Bayesian Sensitivity Analysis and Uncertainty Integration for Robust Optimization (original) (raw)

Probabilistic Sensitivity Analysis Methods for Design under Uncertainty

Sensitivity analysis (SA) is an important procedure in engineering design to obtain valuable information about the model behavior to guide a design process. For design under uncertainty, probabilistic sensitivity analysis (PSA) methods have been developed to provide insight into the probabilistic behavior of a model. In this paper, the goals of PSA at different design stages are investigated. In the prior-design stage, PSA can be utilized to identify those probabilistically non-significant variables and reduce the dimension of a random design space. It can reduce the computational cost associated with uncertainty assessment without much sacrifice on the optimum solution. For post-design analysis, probabilistic sensitivity analysis can be used to identify where to spend design resources for the largest potential improvement of a performance. Based on the interested distribution range of a random response, the PSA methods can be categorized into two types: the global response probabilistic sensitivity analysis (GRPSA) and the regional response probabilistic sensitivity analysis (RRPSA). Four widely-used PSA methods: Sobol' indices, Wu's sensitivity coefficients, the MPP based sensitivity coefficients, and the Kullback-Leibler entropy based method are selected for comparison. The merits behind each method are reviewed in details. Their advantages, limitations, and applicability are investigated. Their effectiveness and applicability under different design scenarios are compared in two numerical examples and two engineering design problems.

Sensitivity Analysis for Uncertainty Quantification in Mathematical Models

Mathematical and Statistical Estimation Approaches in Epidemiology, 2009

All mathematical models are approximate and their usefulness depends on our understanding the uncertainty inherent in the predictions. Uncertainties can affect the reliability of the results at every stage of computation; they may grow or even shrink as the solution of the model evolves. Often these inherent uncertainties cannot be made arbitrarily small by a more complex model or additional computation and we must understand how the uncertainty in the model parameters, the initial conditions, and the model itself, lead to uncertainties in the model predictions. This chapter is an introductory survey of sensitivity analysis and illustrates how to define the derivative of the model solution as a function of the model input and determine the relative importance of the model parameters on the model predictions.

Analytical Variance-Based Global Sensitivity Analysis in Simulation-Based Design Under Uncertainty

Journal of Mechanical Design, 2005

The importance of sensitivity analysis in engineering design cannot be over-emphasized. In design under uncertainty, sensitivity analysis is performed with respect to the probabilistic characteristics. Global sensitivity analysis (GSA), in particular, is used to study the impact of variations in input variables on the variation of a model output. One of the most challenging issues for GSA is the intensive computational demand for assessing the impact of probabilistic variations. Existing variance-based GSA methods are developed for general functional relationships but require a large number of samples. In this work, we develop an efficient and accurate approach to GSA that employs analytic formulations derived from metamodels. The approach is especially applicable to simulation-based design because metamodels are often created to replace expensive simulation programs, and therefore readily available to designers. In this work, we identify the needs of GSA in design under uncertainty...

The NASA Langley Multidisciplinary Uncertainty Quantification Challenge

16th AIAA Non-Deterministic Approaches Conference, 2014

This paper presents the formulation of an uncertainty quantification challenge problem consisting of five subproblems. These problems focus on key aspects of uncertainty characterization, sensitivity analysis, uncertainty propagation, extreme-case analysis, and robust design.

Inclusion of Aleatory and Epistemic Uncertainty in Design Optimization

This paper presents a design optimization methodology under three sources of uncertainty: physical variability (aleatory); data uncertainty (epistemic) due to sparse or imprecise data; and model uncertainty (epistemic) due to modeling errors/approximations. A likelihood-based method is use to fuse multiple formats of information, and a non-parametric probability density function (PDF) is constructed. Two types of model errors are considered: model form error and numerical solution error, each of which is a function of the design variables that changes at every iteration of the optimization. Gaussian process (GP) surrogate models are constructed for efficient computation of model errors in the optimization. The treatment in this paper yields a distribution of the output that accounts for various sources of uncertainty. The use of a probabilistic approach to include both aleatory and epistemic uncertainties allows for their efficient integration into the optimization framework. The proposed methods are illustrated using a three-dimensional wing design problem involving fluid-structure interaction analysis.

Development of a new robust design methodology based on Bayesian perspectives

International Journal of Quality Engineering and Technology, 2012

Robust design (RD), implemented in statistical and mathematical procedures to simultaneously minimise the process bias and variability, is widely used in many areas of engineering and technology to represent complex real-world industrial settings. For RD modelling and optimisation, response surface methodology (RSM) is often utilised as an estimation method to represent the functional relationship between input factors and their associated output responses. Although conventional RSM-based RD methods may offer significant advantages regarding process design, there is room for improvement. In this context, a new RD methodology is developed in this paper by integrating Bayesian principles into the RD procedure. Numerical examples and comparative studies are conducted by using two conventional RSM-based RD models and the proposed model. The results of two numerical examples demonstrate that the proposed RD method provides significantly better RD solutions in terms of the expected quality loss (EQL) than conventional methods.

Analytical Metamodel-Based Global Sensitivity Analysis and Uncertainty Propagation for Robust Design

2004

Metamodeling approach has been widely used due to the high computational cost of using high-fidelity simulations in engineering design. Interpretation of metamodels for the purpose of design, especially design under uncertainty, becomes important. The computational expenses associated with metamodels and the random errors introduced by sample-based methods require the development of analytical methods, such as those for global sensitivity analysis and uncertainty propagation to facilitate a robust design process. In this work, we develop generalized analytical formulations that can provide efficient as well as accurate global sensitivity analysis and uncertainty propagation for a variety of metamodels. The benefits of our proposed techniques are demonstrated through vehicle related robust design applications.

Worst case propagated uncertainty of multidisciplinary systems in robust design optimization

Structural and Multidisciplinary Optimization, 2000

While simulation based design tools continue to be advanced at unprecedented rates, little attention has been paid to how these tools interact with other advanced design tools and how that interaction influences the multidisciplinary system analysis and design processes. In this research an investigation of how uncertainty propagates through a multidisciplinary system analysis subject to the bias errors associated with the disciplinary design tools and the precision errors in the inputs is undertaken. A rigorous derivation for estimating the worst case propagated uncertainty in multidisciplinary systems is developed and validated using Monte Carlo simulation in application to a small analytic problem and an Autonomous HoverCraft (AHC) problem. The method of worst case estimation of uncertainty is then integrated into a robust optimization framework. In robust optimization, both the objective function and the constraints consist of two parts, the original or conventional functions and an estimate of the variation of the functions. In robust optimization the engineer must trade off an increase in the objective function value for a decrease in variation. The robust optimization approach is tested in application to the AHC problem and the corresponding results are discussed.