# A Multi-Fidelity Successive Response Surface Method for Crashworthiness Optimization Problems
## 6 Results and Discussion
### Results on Benchmark Functions
Red dots represent high-fidelity observations; purple dots represent low-fidelity ones. We then analyze the convergence behavior of both problems and include the SRSM approach in the comparison for a more comprehensive evaluation. It is important to note that, for the SRSM method, a quadratic PRS was used. As explained in Section 3, this requires a minimum of six observations when dealing with two variables. Our expectations were indeed met. For both problems, the SRSM method is significantly the slowest to converge and consistently produces qualitatively worse results than the other methods at almost every iteration (see Figure 10). The initial investment in MF-SRS seems to pay off, as it consistently outperforms the other methods in making more accurate predictions, especially in the early steps. The lack of high-fidelity points does not seem to affect its performance, and the efficient use of all available points seems to be a key feature contributing to its effectiveness. The choice of MF-SRS over GP-SRS under these conditions would depend on the computational cost associated with different fidelity levels, as well as the potential for job submissions that utilize multiple computing resources. As our final synthetic problem, we examine the Borehole function, an eight-dimensional problem. This function models the flow of water through a borehole drilled from the ground surface through two different aquifer layers. The flow rate of water, expressed in cubic meters per year (m\({}^{3}\)/year), is described by the properties of both the borehole and the aquifers. The borehole function is often used to compare different types of metamodels and to perform feature importance analysis. Since the optimum of this function is known, we aim to minimize the problem and to compare the predictions of the surface response models at the initial stage. Figure 10: Convergence history of the SRSM, GP-SRS, and MF-SRS methods for the Currin function (**left**) and the constrained Branin function (**right**). Figure 9: First four iterations for the constrained Branin function: MF-SRS (**top row**) and GP-SRS (**bottom row**) are shown. Red dots represent high-fidelity observations, purple dots represent low-fidelity observations, and gray areas represent unfeasible regions. Considering the complexity of this problem, we initiate the process with a dataset of 80 high-fidelity samples for each method. In each subsequent iteration, we add four, six, and nine new samples for the MF-SRS, GP-SRS, and SRSM methods, respectively. It is important to note that, for the SRSM method, a linear polynomial form was employed due to the prohibitively high cost associated with such a large number of function evaluations. Indeed, this form requires \(d+1\) samples, which, although still quite expensive, is more manageable than the \((d+1)(d+2)/2\) samples that would be required otherwise. The initial prediction performance is depicted on the left side of Figure 11, where it is compared with the ground-truth values. The MF-SRS method demonstrates remarkable accuracy across the entire domain represented. The GP-SRS prediction is also strong, although it exhibits a larger average deviation from the ideal diagonal prediction. Notably, toward the minimum values, GP-SRS predictions are affected by a consistent bias that progressively shifts values toward the upper region of the graph. On the other hand, the SRSM method, based on PRS, exhibits the least accuracy, with a consistently higher level of uncertainty across the considered range when compared to the other two methods. These initial prediction results are echoed in the convergence performance, illustrated on the right side of Figure 7:0 While none of the methods manage to reach the global optimum, MF-SRS notably outperforms the other two methods, especially in terms of efficiency, requiring 15 iterations compared to 25 and 34 (according to the convergence criterion defined in Equation (24)) for GP-SRS and SRSM, respectively.