## 4 Numerical Examples
### Nonseparable Spectrum
#### 4.1.1 Parametric Analysis  
The error of the suggested method was mostly due to the approximations of POD interpolation to \(\sqrt{S_{ji}(\omega,t)}\) and the interpolation to [PERSON]'s decomposition results  
Figure 2: Flowchart of the presented approach. \(\beta_{jk}(\omega)\). Thus, two kinds of errors were defined to study the influence of parameter selection on the simulation accuracy. The POD interpolation error is defined as  
\[E_{1}=\frac{1}{n}\sum_{j=1}^{n}\frac{\int_{0}^{T}\int_{0}^{\omega_{\omega_{k}}} \left|\sqrt{S_{jj}(\omega,t)}-\sqrt{\widetilde{S}_{jj}(\omega,t)}\right|d \omega dt}{\int_{0}^{T_{0}}\int_{0}^{\omega_{\omega}}\sqrt{S_{jj}(\omega,t)}d \omega dt} \tag{41}\]  
where \(\sqrt{S_{jj}(\omega,t)}\) is the target value calculated by the spectral model and \(\sqrt{\widetilde{S}_{jj}(\omega,t)}\) is the approximate reconstruction using the POD interpolation. The parameters involved in the POD interpolation were the selected POD mode number \(N_{\Phi}\) and the number of time and frequency interpolation points \(N_{\overline{t}}\) and \(N_{\overline{\omega}}\). The interpolation error of Cholesky decomposition is defined as  
\[E_{2}=\frac{2}{n(n+1)}\sum_{j=1}^{n}\sum_{k=1}^{j}\varepsilon_{jk} \tag{42}\]  
\[\varepsilon_{jk}=\frac{\int_{0}^{\omega_{\omega}}\left|\beta_{jk}(\omega)- \widetilde{\beta}_{jk}(\omega)\right|d\omega}{\int_{0}^{\omega_{\omega}} \beta_{jk}(\omega)d\omega} \tag{43}\]  
where \(\widetilde{\beta}_{jk}(\omega)\) is the interpolation result of the element obtained by the Cholesky decomposition of \(|\mathbf{\Gamma}(\hat{\omega})|\); \(\beta_{jk}(\omega)\) is the element obtained by the [PERSON] decomposition of \(|\mathbf{\Gamma}(\omega)|\); \(\varepsilon_{jk}\) is the relative error between \(\beta_{jk}(\omega)\) and \(\widetilde{\beta}_{jk}(\omega)\). Obviously, the interpolation error of Cholesky decomposition was only related to the number of frequency interpolation points \(N_{\omega}\). Figure 3 shows the effect of the mode number \(N_{\Phi}\) and the time and frequency interpolation point numbers \(N_{\overline{t}}\) and \(N_{\overline{\omega}}\) on the POD interpolation accuracy, where the number of time interpolation points was the same as the number of frequency interpolation points. It was discovered that the considered mode number had a greater impact than the interpolation point number. With the increase in \(N_{\Phi}\), the error decreased rapidly. When \(N_{\Phi}\geq 11\), the POD interpolation error stabilized and kept at a low level (\(E_{1}<1\%\)). Figure 4 shows the effect of different time and frequency interpolation points on the POD interpolation accuracy. As can be seen, the number of time interpolation points and the number of frequency interpolation points had the same effect on POD interpolation, and the more interpolation points, the smaller the error. Within the acceptable error range (\(E_{1}<1\%\)), \(N_{\overline{t}}\geq 60\) and \(N_{\overline{\omega}}\geq 80\) were enough for obtaining a good estimate result. Based on Equations (33) and (34), the approximate reconstruction \(\sqrt{\widetilde{S}_{jj}(\omega,t)}\) could be obtained with POD interpolation, as shown in Figure 5b. The target \(\sqrt{S(\omega,t)}\) could be calculated with Equation (37), as shown in Figure 5a. There was no discernible difference between the two results, as can be seen. Therefore, the decoupled function obtained with POD interpolation could describe the target function well. The interpolation error of the Cholesky decomposition result could be calculated with Equations (42) and (43). Figure 6 shows the relative error \(\varepsilon_{jk}\) and the average relative error \(E_{2}\) for three specific frequency interpolation points \(N_{\overline{\omega}}=10,\ 20\) and \(60\). It can be seen that the interpolation errors of the elements of \(\beta_{j1}(\omega)\) in the first column of the lower triangular matrix \(\mathbf{B}(\omega)\) were evidently larger than that of other elements.