Sequential kriging optimization using multiple-fidelity evaluations

When cost per evaluation on a system of interest is high, surrogate systems can provide cheaper but lower-fidelity information. In the proposed extension of the sequential kriging optimization method, surrogate systems are exploited to reduce the total evaluation cost. The method utilizes data on all systems to build a kriging metamodel that provides a global prediction of the objective function and a measure of prediction uncertainty. The location and fidelity level of the next evaluation are selected by maximizing an augmented expected improvement function, which is connected with the evaluation costs. The proposed method was applied to test functions from the literature and a metal-forming process design problem via finite element simulations. The method manifests sensible search patterns, robust performance, and appreciable reduction in total evaluation cost as compared to the original method.     

一种基于自适应代理模型的并行贝叶斯优化方法

提出一种基于自适应代理模型的并行贝叶斯优化方法,用于求解计算成本高的复杂优化问题.该方法基于多点期望改进判据,通过批次采样实现并行优化.针对并行优化产生的大量历史数据会导致全局代理模型建模成本高的问题,提出一种改进的基于数据并行的高斯过程建模方法,在线构造局部代理模型.此外,针对多点期望改进判据计算成本高的问题,提出一种启发式的分层优化策略,通过序贯优化基于自适应代理模型的单点期望改进判据,近似计算多点期望改进判据.最后通过5个测试问题验证所提出方法的有效性.     

Surrogate modeling of impedance matching transformers by means of variable‐fidelity electromagnetic simulations and nested cokriging

Accurate performance evaluation of microwave components can be carried out using full‐wave electromagnetic (EM) simulation tools, routinely employed for circuit verification but also in the design process itself. Unfortunately, the computational cost of EM‐driven design may be high. This is especially pertinent to tasks entailing considerable number of simulations (eg, parametric optimization, statistical analysis). A possible way of alleviating these difficulties is utilization of fast replacement models, also referred to as surrogates. Notwithstanding, conventional modeling methods exhibit serious limitations when it comes to handling microwave components. The principal challenges include large number of geometry and material parameters, highly nonlinear characteristics, as well as the necessity of covering wide ranges of operating conditions. The latter is mandatory from the point of view of the surrogate model utility. This article presents a novel modeling approach that incorporates variable‐fidelity EM simulations into the recently reported nested kriging framework. A combination of domain confinement due to nested kriging, and low‐/high‐fidelity EM data blending through cokriging, enables the construction of reliable surrogates at a fraction of cost required by single‐fidelity nested kriging. Our technique is validated using a three‐section miniaturized impedance matching transformer with its surrogate model rendered over wide range of operating frequencies. Comprehensive benchmarking demonstrates superiority of the proposed method over both conventional models and nested kriging.     

Variable-fidelity expected improvement method for efficient global optimization of expensive functions

The efficient global optimization method (EGO) based on kriging surrogate model and expected improvement (EI) has received much attention for optimization of high-fidelity, expensive functions. However, when the standard EI method is directly applied to a variable-fidelity optimization (VFO) introducing assistance from cheap, low-fidelity functions via hierarchical kriging (HK) or cokriging, only high-fidelity samples can be chosen to update the variable-fidelity surrogate model. The theory of infilling low-fidelity samples towards the improvement of high-fidelity function is still a blank area. This article proposes a variable-fidelity EI (VF-EI) method that can adaptively select new samples of both low and high fidelity. Based on the theory of HK model, the EI of the high-fidelity function associated with adding low- and high-fidelity sample points are analytically derived, and the resulting VF-EI is a function of both the design variables x and the fidelity level l. Through maximizing the VF-EI, both the sample location and fidelity level of next numerical evaluation are determined, which in turn drives the optimization converging to the global optimum of high-fidelity function. The proposed VF-EI is verified by six analytical test cases and demonstrated by two engineering problems, including aerodynamic shape optimizations of RAE 2822 airfoil and ONERA M6 wing. The results show that it can remarkably improve the optimization efficiency and compares favorably to the existing methods.     

A multi-fidelity surrogate modeling approach for incorporating multiple non-hierarchical low-fidelity data

Multi-fidelity (MF) surrogate models have been widely used in simulation-based design problems to reduce the computational cost by integrating the data with different fidelity levels. Most of the existing MF modeling methods are only applicable to the problems with hierarchical low-fidelity (LF) models, namely the fidelity levels of multiple LF models can be identified. However, the fidelity levels of the LF models that are obtained from different simplification methods often vary over the design space. To address this challenge, a non-hierarchical Co-Kriging modeling (NHLF-Co-Kriging) method that can flexibly handle multiple non-hierarchical LF models is developed in this work. In the proposed method, multiple LF models are scaled by different scale factors, and a discrepancy model is utilized to depict the differences between the HF model and the ensembled LF models. To make the discrepancy Gaussian process (GP) model easy to be fitted, an optimization problem whose objective is to minimize the second derivative of the prediction values of the discrepancy GP model is defined to obtain optimal scale factors of the LF models. The performance of the NHLF-Co-Kriging method is compared with the extended Co-Kriging model and linear regression MF surrogate model through several analytical examples and an engineering case. Results show that the proposed method selects more reasonable scale factors for the multiple LF models and provides more accurate MF surrogate models under a limited computational budget.     

Optimal design of aeroengine turbine disc based on kriging surrogate models

A design optimization method based on kriging surrogate models is proposed and applied to the shape optimization of an aeroengine turbine disc. The kriging surrogate model is built to provide rapid approximations of time-consuming computations. For improving the accuracy of surrogate models without significantly increasing computational cost, a rigorous sample selection is employed to reduce additional design samples based on design of experiments over a sequential trust region. The minimum-mass shape design of turbine discs under thermal and mechanical loads has demonstrated the effectiveness and efficiency of the presented optimization approach.     

数据驱动的贝叶斯SVR自适应建模及昂贵约束多目标代理优化

实际工程中的多目标优化问题往往具有黑箱特性且需要耗时的功能性评估,采用传统的进化优化方法求解,存在计算成本高昂且难以实现的问题.考虑代理优化方法在处理需要功能性评估工程设计问题中的高效性,提出一种小样本数据驱动下的贝叶斯SVR自适应建模及昂贵约束多目标代理优化方法.该方法在实现过程中选取贝叶斯SVR模型以减少功能性评估过程的昂贵仿真成本,利用最大化约束期望改进矩阵聚合策略进行新设计方案选取,并通过小样本信息的不断更新实现数据驱动下的贝叶斯SVR模型自适应更新和逐步优化.贝叶斯SVR模型具有强的边界刻画能力及预测不确定性度量功能,可为新样本挑选提供预测精度保障及潜在的改进方向.所提出的切比雪夫距离和曼哈顿距离聚合策略从样本填充的改进范围考虑,使其具有较强的改进边界探索能力,在多变量优化问题中具有计算复杂度低、适用性强的特点.测试函数及工程实例结果表明:1)所提出的方法可在小样本条件下有效减少昂贵仿真成本,提升昂贵约束多目标问题的优化效率;2)获取昂贵约束多目标问题的Pareto前沿在收敛性、多样性及空间分布性方面均具有一定优势.     

Metamodel-based collaborative optimization framework

This paper focuses on the metamodel-based collaborative optimization (CO). The objective is to improve the computational efficiency of CO in order to handle multidisciplinary design optimization problems utilising high fidelity models. To address these issues, two levels of metamodel building techniques are proposed: metamodels in the disciplinary optimization are based on multi-fidelity modelling (the interaction of low and high fidelity models) and for the system level optimization a combination of a global metamodel based on the moving least squares method and trust region strategy is introduced. The proposed method is demonstrated on a continuous fiber-reinforced composite beam test problem. Results show that methods introduced in this paper provide an effective way of improving computational efficiency of CO based on high fidelity simulation models.     

Surrogate‐assisted EM‐driven miniaturization of wideband microwave couplers by means of co‐simulation low‐fidelity models

This article proposes a methodology for rapid design optimization of miniaturized wideband couplers. More specifically, a class of circuits is considered, in which conventional transmission lines are replaced by their abbreviated counterparts referred to as slow‐wave compact cells. Our focus is on explicit reduction of the structure size as well as on reducing the CPU cost of the design process. For the sake of computational feasibility, a surrogate‐based optimization paradigm involving a co‐simulation low‐fidelity model is used. The latter is a fundamental component of the proposed technique. The low‐fidelity model represents cascaded slow‐wave cells replacing the low‐impedance lines of the original coupler circuit. It is implemented in a circuit simulator (here, ADS) and consists of duplicated compact cell EM simulation data as well as circuit theory‐based feeding line models. Our primary optimization routine is a trust‐region‐embedded gradient search algorithm. To further reduce the design cost, the system response Jacobian is estimated at the level of the low‐fidelity model, which is sufficient due to good correlation between the low‐ and high‐fidelity models. The coupler is explicitly optimized for size reduction, whereas electrical performance parameters are controlled using a penalty function approach. The presented methodology is demonstrated through the design of a 1‐GHz wideband microstrip branch‐line coupler. Numerical results are supported by experimental validation of the fabricated coupler prototype.     

Reduced‐cost microwave filter modeling using a two‐stage Gaussian process regression approach

A technique for the reduced‐cost modeling of microwave filters is presented. Our approach exploits variable‐fidelity electromagnetic (EM) simulations, and Gaussian process regression (GPR) carried out in two stages. In the first stage of the modeling process, a mapping between EM simulation filter models of low and high fidelity is established. The mapping is subsequently used in the second stage, making it possible for the final surrogate model to be constructed from training data obtained using only a fraction of the number of high‐fidelity simulations normally required. As demonstrated using three examples of microstrip filters, the proposed technique allows us to reduce substantially (by up to 80%) the central processing unit (CPU) cost of the filter model setup, as compared to conventional (single‐stage) GPR—the benchmark modeling method in this study. This is achieved without degrading the model generalization capability. The reliability of the two‐stage modeling method is demonstrated through the successful application of the surrogates to surrogate‐based filter design optimization. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:453–462, 2015.     

Multidisciplinary design of a small satellite launch vehicle using particle swarm optimization

In the present paper, particle swarm optimization, a relatively new population based optimization technique, is applied to optimize the multidisciplinary design of a solid propellant launch vehicle. Propulsion, structure, aerodynamic (geometry) and three-degree of freedom trajectory simulation disciplines are used in an appropriate combination and minimum launch weight is considered as an objective function. In order to reduce the high computational cost and improve the performance of particle swarm optimization, an enhancement technique called fitness inheritance is proposed. Firstly, the conducted experiments over a set of benchmark functions demonstrate that the proposed method can preserve the quality of solutions while decreasing the computational cost considerably. Then, a comparison of the proposed algorithm against the original version of particle swarm optimization, sequential quadratic programming, and method of centers carried out over multidisciplinary design optimization of the design problem. The obtained results show a very good performance of the enhancement technique to find the global optimum with considerable decrease in number of function evaluations.     

基于均值改进控制策略的昂贵约束并行代理优化算法

针对具有黑箱特性的昂贵约束优化问题及工程中计算资源利用率不高问题,提出了新的基于均值改进控制策略的并行代理优化算法.该算法为了减少仿真建模计算负担,选取Kriging近似模型对目标函数和约束函数进行近似估计.在Kriging模型基础上,利用均值改进与新增试验样本间的不等关系构建具有距离特性的控制函数.算法的均值改进控制...     

用于测试序列优化的DPSO-WAO*算法研究

针对现有测试序列优化算法所存在的计算效率及优化性能间的矛盾,结合离散粒子群算法(DPSO),提出了基于加权Huffman编码的启发式评估函数,对传统AO*算法进行改进,提出了DPSO-WAO*(DPSO-Weight_AO*)算法。实例证明,基于加权Huffman编码的启发式评估函数更为准确地评估了全局测试成本,在取消了成本回溯的情况下,算法仍能保持较高的优化性能,且有效地降低了计算复杂度,对于大型系统的测试序列设计、可测试性分析及故障诊断等具有重要意义。     

Computational‐budget‐driven automated microwave design optimization using variable‐fidelity electromagnetic simulations

A robust technique for microwave design optimization is presented. It is based on variable‐fidelity electromagnetic (EM) simulations where the approximate optimum of the “coarser” model becomes an initial design for finding the optimum of the “finer” one. The algorithm automatically switches between the models of different fidelity taking into account the computational budget assumed for the design process. Additional mechanisms enhancing the algorithm include: frequency scaling to reduce the misalignment between the models of different fidelity, as well as the local response surface approximation to reduce the number of EM simulations. The presented technique is particularly suitable for problems where simulation‐driven design is the only option, for example, for wideband antennas and dielectric resonator filters. Our method is demonstrated using two filters and one antenna example. In all cases, the optimal design is obtained at a low computational cost corresponding to a few high‐fidelity simulations of the structure. © 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2013.     

Update strategies for kriging models used in variable fidelity optimization

Many optimization methods for simulation-based design rely on the sequential use of metamodels to reduce the associated computational burden. In particular, kriging models are frequently used in variable fidelity optimization. Nevertheless, such methods may become computationally inefficient when solving problems with large numbers of design variables and/or sampled data points due to the expensive process of optimizing the kriging model parameters in each iteration. One solution to this problem would be to replace the kriging models with traditional Taylor series response surface models. Kriging models, however, were shown to provide good approximations of computer simulations that incorporate larger amounts of data, resulting in better global accuracy. In this paper, a metamodel update management scheme (MUMS) is proposed to reduce the cost of using kriging models sequentially by updating the kriging model parameters only when they produce a poor approximation. The scheme uses the trust region ratio (TR-MUMS), which is a ratio that compares the approximation to the true model. Two demonstration problems are used to evaluate the proposed method: an internal combustion engine sizing problem and a control-augmented structural design problem. The results indicate that the TR-MUMS approach is very effective; on the demonstration problems, it reduced the number of likelihood evaluations by three orders of magnitude compared to using a global optimizer to find the kriging parameters in every iteration. It was also found that in trust region-based method, the kriging model parameters need not be updated using a global optimizer—local methods perform just as well in terms of providing a good approximation without affecting the overall convergence rate, which, in turn, results in a faster execution time.     

Multilevel microwave design optimization with automated model fidelity adjustment

A computationally efficient algorithm for electromagnetic (EM)‐simulation‐driven design optimization of microwave structures is proposed. Our technique exploits variable‐fidelity EM simulations and the multilevel design approach where an approximate optimum of the lower accuracy but faster EM model of the structure under design is used as a starting point for optimizing a more accurate model. Several enhancements of the basic multifidelity method are introduced, including an efficient algorithm of optimizing EM models that is based on local response surface approximations, as well as automated adjustment of model fidelity. Convergence of the procedure to the optimum design is ensured by defaulting to the higher fidelity model whenever the prediction given by the lower fidelity fails to improve the design. Distribution of the computational effort between the models of different fidelity allows for making larger steps in the design space at a low cost, as well as substantial reduction of the number of high‐fidelity model evaluations, because the high‐fidelity model is only referred to in the last design stage. The article provides comprehensive numerical verification of our technique. Substantial computational savings are demonstrated in comparison to the benchmark methods: over 40% on average as compared to a basic version of the multifidelity optimization approach and over 95% as compared to direct optimization of the high‐fidelity model. © 2013 Wiley Periodicals, Inc. Int J RF and Microwave CAE 24:281–288, 2014.     

基于粗糙集和改进鲸鱼优化算法的特征选择方法

随着互联网和物联网技术的发展,数据的收集变得越发容易。但是,高维数据中包含了很多冗余和不相关的特征,直接使用会徒增模型的计算量,甚至会降低模型的表现性能,故很有必要对高维数据进行降维处理。特征选择可以通过减少特征维度来降低计算开销和去除冗余特征,以提高机器学习模型的性能,并保留了数据的原始特征,具有良好的可解释性。特征选择已经成为机器学习领域中重要的数据预处理步骤之一。粗糙集理论是一种可用于特征选择的有效方法,它可以通过去除冗余信息来保留原始特征的特性。然而,由于计算所有的特征子集组合的开销较大,传统的基于粗糙集的特征选择方法很难找到全局最优的特征子集。针对上述问题,文中提出了一种基于粗糙集和改进鲸鱼优化算法的特征选择方法。为避免鲸鱼算法陷入局部优化,文中提出了种群优化和扰动策略的改进鲸鱼算法。该算法首先随机初始化一系列特征子集,然后用基于粗糙集属性依赖度的目标函数来评价各子集的优劣,最后使用改进鲸鱼优化算法,通过不断迭代找到可接受的近似最优特征子集。在UCI数据集上的实验结果表明,当以支持向量机为评价所用的分类器时,文中提出的算法能找到具有较少信息损失的特征子集,且具有较高的分类精度。因此,所提算法在特征选择方面具有一定的优势。     

基于PRS元模型的粒子群优化算法及其应用*

针对传统粒子群算法优化黑箱模型过程中存在巨大计算开销的问题,提出一种基于PRS元模型的改进粒子群优化算法—PPSO算法。在该算法迭代过程中,构建PRS元模型,利用其最优值点辅助粒子种群的更新,此外仅选择元模型预估集中优值集的粒子进行目标函数的计算仿真。将PPSO算法与基本粒子群算法、混沌粒子群算法进行数值测试对比,并应用于模糊控制器的优化设计,仿真结果表明该算法可减少真实估值次数,提高优化搜索能力。     

Review of topology optimisation refinement processes for sheet metal manufacturing in the automotive industry

Design can be viewed a sequential decision process that increases the detail of modeling and analysis while simultaneously decreasing the space of alternatives considered. In a decision theoretic framework, low-fidelity models help decision-makers identify regions of feasibility and interest in the tradespace and cull others prior to constructing more computationally expensive models of higher fidelity. The method presented herein demonstrates design as a sequence of finite decision epochs through a search space defined by the extent of the set of designs under consideration, and the level of analytic fidelity subjected to each design. Previous work has shown that multi-fidelity modeling can aid in rapid optimization of the design space when high-fidelity models are coupled with low-fidelity models. This paper offers two contributions to the design community: (1) a model of design as a sequential decision process of refinement using progressively more accurate and expensive models, and (2) a connected approach for how conceptual models couple with detailed models. Formal definitions of the process are provided, and several structural design examples are presented to demonstrate the use of sequential multi-fidelity modeling in determining an optimal modeling selection policy.     

Aerodynamic shape optimization using efficient evolutionary algorithms and unstructured CFD solver

An efficient evolutionary algorithm is presented for shape optimization of transonic airfoils. Several techniques have been used to improve the efficiency and convergence rate of the optimization Genetic Algorithm (GA). A new airfoil shape parameterization method is used which is capable of producing more efficient shapes at viscous flow conditions. A Real-Coded Population Dispersion (PD) Genetic Algorithm is developed in order to increase the robustness and convergence rate of the Genetic Algorithm. A Multi-Layer Perceptron Neural Network (NN) is utilized to reduce the huge computational cost of the objective function evaluation. Further improvement in the performance of NN is obtained by using dynamic retraining and normal distribution of the training data to determine well trained parts of the design space to NN. Using the above techniques, the total computational time of optimization algorithm is reduced up to 60% compared with the conventional GA.