## 4 Numerical Test
### Griewank Function Test
However, when the Griewank function is 30D and 220 sample points, KMDS is faster than KPLS-1 by 4.077 s, but the difference in the error score is only 1.2166. For the Griewank function of 70D and 110 sample points, KMDS is 2.3 times faster than KPLS-1, and the difference in the error score is only 0.1328. Therefore, with the increase in the number of sample points or the increase in the number of test dimensions, the test time of the KPLS-1 method is much longer than that of the KMDS method. At this time, the advantage of the KMDS method in terms of time is far greater than the disadvantage in terms of accuracy. For KPLS-2 and KPLS-3, although the accuracy of these two methods is a bit higher than that of KMDS, the modeling efficiency cannot match the KMDS method. The reason may be that the calculation burden of the correlation matrix and hyper-parameters in the KMDS method is greatly reduced during the modeling process, while the KPLS method mainly reduces the calculation cost of hyper-parameters to improve efficiency. Therefore, the efficiency of the KPLS method is lower than that of KMDS. Therefore, the KMDS method can effectively improve modeling efficiency while maintaining a small loss of accuracy. In addition, the standard deviation also reflects the stability of the KMDS method, especially in terms of modeling efficiency. Figure 4 shows the average change trend of the modeling time and modeling error in ten tests of five different modeling methods for the 70D Griewank function. For each test, the Kriging model is firstly established through 10 initial sample points, and the corresponding modeling time is used as the first time. In the next iteration process, whenever a new sampling point is added, the Kriging model is updated again, and the total time spent executing the algorithm is recorded. When the number of newly added sampling points reaches 100, the entire iterative process is stopped. Figure 4 clearly shows how as the number of sample points increases, the modeling error scores of the five modeling methods gradually decrease, and the time it takes for KMDS is much less than that of the other four methods. In addition, when the modeling accuracy of the five different methods gradually stabilized, their RMSE values differed slightly, while the RMSE obtained by the KMDS method was generally small at first. Therefore, the KMDS method is superior to the other four methods in terms of efficiency under the premise of ensuring accuracy for high-dimensional problems. Figure 4: (**a**) The average change trend for the modeling time for the 70D Griewank. (**b**) The average change trend for RMSE for the 70D Griewank. The red and blue curves in Figures 5 and 6 are error bar graphs for the ten test results. The point on the solid line is the average of the test results of each method. The upper end of the error bar is the maximum value of the test result of each method. Similarly, the lower end of the error bar is minimum of the test result of each method. The green and purple curves are the standard deviations of ten tests with different methods. Compared with Kriging, KPLS-1, KPLS-2 and KPLS-3, the proposed method uses the least modeling time, and the error accuracy obtained is not much different from that of KPLS1, KPLS-2 and KPLS-3, which is exactly what we expect. Moreover, the standard deviation chart also reflects the stability of the proposed method.