable correlation information, leading to significant savings in function calls. A family of universal electric motors is designed for product performance and the efficiency of this method is studied. The impact of the employed parameters is also analysed. Then, the proposed method is modified for obtaining higher commonality. The proposed method is shown to yield design solutions with better objective function values, allowable performance loss and higher commonality than the previously developed methods in the literature.

A global optimization algorithm for simulation-based problems via the extended DIRECT scheme

This article presents a global optimization algorithm via the extension of the DIviding RECTangles (DIRECT) scheme to handle problems with computationally expensive simulations efficiently. The new optimization strategy improves the regular partition scheme of DIRECT to a flexible irregular partition scheme in order to utilize information from irregular points. The metamodelling technique is introduced to work with the flexible partition scheme to speed up the convergence, which is meaningful for simulation-based problems. Comparative results on eight representative benchmark problems and an engineering application with some existing global optimization algorithms indicate that the proposed global optimization strategy is promising for simulation-based problems in terms of efficiency and accuracy.

Comparative study of metamodelling techniques in building energy simulation: Guidelines for practitioners

Computer simulation of real system behaviour is increasingly used in research and development. As simulation models become more reliable, they also often become more complex to capture the progressive complexity of the real system. Calculation time can be a limiting factor for using simulation models in optimisation studies, for example, which generally require multiple simulations. Instead of using these time-consuming simulation models, the use of metamodels can be considered. A metamodel approximates the original simulation model with high confidence via a simplified mathematical model. A series of simulations then only takes a fraction of the original simulation time, hence allowing significant computational savings.In this paper, a strategy that is both reliable and time-efficient is provided in order to guide users in their metamodelling problems. Furthermore, polynomial regression (PR), multivariate adaptive regression splines (MARS), kriging (KR), radial basis function networks (RBF), and neural networks (NN) are compared on a building energy simulation problem. We find that for the outputs of this example and based on Root Mean Squared Error (RMSE), coefficient of determination (R2), and Maximal Absolute Error (MAE), KR and NN are the overall best techniques. Although MARS perform slightly worse than KR and NN, it is preferred because of its simplicity. For different applications, other techniques might be optimal.

Extending the use of ecological models without sacrificing details: a generic and parsimonious meta‐modelling approach

Summary Difficulties in accounting for the fine scale nature of ecological processes in large‐scale simulations constitute an important issue in ecology. Among existing methods, meta‐modelling, that is creating a statistical emulator of a model, has seen very few applications in ecology. Yet, meta‐modelling methods are well advanced in the field of engineering. We adapted and applied a meta‐modelling approach to a case study typical of the complexity found in ecosystems. It involved a highly detailed, individual‐based and spatially explicit biophysical model (noTG). The model was parameterized for a multi‐specific, spatially heterogeneous forest. Our goal was to increase its temporal domain of applicability by obtaining a meta‐model of its light interception module many times faster. The meta‐model was constructed from a series of simulations with noTG, following a latin hypercube design. Several meta‐modelling techniques were compared, with neural networks providing the best results. The meta‐model accurately reproduced the behaviour of noTG across a range of variables and organization levels. It was also 62 times faster. These result show that meta‐modelling can be a practical tool in ecology and represents a highly powerful way to change the scope of a model while still accounting for fine details.

Boosting Kernel-Based Dimension Reduction for Jointly Propagating Spatial Variability and Parameter Uncertainty in Long-Running Flow Simulators

Assessing the impact of multiple sources of uncertainty in flow models requires simulating the response associated with a large set of stochastic realizations ( $$>\!\!1,\!000$$ ) of both the spatial variable (e.g. permeability field) and of the large number of model parameters (e.g., of the relative permeability curve). Yet, this Monte Carlo approach may be hindered by the high computation time cost of long-running flow simulators (CPU time $$> $$ several hours). A possible strategy can then rely on the combination of meta-modelling techniques and of basis set expansion like kernel principal component analysis to reduce the high-dimensionality ( $$>\!\!10,\!000$$ grid cells) of the spatial variable. Using a synthetic heterogeneous channelized reservoir, it is shown that $$\sim $$ 50 principal components PCs are still necessary to retain $$\sim $$ 80 % of the total variability: this might pose difficulties for the construction of the meta-model by imposing a large set of training samples. This computational limitation can be overcome by improving the meta-modelling phase through boosting techniques, which allow selecting during the fitting process the input variables (PCs or model parameters), which are the most informative regarding the prediction accuracy, and allow in this manner reducing the number of necessary model runs. The procedure is applied to a synthetic flow model of a CO $$_{2}$$ geological storage with low CPU time. By comparing the P10, P50 and P90 quantiles of the pressure build-up estimated using the true simulator (1,500 simulations) with the ones estimated using the meta-model. The joint impact of spatial variability and of 12 uncertain parameters is shown to be accurately captured using a number of training samples of the order of the number of PCs ( $$\sim $$ a few tens). The level of performance proved to be as high as the one of the distance-based alternative proposed in the literature to solve the problem of joint propagation.

Meta-Modeling Based Secure Software Development Processes

This work suggests a multilevel support to software developers, who often lack knowledge and skills on how to proceed to develop secure software. In fact, developing software with such quality is a hard and complex task that involves many additional security-dedicated activities which are usually omitted in traditional software development lifecycles or integrated but not efficiently and appropriately deployed in some others. To federate all these software security-assurance activities in a structured way and provide the required guidelines for choosing and using them in a flexible development process, authors used meta-modeling techniques and dynamic process execution that consider developer's affinities and product's states. The proposed approach formalizes existing secure software development processes, allows integration of new ones, prevents ad-hoc executions and is supported by a tool to facilitate its deployment. A case study is given here to exemplify the proposed approach application and to illustrate some of its advantages.

Global structure of sloppiness in a nonlinear model

The problem of structural ambiguity or “sloppiness” of a mathematical model is here studied by multivariate metamodeling techniques. If a given model is “sloppy”, it means that a number of different parameter combinations—“a neutral parameter set”—can give more or less the same model behavior and thus equally good fit to data. This paper presents a way to characterize the structure of such sloppiness. The model used for illustration is a nonlinear dynamic model of reaction kinetics—a simple version of the S‐system model. When fitted to time series data by various nonlinear curve fitting methods, an unexpected problem was discovered: For every time series, a large neutral parameter set was observed. Each of these sets was analyzed by principal component analysis and found to have clear, but nonlinear, subspace structure. The neutral parameter sets were found for many different time series data, and the global sloppiness structure of the model was characterized. This global sloppiness structure of the model allowed us to find strong correlations between parameters, and on this basis to simplify the original model. A method to reduce the ambiguity in kinetic model parameter estimates based on combining several time series is suggested. Copyright © 2014 John Wiley & Sons, Ltd.

Classifier-guided sampling for discrete variable, discontinuous design space exploration: Convergence and computational performance

A classifier-guided sampling (CGS) method is introduced for solving engineering design optimization problems with discrete and/or continuous variables and continuous and/or discontinuous responses. The method merges concepts from metamodel-guided sampling and population-based optimization algorithms. The CGS method uses a Bayesian network classifier for predicting the performance of new designs based on a set of known observations or training points. Unlike most metamodelling techniques, however, the classifier assigns a categorical class label to a new design, rather than predicting the resulting response in continuous space, and thereby accommodates non-differentiable and discontinuous functions of discrete or categorical variables. The CGS method uses these classifiers to guide a population-based sampling process towards combinations of discrete and/or continuous variable values with a high probability of yielding preferred performance. Accordingly, the CGS method is appropriate for discrete/discontinuous design problems that are ill suited for conventional metamodelling techniques and too computationally expensive to be solved by population-based algorithms alone. The rates of convergence and computational properties of the CGS method are investigated when applied to a set of discrete variable optimization problems. Results show that the CGS method significantly improves the rate of convergence towards known global optima, on average, compared with genetic algorithms.