# A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems
## 6 Results and Discussion
### Results on Benchmark Functions  
We continue our experiments with tests on benchmark functions, aiming to evaluate the convergence performance of the MF-SRS method in terms of speed and effectiveness. We aim to compare our method with its simplified variant, GP-SRS, which relies solely on GP and high-fidelity, and with the SRSM method based on PRS. Given our primary focus on crashworthiness problems, the objective remains to identify the best achievable local optimum with the least number of function evaluations. Our general expectation is to see a clear predominance of Gaussian process-based methods over polynomial regression-based approaches. We also hope to see possibly more efficient convergence of MF-SRS than GP  
Figure 6: From left to right: GP, AR1, and NARGP reconstructions of the Forrester function. All use 5 high-fidelity observations; multi-fidelity methods also use 10 low-fidelity observations. Figure 7: From left to right: GP, AR1, and NARGP reconstructions of the Sinusoidal Wave function. All use 7 high-fidelity observations; multi-fidelity methods also use 14 low-fidelity observations. SRS, especially in the first few iterations, although we do not necessarily expect superior performance in the long run. For this evaluation, we leverage the work of [PERSON] and [PERSON] [53], who have gathered a set of well-known benchmark functions from the literature, each with at least two levels of fidelity. The mathematical formula of those functions and further details regarding the input variables can be found in Appendix B. Specifically, we investigate:  
* The Currin function [54]: a 2D single-objective problem. * The Branin and Himmelblau functions [55]: a 2D double-constrained single-objective problem. * The Borehole function [54]: an 8D single-objective problem. For the first problem, the goal is to either maximize or minimize the negative counterpart of the Currin function. We specify that each method starts with an equal set of high-fidelity points, 10 in this case. After each iteration, both GP-SRS and SRSM add six more high-fidelity samples, while MF-SRS introduces only four. The ratio of low-fidelity to high-fidelity samples remains constant at 2:1. For these analytical functions, which do not come with any evaluation cost, we will consistently use twice the number of low-fidelity samples compared to high-fidelity samples, similar to what is shown in [36]. The comparison between MF-SRS and GP-SRS of the first four iterations is shown in Figure 8. The first iteration highlights the value of low-fidelity samples; while GP-SRS suffers from severe overfitting complications, making it difficult to extract valuable information, MF-SRS shows a more stable progression, guiding the optimization toward the optimum despite having fewer high-fidelity samples. Both methods appear to converge quickly to the global minimum. In a second two-dimensional test, we focus on minimizing a double-constrained Branin function. We impose a constraint on the objective function itself, thereby preserving the nearly flat region where the three global minima of the function lie. In addition, we further narrow down this region through a constraint applied to the Himmelblau function, as shown by Equations (30)-(33):  
\[\min_{\mathbf{x}} f_{branin}(\mathbf{x})\] (30) subject to \[g_{himmelblau}(\mathbf{x})\geq 60, \tag{31}\] \[g_{branin}(\mathbf{x})\leq 80,\] (32) \[-5\leq x_{i}\leq 15,\ i=1,2. \tag{33}\]  
From the first four iterations shown in Figure 9, it is quite clear that the combination of the given constraints poses a significant challenge for accurate representation, especially with only 15 high-fidelity observations available. However, while the GP-SRS method bounces between opposite sides of the variable domain during the initial iteration steps, the MF-SRS method exhibits more stable behavior. It guides the RoI toward a global minimum from the first iteration, consistently avoiding the infeasible region. Figure 8: First four iterations for the Currin function: MF-SRS (top row) and GP-SRS (bottom row) are shown.