# Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics
###### Abstract  
Efficient Robust Design Optimization (RDO) strategies coupling a parsimonious uncertainty quantification (UQ) method with a surrogate-based multi-objective genetic algorithm (SMOGA) are investigated for a test problem in computational fluid dynamics (CFD), namely the inverse robust design of an expansion nozzle. The low-order statistics (mean and variance) of the stochastic cost function are computed through either a gradient-enhanced kriging (GEK) surrogate or through the less expensive, lower fidelity, first-order method of moments (MoM). Both the continuous (non-intrusive) and discrete (intrusive) adjoint methods are evaluated for computing the gradients required for GEK and MoM. In all cases, the results are assessed against a reference kriging UQ surrogate not using gradient information. Subsequently, the GEK and MoM UQ solvers are fused together to build a multi-fidelity surrogate with adaptive infill enrichment for the SMOGA optimizer. The resulting hybrid multi-fidelity SMOGA RDO strategy ensures a good tradeoff between cost and accuracy, thus representing an efficient approach for complex RDO problems. robust design optimization; uncertainty quantification; gradient enhanced kriging; method of moments; multi-fidelity surrogate; continuous adjoint; discrete adjoint [color=orange!10,colback=orange! computational effort for problems described by costly computer models, such as those encountered in computational fluid dynamics (CFD), despite the use of massive parallelization [8; 9; 10]. In order to reduce the number of costly function calls, it is crucial to select parsimonious UQ methods and optimizers, the overall cost of RDO being typically the product of the cost of the two approaches [11]. Past examples of RDO in CFD include various forms of UQ solvers based on non-intrusive polynomial chaos expansion [11; 12] or surrogate models such as simplex stochastic collocation [13] or kriging [9]. All of them require a number of CFD solves quickly increasing with the number of uncertain parameters, and their direct coupling with MOGA optimizers is not computationally affordable for industrial applications, especially if massively parallel computers are not available. An interesting option for reducing the cost of UQ solves is to use gradient information. A simple method for approximating statistical moments of the cost function by Taylor series expansions is the so-called first-order method of moments (MoM) [14]. Such a method can be remarkably fast if the derivatives of the cost function with respect to the uncertain variables are readily available by means of a discrete or continuous adjoint solver [15; 16; 17; 18; 19; 20]. Nevertheless, its accuracy is limited to Gaussian or weakly non-Gaussian processes with small uncertainties, since higher-order terms become increasingly important for strongly non-Gaussian input distributions. Some improvement can be achieved by using higher-order expansions, but these require information about higher-order sensitivity derivatives, which may represent a delicate and highly intrusive task. A more complete discussion can be found in [21]. An alternative to MoM, better suited for high uncertainty ranges and generic pdf, consists of leveraging gradient information to construct a high-quality surrogate from a reduced number of samples. Such an approach is used for instance in gradient-enhanced kriging (GEK) surrogates [22; 23; 24]. Once the surrogate is available, inexpensive Monte Carlo sampling on the response surface can be used to estimate the required statistics. Massive parallelization is of great help for speeding up the RDO process [10; 11; 25], but it is not promptly applicable for routine industrial use. A way of drastically reducing the required number of function calls consists of replacing the costly CFD or UQ solvers with surrogate models, such as radial basis functions [26], artificial neural networks [27], and kriging [9; 10], approximating variations of the cost functions through the design space.