## 4 Results
### Coupled Spring-Mass-System  
For the MF SPCE models of the coupled spring-mass-system, many different orders for the polynomials were combined with several values of \(\lambda\) to find the combination with the lowest root-mean-square error (RMSE). The resulting best values for the two-dimensional case are shown in Table 6.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**\# of HF Points** & **\# of LF Points** & \(\lambda\) & **Order, P** & **Additive, \(R\)** & **Multiplicative, \(Q\)** \\ \hline
5 & 0 & \(10^{-1}\) & 6 & – & – \\
9 & 0 & \(10^{-4}\) & 9 & – & – \\
17 & 0 & \(10^{-2}\) & 9 & – & – \\
33 & 0 & \(10^{-4}\) & 6 & – & – \\
65 & 0 & \(10^{-8}\) & 7 & – & – \\
97 & 0 & \(10^{-7}\) & 12 & – & – \\
161 & 0 & \(10^{-8}\) & 11 & – & – \\
257 & 0 & \(10^{-8}\) & 11 & – & – \\ \hline
5 & 33 & \(10^{-4}\) & 6 & 1 & 1 \\
9 & 65 & \(10^{-2}\) & 12 & 2 & 4 \\
17 & 97 & \(10^{-8}\) & 10 & 4 & 3 \\
33 & 97 & \(10^{-8}\) & 12 & 6 & 1 \\
33 & 161 & \(10^{-8}\) & 12 & 6 & 1 \\
65 & 161 & \(10^{-8}\) & 12 & 6 & 4 \\
97 & 161 & \(10^{-8}\) & 12 & 6 & 6 \\
161 & 321 & \(10^{-8}\) & 11 & 6 & 6 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Best order and \(\lambda\) values for the two-dimensional coupled spring-mass-system using MF SPCE.  
Figure 16: Kriging: ALOS function RMSE (from left to right, one, two and three dimensions). Ordinary and SPCE-Kriging in dashed and solid lines, respectively.  
Figure 15: PCE: ALOS function RMSE (from left to right, one, two and three dimensions). PCE and SPCE in dashed and solid lines, respectively.  
The results for these settings are plotted on the left in Figure 17, whereas the performance of a MF kriging model is shown on the right. In general, one can see that the SPCE surrogate models outperform the full order ones (compare solid to dashed lines) again up to a point, and the mono-fidelity models are inferior to the multi-fidelity ones (compare black to red lines) for both the MF SPCE and MF kriging models. Overall, the performance of the multi-fidelity ordinary kriging is comparable to the one of MF SPCE and MF SPCE-Kriging.  
In addition to the two-dimensional case, a four-dimensional one where both spring constants and both masses are varied between one and four was considered as well. Again, many different orders for the SPCE are combined with various values for \(\lambda\) to find the combination with the lowest RMSE. The resulting best values are given in Table 7, and the results are shown in Figure 18.  
Once again, one can see that the SPCE surrogate model outperforms the full order one (compare solid to dashed line), and the MF models are better than the mono-fidelity ones (compare red to black lines) for both the MF SPCE and MF kriging models. Overall, the performance of the multi-fidelity ordinary kriging is somewhat better than MF SPCE.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**\# of HF Points** & **\# of LF Points** & \(\lambda\) & **Order, \(P\)** & **Additive, \(R\)** & **Multiplicative, \(Q\)** \\ \hline
9 & 0 & \(10^{0}\) & 6 & – & – \\
33 & 0 & \(10^{-1}\) & 10 & – & – \\
81 & 0 & \(10^{-1}\) & 10 & – & – \\
193 & 0 & \(10^{-1}\) & 7 & – & – \\
385 & 0 & \(10^{-3}\) & 7 & – & – \\
641 & 0 & \(10^{-4}\) & 8 & – & – \\ \hline \hline \end{tabular}
\end{table}
Table 7: Best order and \(\lambda\) values for the four-dimensional coupled spring-mass-system using MF SPCE.  
Figure 17: Spring-mass system (only springs) RMSE (from left to right, PCE and Kriging).  
Figure 18: Spring-Mass System RMSE (from left to right, PCE and Kriging).