As a common concept in multi-objective optimization, minimizing a weighted sum constitutes an independent method as well as a component of other methods. Consequently, insight into characteristics of the weighted sum method has far reaching implications. However, despite the many published applications for this method and the literature addressing its pitfalls with respect to depicting the Pareto optimal set, there is little comprehensive discussion concerning the conceptual significance of the weights and techniques for maximizing the effectiveness of the method with respect to a priori articulation of preferences. Thus, in this paper, we investigate the fundamental significance of the weights in terms of preferences, the Pareto optimal set, and objective-function values. We determine the factors that dictate which solution point results from a particular set of weights. Fundamental deficiencies are identified in terms of a priori articulation of preferences, and guidelines are provided to help avoid blind use of the method.

The weighted sum method for multi-objective optimization: new insights

Multidisciplinary design optimization is a field of research that studies the application of numerical optimization techniques to the design of engineering systems involving multiple disciplines or components. Since the inception of multidisciplinary design optimization, various methods (architectures) have been developed and applied to solve multidisciplinary design-optimization problems. This paper provides a survey of all the architectures that have been presented in the literature so far. All architectures are explained in detail using a unified description that includes optimization problem statements, diagrams, and detailed algorithms. The diagrams show both data and process flow through the multidisciplinary system and computational elements, which facilitate the understanding of the various architectures, and how they relate to each other. A classification of the multidisciplinary design-optimization architectures based on their problem formulations and decomposition strategies is also provided, a...

Multidisciplinary design optimization: A survey of architectures

In many situations across computational science and engineering, multiple computational models are available that describe a system of interest. These different models have varying evaluation costs...

https://epubs.siam.org/doi/pdf/10.1137/16M1082469

Survey of Multifidelity Methods in Uncertainty Propagation, Inference, and Optimization

This report documents the results of a study to address the long range, strategic planning required by NASA's Revolutionary Computational Aerosciences (RCA) program in the area of computational fluid dynamics (CFD), including future software and hardware requirements for High Performance Computing (HPC). Specifically, the "Vision 2030" CFD study is to provide a knowledge-based forecast of the future computational capabilities required for turbulent, transitional, and reacting flow simulations across a broad Mach number regime, and to lay the foundation for the development of a future framework and/or environment where physics-based, accurate predictions of complex turbulent flows, including flow separation, can be accomplished routinely and efficiently in cooperation with other physics-based simulations to enable multi-physics analysis and design. Specific technical requirements from the aerospace industrial and scientific communities were obtained to determine critical capability gaps, anticipated technical challenges, and impediments to achieving the target CFD capability in 2030. A preliminary development plan and roadmap were created to help focus investments in technology development to help achieve the CFD vision in 2030.

/pdf/cfd-vision-2030-study-a-path-to-revolutionary-computational-3hk79p0tz6.pdf

CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences

Adaptive control processes

Improving variable-fidelity surrogate modeling via gradient-enhanced kriging and a generalized hybrid bridge function

The efficiency of building a surrogate model for the output of a computer code can be dramatically improved via variable-fidelity surrogate modeling techniques. In this article, a hierarchical kriging model is proposed and used for variable-fidelity surrogate modeling problems. Here, hierarchical kriging refers to a surrogate model of a highfidelity function that uses a kriging model of a sampled lower-fidelity function as a model trend. As a consequence, the variation in the lower-fidelity data is mapped to the high-fidelity data, and a more accurate surrogate model for the high-fidelity function is obtained. A self-contained derivation of the hierarchical kriging model is presented. The proposed method is demonstrated with an analytical example and used for modeling the aerodynamic data of an RAE 2822 airfoil and an industrial transport aircraft configuration. The numerical examples show that it is efficient, accurate, and robust. It is also observed that hierarchical kriging provides a more reasonable mean-squared-error estimation than traditional cokriging. It can be applied to the efficient aerodynamic analysis and shape optimization of aircraft or any other research areas where computer codes of varying fidelity are in use.

Hierarchical Kriging Model for Variable-Fidelity Surrogate Modeling

Surrogate modeling plays an increasingly important role in
different areas of aerospace engineering, such as erodynamic
shape optimization, aerodynamic data production, structural design, and multidisciplinary design optimization of aircraft or spacecraft. Cokriging provides an attractive alternative approach to conventional kriging to improve the efficiency of building a surrogate model. It was initially proposed and applied in the geostatistics community for the enhanced prediction of less intensively sampled primary variables of interest with the assistance of intensively sampled auxiliary variables. As the underlying theory of cokriging is that of two-variable or multivariable kriging, it can be regarded as a general extension of (one-variable) kriging to a model that is assisted by auxiliary variables or secondary information.
In an attempt to apply cokriging to the surrogate modeling
problems associated with deterministic computer experiments, this article is motivated by the development of an alternative cokriging method to address the challenge related to the construction of the covariance matrix of cokriging [7]. Earlier work done by other authors related to this study can be found in the statistical community.
For example, Kennedy and O’Hagan (KOH) proposed an
autoregressive model to calculate the covariances and crosscovariances in the covariance matrix and developed a Bayesian approach to predict the output from an expensive high-fidelity simulation code with the assistance of lower-fidelity simulation codes. This Bayesian approach is identical to a form of cokriging suitable for computer experiments. Later, Qian andWu proposed a similar method, in which a random function (Gaussian process model) was used to replace the constant multiplicative factor of KOH’s method to account for the nonlinear scale change. KOH’s method was applied to multifidelity analysis and design optimization
in the context of aerospace engineering by Forrester et al. and Kuya et al. More recently, Zimmerman and Han proposed a cokriging method with simplified cross-correlation estimation.
In this article, we propose an alternative approach for the
construction of the cokriging covariance matrix and develop a more practical cokriging method in the context of surrogate-based analysis and optimization. The developed cokriging method is validated against an analytical problem and applied to construct global approximation models of the aerodynamic coefficients as well as the drag polar of an RAE 2822 airfoil.

Alternative Cokriging Method for Variable-Fidelity Surrogate Modeling

Cokriging is a statistical interpolation method for the enhanced prediction of a less intensively sampled primary variable of interest with assistance of intensively sampled auxiliary variables. In the geostatistics community it is referred to as two- or multi-variable kriging. In this paper, a new cokriging method is proposed and used for variable-fidelity surrogate modeling of aerodynamic data obtained with an expensive high-fidelity CFD code, assisted by data computed with cheaper lower-fidelity codes or by gradients computed with an adjoint version of the high-fidelity CFD code, or both. A self-contained derivation as well as the numerical implementation of this new cokriging method is presented and the comparison with the autoregressive model of Kennedy and O’Hagan is discussed. The developed cokriging method is validated against an analytical problem and applied to construct global approximation models of the aerodynamic coefficients as well as the drag polar of an RAE 2822 airfoil based on sampled CFD data. The numerical examples show
that it is efficient, robust and practical for the surrogate modeling of aerodynamic data based on a set of CFD methods with varying degrees of fidelity and computational expense. It can potentially be applied in the efficient CFD-based aerodynamic analysis and design optimization of aircraft.

/pdf/a-new-cokriging-method-for-variable-fidelity-surrogate-mcihecd44b.pdf

A New Cokriging Method for Variable-Fidelity Surrogate Modeling of Aerodynamic Data

Surrogate-based optimization (Queipo et al. 2005, Simpson et al. 2008) represents a class of optimization methodologies that make use of surrogate modeling techniques to quickly find the local or global optima. It provides us a novel optimization framework in which the conventional optimization algorithms, e.g. gradient-based or evolutionary algorithms are used for sub-optimization(s). Surrogate modeling techniques are of particular interest for engineering design when high-fidelity, thus expensive analysis codes (e.g. Computation Fluid Dynamics (CFD) or Computational Structural Dynamics (CSD)) are used. They can be used to greatly improve the design efficiency and be very helpful in finding global optima, filtering numerical noise, realizing parallel design optimization and integrating simulation codes of different disciplines into a process chain. Here the term “surrogate model” has the same meaning as “response surface model”, “metamodel”, “approximation model”, “emulator” etc. This chapter aims to give an overview of existing surrogate modeling techniques and issues about how to use them for optimization.

/pdf/surrogate-based-optimization-3aaovo4gra.pdf

Zhong-Hua Han

Papers

Improving variable-fidelity surrogate modeling via gradient-enhanced kriging and a generalized hybrid bridge function

Hierarchical Kriging Model for Variable-Fidelity Surrogate Modeling

Alternative Cokriging Method for Variable-Fidelity Surrogate Modeling

A New Cokriging Method for Variable-Fidelity Surrogate Modeling of Aerodynamic Data

Surrogate-Based Optimization