Robust design under mixed aleatory/epistemic uncertainties using gradients and surrogates
 Markus P Rumpfkeil^{1}Email author
DOI: 10.1186/2195546817
© Rumpfkeil; licensee Springer. 2013
Received: 19 April 2013
Accepted: 10 October 2013
Published: 30 October 2013
Abstract
In this paper, mixed aleatory/epistemic uncertainties in a robust design problem are propagated via the use of boxconstrained optimizations and surrogate models. The assumption is that the uncertain input parameters can be divided into a set only containing aleatory uncertainties and a set with only epistemic uncertainties. Uncertainties due to the epistemic inputs can then be propagated via a boxconstrained optimization approach, while the uncertainties due to aleatory inputs can be propagated via sampling. A statisticsofintervals approach is used in which the boxconstrained optimization results are treated as a random variable and multiple optimizations need to be performed to quantify the aleatory uncertainties via sampling. A Kriging surrogate is employed to model the variation of the optimization results with respect to the aleatory variables enabling exhaustive MonteCarlo sampling to determine the desired statistics for each robust design iteration. This approach is applied to the robust design of a transonic NACA 0012 airfoil where shape design variables are assumed to have epistemic uncertainties and the angle of attack and Mach number are considered to have aleatory uncertainties. The very good scalability of the framework in the number of epistemic variables is demonstrated as well.
Introduction and motivation
Computational methods have been playing an increasingly important role in science and engineering analysis and design over the last several decades, due to the rapidly advancing capabilities of computer hardware, as well as increasingly sophisticated and capable numerical algorithms. However, in spite of the rapid advances and acceptance of numerical simulations, serious deficiencies remain in terms of accuracy, uncertainty, and validation for many applications. Many realworld problems involve input data that is noisy or uncertain, due to measurement or modeling errors, approximate modeling parameters [1], manufacturing tolerances [2], inservice wearandtear, or simply the unavailability of the information at the time of the decision [3]. These imprecise or unknown inputs are important in the design process and need to be quantified in some fashion. To this end, uncertainty quantification (UQ) has emerged as an important area in modern computational engineering. Today, it is no longer sufficient to predict specific objectives using a particular physical model with deterministic inputs. Rather, a probability distribution function (PDF) or interval bound of the simulation objectives is required depending on whether aleatory or epistemic uncertainties are involved [4]. Epistemic uncertainty (or type B, or reducible uncertainty) represents a lack of knowledge about the appropriate value to use for a quantity, i.e. there is a single correct (but unknown value) [5]. This may be, for example, because a quantity has not been measured sufficiently accurately or because the model neglects certain effects. In contrast, uncertainty characterized by inherent natural randomness is called aleatory uncertainty (or type A, or irreducible uncertainty). For discrete variables, this randomness is parameterized by the probability of each possible value. For continuous variables, the randomness is parameterized by a PDF. Regulatory agencies and design teams are increasingly being asked to specifically characterize and quantify epistemic uncertainty and separate its effect from that of aleatory uncertainty [6].
 1.
Data assimilation in which the input parameters are characterized as aleatory or epistemic (via appropriate PDFs or interval bounds) from observations and physical evidence
 2.
Uncertainty propagation in which the input variabilities are propagated through the mathematical model
 3.
Characterization of the outputs of the numerical simulation in terms of their statistical properties
In principle, these mixed sampling/optimization approaches may be posed in two ways: determining intervals of statistics and determining statistics of intervals:

Intervalsofstatistics can be viewed as an optimization under uncertainty problem with the metric of the optimization defined as a relevant statistic of the aleatory distribution, such as the mean and variance, bounds on a confidence interval, or a reliability index [9, 13]. For each step in the optimization, the aleatory uncertainty is quantified, and the relevant statistics of the distribution are calculated and used as the objective function for the optimization.

Statisticsofinterval poses an optimization problem for each set of aleatory variables, and repeated optimization evaluations over the epistemic design space can be used to determine the relevant statistics of the interval [12].
In the statisticsofinterval approach, gradientbased optimization methods can be employed, assuming that the global extrema in the epistemic design space can be found this way, reducing the cost of each optimization and ensuring very good scaling as the number of epistemic variables increases if adjoint capabilities [14, 15] are used. To reduce the number of required optimizations for low statistical errors, a surrogate model of the optimization results can be constructed with respect to the aleatory variables which can then be sampled exhaustively, ensuring that fewer optimizations are required to characterize the statistics of the interval accurately.
A last important observation for the work in this paper is that deterministic optimization tools are widely used in engineering practice; however, engineering designs do not operate exactly at their design point due to physical variability in the environment. These small variations can deteriorate the performance of deterministically optimized designs. It is, therefore, necessary to account for these uncertainties in the optimization process using optimization under uncertainty (OUU) techniques, which implies that UQ is used in the optimization loop instead of a deterministic simulation. Beginning with the seminal works of Beale [16], Dantzig [17], and Tintner [18], OUU has experienced rapid development in both theory and algorithms. Dantzig considers planning under uncertainty as one of the most important open problems in optimization [19, 20]. Good overviews of the state of the art in the field of OUU are provided by Beyera et al. [21], Sahinidis [19], Giunta et al. [22] and Li [23]. An important subfield in OUU is robust optimization (RO) [24, 25], which can be subdivided into robustdesignbased methods and reliabilitybased methods [26]. Robust design improves the quality of a product by minimizing the effect of the causes of variation without eliminating these causes. The objective here is to optimize the mean performance and minimize its variation, while maintaining feasibility with probabilistic constraints, hence the robust design concentrates on the probability distribution near the mean values. The ability to identify and catalog overly conservative design margins resulting from applying safety factors on top of other safety factors, for example, is an important application for the robust design, which is being increasingly viewed as an enabling technology for design of aerospace, civil, and automotive structures subject to uncertainty [27–30]. The reliabilitybased methods, on the other hand, are predominantly used for risk analysis by computing the probability of failure of a system. Thus, reliability approaches concentrate on the rare events at the tails of the probability distribution.
The outline of the remainder of this paper is as follows: section 'Optimization with mixed aleatory/epistemic uncertainty’ describes the employed OUU approach for mixed aleatory/epistemic uncertainties in detail. Application results of the presented approach are given in section 'Robust design of a transonic airfoil’ and section 'Conclusions’ concludes this paper.
Optimization with mixed aleatory/epistemic uncertainty
Here, the state equation residuals, R, are expressed as an equality constraint, and other system constraints, g, are represented as general inequality constraints. Note, that R (and g) could represent any class of models, however, if gradient information is to be used the models must be differentiable and if surrogate models are to be employed successfully the models must also be relatively smooth. In the case where the input variables are precisely known, all functions dependent on D are deterministic. However, given uncertain inputs all functions in Equation (1) can no longer be treated deterministically.
Objective function evaluation
In this work, the design variables are assumed to have only aleatory or only epistemic uncertainty. Let α represent the variables associated with aleatory uncertainties and β represent variables with epistemic uncertainties, for example, geometric shape variables subject to manufacturing tolerances, or flow boundary conditions subject to random fluctuations. The design variables D = (D _{ α },D _{ β }) are considered to be either the mean values of aleatory uncertainties which are assumed to be statistically independent and normally distributed with $\alpha \sim \mathcal{N}({D}_{\alpha},{\sigma}_{D}^{2})$, or the midpoint of bounds on epistemic uncertainties with β∈I(D), where I(D) = [ D _{ β }s _{ D },D _{ β } + s _{ D }]. Note that σ _{ D } and s _{ D } are treated as fixed but this could be easily changed. One could also derive equations for correlated and/or nonnormally distributed aleatory variables; however, the analysis and resulting equations become more complex [31] and are beyond the scope of this paper.
The functional outputs f _{max} and f _{min} can now be treated as random variables, since their only inputs are random variables with associated probability distributions. In the remainder of this paper, the subscript ext (for extrema) will be used as a placeholder for either max or min. To characterize the probability distribution of f _{ext}, one must extract repeated samples of f _{ext} according to the underlying PDF of α. Each sampling entails solving the appropriate optimization problem, Equation (2) or (3), for the specified sample of α. For these optimizations, an LBFGS [32, 33] algorithm that can utilize function and gradient information is used in this work, thereby reducing the cost of each optimization and ensuring excellent scaling in the number of variables with epistemic uncertainties.
Nonetheless, because of the expense of these optimizations, strategies to reduce the number of samples and thus the computational cost associated with sampling must be employed. For this work, a surrogate is created for f _{ext} as a function of the aleatory variables, which enables the extraction of a large number of samples in order to obtain accurate statistics for very low computational cost. Because the number of aleatory variables used here is relatively small, the required number of training points for an accurate surrogate is small, necessitating only a small amount of optimizations. Because the optimization results are viewed as general random variables, any surrogate can be used to represent the aleatory dependence of the variables. A Kriging surrogate model is employed in this work. The details of the construction of this particular Kriging model, which can utilize gradient and Hessian information and employ a dynamic training point selection, is described in previously published papers [34–37]. The center of the Kriging domain is prescribed by the mean value of α, D _{ α }, and the boundary is taken to be two standard deviations σ _{ D } away in all aleatory input dimensions. This implies that for the normally distributed input variables α more than 97% of all MC samples fall within the Kriging domain and the less accurate extrapolation capabilities of the Kriging surrogate model only need to be used for a small fraction of the samples. Since the purpose of this article is a robust design and not the accurate prediction of the tail statistics, this approach leads to very good results as demonstrated in section 'Robust design of a transonic airfoil’.
where k is the number of standard deviations, σ _{ g }, that the constraint g must be displaced in order to achieve P _{ k }. The software package Ipopt (Interior Point Optimizer) [41] for largescale nonlinear optimization with constraints is used for the solution of the OUU problem given by Equation (5). Ipopt also allows users to impose bound or box constraints on the design variables which can be very helpful in ensuring the stability of the flow analysis by preventing the exploration of too extreme regions of the design space.
Gradient evaluation
where it is relatively straightforward to calculate $\frac{d{\widehat{f}}_{\text{ext}}\left({\alpha}^{k}\right)}{d{\alpha}^{k}}$ from the Kriging surrogate model [42, 43]. This is especially true if the Kriging construction process can be gradientenhanced since this derivative needs to be readily available for this.
that is the derivative of the mean value is approximated by the derivative of just f with respect to D _{ β } at the mean values of the aleatory uncertainty variables α and midpoints of the intervals for the epistemic variables β. This derivative is, in general, nonzero since for the epistemic optimizations, the extreme value is typically encountered on the interval bound. The variances for the problems studied in this paper are much smaller than the mean values which allow the neglection of $\frac{d{\text{Var}}_{{f}_{\text{ext}}}}{d{D}_{\beta}}$. The following section will demonstrate that the presented approach can lead to successful robust designs.
Robust design of a transonic airfoil
Even though one flow solve takes only about 10 s on 12 Intel Xeon processors with 3.33 GHz each it is still prohibitively expensive to obtain the mixed aleatory/epistemic optimization under uncertainty results through either nested sampling or exhaustive sampling of optimization results. In order to provide validation for the OUU framework with mixed aleatory/epistemic uncertainty the uncertainty propagations of aleatory and epistemic variables are validated only for the initial and optimized points and also only using 3,000 sample points. But before these combined results are shown, the uncertainty propagations of aleatory and epistemic variables are validated separately.
Validation of epistemic uncertainty propagation
Method  ${C}_{{l}_{\text{min}}}$  ${C}_{{l}_{\text{max}}}$  ${C}_{{d}_{\text{min}}}$  ${C}_{{d}_{\text{max}}}$ 

Optimization  0.195  0.344  3.56×10^{3}  6.90×10^{3} 
LHS sampling  0.195  0.344  3.56×10^{3}  6.90×10^{3} 
Comparison of NLMC and Kriging aleatory uncertainty propagation
Method  ${\stackrel{\u0304}{C}}_{l}$  ${\sigma}_{{C}_{l}}$  ${\stackrel{\u0304}{C}}_{d}$  ${\sigma}_{{C}_{d}}^{2}$ 

NLMC (Ñ=3,000)  0.269  2.3×10^{2}  5.54×10^{3}  6.1×10^{6} 
Kriging (N=5)  0.270  2.2×10^{2}  5.65×10^{3}  7.5×10^{6} 
Kriging (N=13)  0.269  2.4×10^{2}  5.53×10^{3}  6.1×10^{6} 
Kriging (N=19)  0.269  2.3×10^{2}  5.54×10^{3}  6.1×10^{6} 
Comparison of NLMC and Kriging predictions for the initial guess with two shape design variables
${\stackrel{\u0304}{C}}_{{l}_{\text{min}}}$  ${\sigma}_{{C}_{{l}_{\text{min}}}}$  ${\stackrel{\u0304}{C}}_{{d}_{\text{max}}}$  ${\sigma}_{{C}_{{d}_{\text{max}}}}^{2}$  

NLMC (3,000 optimizations)  0.195  2.1×10^{2}  7.22×10^{3}  7.9×10^{6} 
Kriging (13 optimizations)  0.195  2.1×10^{2}  7.22×10^{3}  7.9×10^{6} 
Robust design results with two shape design variables
k  P _{ k }  ${\stackrel{\u0304}{C}}_{{d}_{\text{max}}}$  ${\sigma}_{{C}_{{d}_{\text{max}}}}^{2}$  ${\stackrel{\u0304}{C}}_{{l}_{\text{min}}}$  ${\sigma}_{{C}_{{l}_{\text{min}}}}$  D _{ u }  D _{ l }  D _{AoA}  D _{ M } 

0  0.5000  7.94×10^{3}  8.1×10^{6}  0.600  3.1×10^{2}  2.50×10^{2}  2.43×10^{2}  1.85  0.711 
1  0.8413  9.95×10^{3}  1.1×10^{5}  0.631  3.4×10^{2}  2.50×10^{2}  2.28×10^{2}  1.85  0.730 
2  0.9772  1.36×10^{2}  1.0×10^{5}  0.657  2.3×10^{2}  2.49×10^{2}  2.40×10^{2}  1.84  0.736 
3  0.9986  1.91×10^{2}  1.8×10^{5}  0.677  2.5×10^{2}  2.49×10^{2}  2.50×10^{2}  1.85  0.745 
4  0.9999  2.10×10^{2}  2.7×10^{5}  0.673  2.0×10^{2}  2.44×10^{2}  2.44×10^{2}  1.85  0.750 
Deterministic  1.36×10^{3}    0.600    1.76×10^{2}  2.06×10^{2}  1.58  0.734 
Comparison of NLMC and Kriging predictions for optimal design with two shape design variables obtained for k =1
${\stackrel{\u0304}{C}}_{{d}_{\text{max}}}$  ${\sigma}_{{C}_{{d}_{\text{max}}}}^{2}$  ${\stackrel{\u0304}{C}}_{{l}_{\text{min}}}$  ${\sigma}_{{C}_{{l}_{\text{min}}}}$  

NLMC (3,000 optimizations)  9.95×10^{3}  1.0×10^{5}  0.636  4.0×10^{2} 
Kriging (13 optimizations) (Ñ=3,000)  1.02×10^{2}  4.9×10^{6}  0.634  3.3×10^{2} 
Kriging (13 optimizations) (Ñ=10^{5})  9.95×10^{3}  1.1×10^{5}  0.631  3.4×10^{2} 
The total number of CFD function equivalent evaluations is approximately:
Number of optimization iterations × 2 (one for minimum lift and one for maximum drag) × 13 (number of training points) × number of optimization iterations per epistemic optimization × 2 (one function and one gradient call) ≈1,600.
Scalability of the framework
Robust design results with six shape design variables
k  P _{ k }  ${\stackrel{\u0304}{C}}_{{d}_{\text{max}}}$  ${\sigma}_{{C}_{{d}_{\text{max}}}}^{2}$  ${\stackrel{\u0304}{C}}_{{l}_{\text{min}}}$  ${\sigma}_{{C}_{{l}_{\text{min}}}}$  D _{AoA}  D _{ M } 

0  0.5000  2.75×10^{3}  2.0×10^{8}  0.600  1.8×10^{2}  1.75  0.600 
1  0.8413  2.87×10^{3}  2.3×10^{8}  0.618  1.8×10^{2}  1.85  0.602 
2  0.9772  3.28×10^{3}  2.0×10^{7}  0.640  2.0×10^{2}  1.85  0.623 
3  0.9986  5.60×10^{3}  5.5×10^{6}  0.666  2.2×10^{2}  1.85  0.645 
4  0.9999  1.31×10^{2}  2.4×10^{5}  0.700  2.5×10^{2}  1.85  0.668 
Deterministic  1.21×10^{3}    0.600    1.85  0.600 
Comparison of NLMC and Kriging predictions for the optimal design with six shape design variables obtained for k =2
${\stackrel{\u0304}{C}}_{{d}_{\text{max}}}$  ${\sigma}_{{C}_{{d}_{\text{max}}}}^{2}$  ${\stackrel{\u0304}{C}}_{{l}_{\text{min}}}$  ${\sigma}_{{C}_{{l}_{\text{min}}}}$  

NLMC (3,000 optimizations)  3.33×10^{3}  2.3×10^{7}  0.640  2.0×10^{2} 
Kriging (13 optimizations) (Ñ=3,000)  3.33×10^{3}  1.8×10^{7}  0.640  2.0×10^{2} 
Kriging (13 optimizations) (Ñ=10^{5})  3.28×10^{3}  2.0×10^{7}  0.640  2.0×10^{2} 
Robust design results with fourteen shape design variables
k  P _{ k }  ${\stackrel{\u0304}{C}}_{{d}_{\text{max}}}$  ${\sigma}_{{C}_{{d}_{\text{max}}}}^{2}$  ${\stackrel{\u0304}{C}}_{{l}_{\text{min}}}$  ${\sigma}_{{C}_{{l}_{\text{min}}}}$  D _{AoA}  D _{ M } 

0  0.5000  3.21×10^{3}  1.2×10^{8}  0.800  1.5×10^{2}  4.00  0.461 
1  0.8413  3.40×10^{3}  1.5×10^{8}  0.816  1.6×10^{2}  4.00  0.485 
2  0.9772  3.61×10^{3}  1.8×10^{8}  0.833  1.6×10^{2}  4.00  0.508 
3  0.9986  3.84×10^{3}  2.3×10^{8}  0.852  1.7×10^{2}  4.00  0.531 
4  0.9999  4.23×10^{3}  6.3×10^{8}  0.877  1.9×10^{2}  4.00  0.562 
Deterministic  1.66×10^{3}    0.800    3.76  0.300 
Conclusions
This article describes the use of gradientbased optimizations and Kriging surrogate models for the propagation of mixed aleatory/epistemic uncertainties for a robust liftconstrained drag minimization problem. Uncertainty due to epistemic variables is propagated via a boxconstrained optimization approach, while the uncertainty due to aleatory variables is propagated via sampling of a Kriging surrogate model built with the optimization results. This statisticsofintervals approach makes robust design under mixed aleatory/epistemic uncertainty possible while at the same time keeping the computational cost for these types of problems manageable.
Declarations
Acknowledgments
This work was partially supported by the University of Dayton Research Council seed grants. I would also like to thank Karthik Mani for making his flow and adjoint solver available as well as Wataru Yamazaki for his Kriging model.
Authors’ Affiliations
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