## 2 Methods
### Multi-Fidelity Extension of PCE and LRA  
In multi-fidelity surrogate modelling, by taking advantage of the low-cost computation of the low-fidelity model data, the accuracy of the surrogate model is increased by integrating a limited number of high-fidelity data with a larger number of low-fidelity data. In the present study, the additive and multiplicative correction approaches are implemented in order to use the orthogonal polynomial-based PCE and LRA methods in multi-fidelity surrogate modelling. [PERSON] et al. (2019) used the PCE method with the additive and multiplicative correction approach for multi-fidelity surrogate modelling and demonstrated that more accurate results were achieved with the correction approach as compared  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Model Name** & **Model Shape** \\ \hline Polynomial Chaos Expansion & \(\sum_{j=0}^{p}a_{j}\Psi_{j}(X)\) \\ Low-Rank Approximation & \(\sum_{l=1}^{R}b_{l}\prod_{l=1}^{M}v_{l}^{(i)}(X_{i})\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The general form of the surrogates.  
Figure 1: Comparing number of unknown coefficients between the PCE and LRA methods for high dimensional problems.  
with those of the single-fidelity model. To the best of the authors' knowledge, a multi-fidelity surrogate modelling study in which the LRA method is used does not exist. Thus, the authors are encouraged to investigate the cost and accuracy performance of the LRA method in multi-fidelity modelling in the present study where the LRA method is extended for multi-fidelity surrogate modelling with additive and multiplicative approaches similar to the PCE method.  
The present multi-fidelity surrogate strategy is based on modelling the difference between low-fidelity \(f_{lf}(\mathbf{X})\) and high-fidelity \(f_{hf}(\mathbf{X})\) model responses. The additive and multiplicative correction approach essentially shifts and scales the low-fidelity model according to the high-fidelity model results. The additive (\(C_{a}(\mathbf{X}_{hf}):\mathbf{X}\to\gamma\)) and multiplicative correction (\(C_{m}(\mathbf{X}_{lf}):\mathbf{X}\to\gamma\)) functions are defined as in Equations (18) and (19), respectively.  
\[C_{a}(\mathbf{X}_{hf})=f_{hf}(\mathbf{X}_{hf})-f_{lf}(\mathbf{X}_{hf}) \tag{18}\]  
\[C_{m}(\mathbf{X}_{hf})=\frac{f_{hf}(\mathbf{X}_{hf})}{f_{lf}(\mathbf{X}_{hf})} \tag{19}\]  
Using the combination of the correction functions, the multi-fidelity surrogate model, which shifts the low-fidelity prediction to high-fidelity data, is given in Equation (20).  
\[\tilde{f}_{mf}(\mathbf{X})=\gamma[\tilde{f}_{lf}(\mathbf{X})+\tilde{C}_{a}( \mathbf{X})]+(1-\gamma)\tilde{f}_{lf}(\mathbf{X})\tilde{C}_{m}(\mathbf{X}) \tag{20}\]  
In Equation (20), the\"expression represents the surrogate model, while \(\tilde{f}_{lf}(\mathbf{X})\), \(\tilde{C}_{a}(\mathbf{X})\), and \(\tilde{C}_{m}(\mathbf{X})\) represent the surrogate models established for the low fidelity, additive function, and multiplicative function results, respectively. Here, \(\tilde{f}_{mf}(\mathbf{X})\) refers to the multi-fidelity surrogate model established by the combination of these surrogate models. In this context, \(\gamma\) is called the weight coefficient and represents the weights of the additive and multiplicative models. The authors of [42] proposed an expression for \(\gamma\) that can be calculated using the norm of the additive and multiplicative correction in the mean-square sense as in Equation (21):  
\[\gamma=\frac{\langle\tilde{C}_{a}^{2}(\mathbf{X})\rangle}{\langle\tilde{C}_{a }^{2}(\mathbf{X})\rangle+\langle\tilde{C}_{m}^{2}(\mathbf{X})\rangle} \tag{21}\]