# A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems
## 6 Results and Discussion
### Synthetic Illustrative Problems  
We begin with a visual illustration of two well-known problems in the literature of multi-fidelity functions: The Forrester function and the Sinusoidal Wave function have been presented in the works of [36; 52], respectively. We break down the mathematical formula of the Forrester into its fidelities in Equations (26) and (27), while the high- and low-fidelity functions of the Sinusoidal Wave are described in Equations (28) and (29). The multi-fidelity Forrester function is  
\[f_{high}(x)=(6x-2)^{2}\sin(12x-4) \tag{26}\] \[f_{low}(x)=0.5f_{high}(x)+10(x-0.5)-5. \tag{27}\]  
The multi-fidelity Sinusoidal Wave function is  
\[f_{high}(x)=(x-\sqrt{2})\cdot f_{low}^{2}(x) \tag{28}\] \[f_{low}(x)=\sin(8\pi x). \tag{29}\]  
The inclusion of these formulas is important to show the different types of mappings between the fidelities of these two functions. The dominance of the \"linear mapping\" in the Forrester function is clearly seen in the \(0.5f_{high}\) term, while the quadratic mapping between the fidelities of the sine wave is also recognizable. The upper part of Figure 5 shows the overall trend of these functions. The high-fidelity functions are shown in red, while the low-fidelity ones are shown in violet. The lower part of the same figure plots the correlation between the fidelity levels. It can be seen that, unlike the sine wave, the Forrester function has a predominantly linear type of correlation. To caution readers against drawing simplistic conclusions, we would like to point out that the idea of a linear mapping between the high-fidelity and the low-fidelity representations in the Forrester function is just a simplification, commonly used for illustrative purposes in multi-fidelity contexts. This does not mean that a plot of \(f_{high}(x)\) over \(f_{low}(x)\) will yield a perfectly straight line over the entire range of the function. Indeed, this would have been the case if there were no deviations induced by the bias term \(10(x-0.5)\), which is linear in \(x\) but not in \(f_{high}(x)\). To observe the practical implications of the theory outlined in Section 4, we approach comparing GP, AR1, and NARGP methods on these two functions. As a rule of thumb for these experiments, we always used twice as many high-fidelity observations for the low-fidelity level. Note that this number depends strongly on the problem at hand, and  
Figure 5: The first row shows the ground-truth values of the Forrester (first column) and Sinusoidal Wave (second column). The bottom row illustrates the respective mapping between the fidelities. especially on the \"cost\" ratio of the fidelities. While comparing GP and multi-fidelity models may seem unfair, it serves as an insightful exercise. This comparison highlights the capability of multi-fidelity models to assimilate valuable information from alternative sources, potentially leveraging it for faster convergence. The results for the Forrester function, based on five observations, are shown in Figure 6. Without delving into error metrics, it is evident that the multi-fidelity methods perform similarly, representing the function with greater precision compared to GP. Indeed, NARGP and AR1 show comparable performance in terms of mean and uncertainty. The most noticeable differences with GP appear at domain boundaries and within the valley of the optimum. In these regions, low-fidelity observations contribute valuable information that a conventional GP model cannot exploit. A different scenario can be seen in Figure 7. Based on seven high-fidelity observations, the GP and AR1 methods appear to have a remarkably similar posterior distribution that deviates significantly from the true sinusoidal trend. While the subpar metamodeling of GP can be justified by the low sampling rate with respect to the period of the function, the performance of the AR1 is rather surprising. It seems to struggle to exploit the information from the low-fidelity observations.