# A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems
## 4 Multi-Fidelity Metamodeling
### Non-Linear Multi-Fidelity Metamodeling  
Although [PERSON] and [PERSON] scheme has already been proven successful in the literature [34; 35], there are cases that are very common in realistic modeling scenarios, where cross-correlations between models of different fidelity, while very informative, show a more complex pattern than a simple linear correlation. In such a case, a linear autoregressive scheme might work well only in narrow ranges of the input parameters, but would not be able to learn a more comprehensive non-linear correlation in the global domain. With this in mind, we briefly introduce the non-linear autoregressive multi-fidelity GP regression (NARGP) algorithm introduced by [PERSON] et al. [36], which aims to extend the [PERSON] and O'Hagan scheme to more complex non-linear and space-dependent correlations. We assume the design datasets to have a nested structure, such as \(\mathcal{D}_{1}\subseteq\mathcal{D}_{2}\subseteq,\ldots,\mathcal{D}_{s}\), and we generalize the scheme of Equation (17):  
\[f_{t}(\mathbf{x})=z_{t-1}\cdot f_{t-1}(\mathbf{x})+\delta_{t}(\mathbf{x}) \tag{19}\]  
where the mapping between a low model and its higher fidelity counterpart is denoted by the unknown function \(z_{t-1}\). To enable flexible and non-linear multi-fidelity algorithms, a GP prior is assigned to \(z\). Since \(f_{t-1}\) is also assigned a GP prior, the posterior distribution of \(f_{t}\), because of the functional composition of two priors, is no longer Gaussian.  
The generalization outlined in Equation (19) does not come without implications in terms of computational cost and implementation complexity. This is mainly because of the need to use variational Bayesian methods to approximate intractable integrals. Therefore, to favor an algorithmic complexity closer to that of GP regression, [PERSON] et al. proposed a reformulation of the generalized scheme in which the GP prior \(f_{t-1}\) is replaced by the GP posterior of the first inference level \(f_{st-1}\):  
\[f_{t}(\mathbf{x})=g_{t}(\mathbf{x},f_{st-1}(\mathbf{x})) \tag{20}\]  
where \(g_{t}\) is assigned the prior of Equations (21) and (22):  
\[f(\mathbf{x})\sim\mathcal{GP}(f_{t}|0,k_{t};\mathbf{\theta}_{t}) \tag{21}\]  
\[k_{t_{g}}=k_{t_{p}}(\mathbf{x},\mathbf{x}^{\prime};\mathbf{\theta}_{t_{p}})\cdot k_{t_{f}} (f_{st-1}(\mathbf{x}),f_{st-1}(\mathbf{x}^{\prime});\mathbf{\theta}_{t_{f}})+k_{t_{g}}(\bm {x},\mathbf{x}^{\prime};\mathbf{\theta}_{t_{g}}),\] (22 where \(k_{t_{p}}\), \(k_{t_{f}}\), and \(k_{t_{s}}\) are squared exponential anisotropic kernel functions based on Automatic Relevance Determination (ARD) weights (for additional details, please refer to [7]) and \(\mathbf{\theta}_{t_{p}}\), \(\mathbf{\theta}_{t_{f}}\), \(\mathbf{\theta}_{t_{g}}\) represent their hyperparameters. With this scheme, it is possible to infer the high fidelity response through \(g_{t}\) by projecting the lower fidelity posterior onto a latent manifold of dimension \(d+1\).  
The lowest fidelity level of the proposed recursive scheme is still a common GP regressor with kernel function \(k_{1}(\mathbf{x},\mathbf{x}^{\prime},\mathbf{\theta}_{1})\). Therefore, its posterior distribution is still Gaussian. However, as for all the other levels, the predictions have to be made given a test point \((x_{*}f_{st-1}(\mathbf{x}_{*}))\). Therefore, for \(t\geq 2\), the posterior distribution is no longer Gaussian and is defined by the following integral:  
\[p(f_{st}(\mathbf{x}_{*}))=\int_{X}p(f_{t}(\mathbf{x}_{*},f_{st-1}(\mathbf{x}_{*}))|\mathbf{y}_ {t},\mathbf{x}_{t},\mathbf{x}_{*})p(f_{st-1}(\mathbf{x}_{*}))\,dx_{*}. \tag{23}\]  
Bear in mind that solving the integral of Equation (23) is commonly performed by a recursive Monte Carlo approach [36].