# Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics
###### Abstract
Such an approach is called a surrogate-based multi-objective genetic algorithm (SMOGA). Further reductions of computational time can be achieved by combining models with various levels of fidelity during the optimization. Such so-called multi-fidelity (MF) models [28; 29] leverage the use of inexpensive, but low-fidelity (LF) models for efficiently exploring the design or stochastic spaces, while using parsimonious high-fidelity (HF) samples to improve model accuracy. Examples of LF models are given by coarse-grid approximations [30; 31; 32; 33], data-fit interpolation and regression models [34], projection-based reduced models [35; 36], machine-learning-based models [37], or simplified models relying on approximations of the underlying physics [38; 39; 40]. In addition to an LF and an HF model, a correlation model is also required for combining data with various fidelity levels. A simple approach consists of linking the HF and the LF models by means of an additive correlation [41]: given an LF model \(f_{LF}(\xi)\) and an HF model \(f_{HF}(\xi)\), it is assumed that \(f_{HF}(\xi)=f_{LF}(\xi)+\delta(\xi)\) where \(\delta(\xi)\) is an error function to be estimated. This approach is accurate enough when HF and LF models have similar scales and a good correlation, as is the case for coarse-grid approximations [42]. As an alternative, a multiplicative correlation can be used [43; 44; 45]; \(f_{HF}(\xi)=\rho(\xi)f_{LF}(\xi)\), with \(\rho(\xi)\) a constant scalar multiplier. A more comprehensive formulation combining the preceding ones was proposed in [32]: \(f_{HF}(\xi)=\rho f_{LF}(\xi)+\delta(\xi)\). This approach is considerably more robust, and it has been extensively used in conjunction with Gaussian process approximation (including kriging) and Bayesian inference [46; 47; 48; 49]). More sophisticated correlations exist [50; 51; 52], but they may be difficult to implement for complex problems. In the present paper, we build on a SMOGA-based RDO technique introduced in [10], relying on the coupling of two nested Bayesian kriging (BK) surrogates: the first one is used to compute the required statistics of the objective functions in the uncertain parameter space, while the second one is used to model the response of these statistics to the design variables. Such an approach is called \"combined kriging\" [53]. An expected improvement criterion is used to update the second kriging surrogate during convergence towards the optimum. This technique has been successfully applied to the design of turbine blades for organic Rankine cycles [9] and to the RDO of the thermodynamic cycle [54]. Assuming that each kriging surrogate requires a number of samples approximately equal to 10 times the cardinality of the parameter space to build a reasonably accurate approximation [55], the nested BK RDO strategy needs \(\mathcal{O}(100\times n_{unc}\times n_{des})\) function evaluations (with \(n_{unc}\) the number of uncertain parameters and \(n_{des}\) the number of design variables) in the first generation of the GA to generate the initial kriging surrogates for the statistical moments. Additional \(\mathcal{O}(10\times n_{unc})\) evaluations are required for each update of the external kriging surrogate. Such a number of function calls is still too expensive for industrial CFD problems, even for uncertain or design spaces of low to moderate dimensionality (up to about eight uncertain or design parameters). GEK surrogates can be used to reduce the number of samples for the UQ step, but GEK-based MOGA is not straightforward in the context of RDO problems, since it requires also the gradient of the statistical moments of the QoI's pdf with respect to the design variables. Obtaining such a piece of information by using efficient adjoint methods is not a trivial task; on the other hand, finite difference approximations are easily applicable, but at the price of a considerable computational expense for high-dimensional design spaces.