# Low-Rank Approximation of Frequency Response Analysis of Perforated Cylinders under Uncertainty
## 5 Numerical Experiments: Trommel Screen
### Low-Rank Approximation Concerns  
As demonstrated above, for perforated shells, the interactions between loading, perforation patterns, and deterministic parameters, such as thickness and frequency in this context, are complex. In particular, in those situations where the local features become dominant, it is intuitively clear that the low-rank approximation may lead to unexpected errors. On the other hand, in the global feature range of problems, there is no reason not to employ low-rank approximations. Given the mesh of Figure 2 with uniform \(p=6\), one ends up with a system of 481,140 degrees of freedom after the boundary conditions are deployed. Remarkably, the Krylov basis has a rank \(=4\) after orthogonalisation for every instance tested here.  
In Figure 5, the frequency responses of the cases \(K=20\) and the concentrated load at \(t=1/1000\) are shown. For the Fourier case, the relative errors in the total energy are fairly large, but the frequencies correct, with the exception of the secondary peak at the low frequencies. In the concentrated load case, the low-rank approximation fails and the dominant frequency is not correct.  
Let us further focus on the case of Figure 5b. In Figure 6, the convergence of the low-rank approximation as the number of basis vectors grows has been added in two cases. The basis has a rank \(=4\), so the sequence rank \(=2,3,4\) is complete. In the case of \(K=5\), \(t=1/100\), already rank \(=3\) is sufficient. However, in Figure 6b, strong variation within the sequence is observed. In particular, in the case of rank \(=3\), the observed energy is very small \(=0.001\) exactly at the frequency \(f=70\), where the full version has its maximum \(=0.072\). In this case, the \(L^{2}\)-norms of the different vector field components of the solution (see Appendix A.2) reveal that, in the low-rank approximation, the boundary layers in the axial direction have amplitudes that are an order of magnitude smaller than those of the full solution. These data are tabulated in Figure 6c.  
Figure 5: Frequency response. Comparison of the full (solid line) and low-rank approximation (dashed line) over a range of frequencies. In the detail plot, one standard deviation (std) from the collocation solution (\(M=6\)) is added. Green indicates higher energies for the full and red for the low rank with one standard deviation.  
An interesting question is whether stochastic modelling of the material parameters could recover or contain the right solution in the case shown in Figure 5d. In order to test this, the collocation with \(M=6\) and second-order polynomials (729 collocation points in tensor quadrature) was used with pointwise low-rank approximation. It appears that the approximation error is orders of magnitude larger than anything that could result from simple perturbations of the material parameters. This is not necessarily a general phenomenon; in Figure 7, an example where this compensation actually happens is shown.  
It has to be emphasised that the message here is that, indeed, there are extreme configurations where the low-rank approximations may fail. It is reasonable to expect that, in usual design situations, these cases can be identified a priori.