Reduction of Benchmarks
3.4 Advection - static
3.4.2 Results and Discussion
The full-order model the response can be viewed at figure 3.14. It is clear that we do not have the same amplitude as the response 3.13b. However, since the loaduwas not given, it is impossible for us to check if our HDM corresponds to the same model as the author’s. Given the fact that the functions for reading the matrices have been verified and the function for constructing the problem was supplied by the authors, we see no way to obtain the original author’s response. We will thus keep the model, and for the sake of consistency apply the same problem used in the HDM to the ROMs. The eigenvalues for the POD of different snapshot sets is given in figure 3.15. The magnitude of the eigenvalues does not go down as fast as the eigenvalues for the dynamical problems. Therefore, for parametric problems, more snapshots and higher order ROMs may be required in order to maintain low error.
1.2 1 1.6 1.4
1.8
0
−0.1 1
−0.05 0
k
vc
∆T
HDM ROM
Figure 3.14: Computed HDM and chosen ROM of order 6 ∆S = 10 response for the Static Advection problem.
0 20 40 60 80 100
10−11 10−8 10−5 10−2 101 104
i
POD ordered EigenValues
∆S = 1
∆S= 10
Figure 3.15: Magnitude of the ordered Eigen- values of the POD of the Static Advection prob- lem.
Table 3.6: Relative errors for a ROM of order 6 for the Static Advection Problem. The errors for the System Induced POD (not shown), are all superior to 10.000 %
k=6 ∆S=1 ∆S=5 ∆S=5 LMS ∆S= 10 ∆S=10 LMS ∆S=10 Output
Output 6.77 % 10.00 % 22.50 % 8.94 % 66.40 % 100.00 %
Total 46.00 % 45.00 % 45.00 % 44.00 % 49.00 % 100.00 %
Orthogonal 0.05 % 0.33 % NA 0.45 % NA 100.00 %
Predicted orth. 0.05 % 0.01 % NA 8.89E-06 % NA NA
Collinear 46.00 % 45.00 % NA 44.00 % NA 0.02 %
For the Galerkin Projection the errors can be viewed in figures 3.16, 3.18, 3.21 for∆S= 1,∆S = 5 and∆S = 10, respectively. For ROMs of order 6, table 3.6 gives the actual values of the relative errors for the several snapshot sets and model reduction methods. We can see in figure 3.16, equation 3.6 is well verified. One of the most important lessons in these figures is that even with all the snapshots and with high order models the Output error is still superior to 5%. It seems that we have reached the end of the capabilities of the method: domain decomposition may be required. The problem is specially severe for∆S = 10, where the Output error is close to 10%. For any of the cases the Total error is quite high and its value dominated by the Collinear error. Thea prioriorthogonal error measure is therefore of little use. Trying to use a different formulation for the POD problem nets no improvement. It can be seen from either figure 3.20 (Output Oriented POD) or 3.19 (System Oriented POD) that the Output error for these methods is always close to 100% or far more superior, even if the Output Oriented POD managed to give a Collinear error inferior to the Orthogonal Error. It seems therefore useless to use these methods as a way to decrease the error of the Galerkin Projection. Finally, we attempted to estimate the total error using the method presented in section 2.3.4, but no matter the implementation, the computation of this error estimate took more than one hour to complete, making it impractical for MOR.
For the Least Mean Squares reduction the errors can be seen in figures 3.17 and 3.22 for ∆S = 5 and∆S = 10. The error achieved is quite similar to the normal Galerkin projection for∆S = 5, but for
∆S = 10this is clearly not the case. Here, the error is far superior to that of the Galerkin ROM. This makes it clear some of the limitations of the Galerkin and Least Mean Squares ROM in terms of error : even if we manage to decrease orthogonal error (for the Galerkin ROM) by increasing model order, the actual Output and Total error might not vary at all. This might be solved by a more clever choice of snapshots and hence, the use of a sampling algorithm. The Galerkin ROM appears to have lower error or deviation than the Least Mean Squares method.
In terms of Wall Clock time the HDM took 19 s to complete, with the ROM and SVD solution times shown in table 3.7. As seen before, it seems that for linear systems the most penalizing step in MOR is computing the SVD, as the higher order ROMs are still about 10×faster than the HDM. Each Galerkin ROM test case or an equivalent test case with the Least Mean Squares Reduction takes about 1 second to compute. The modified POD methods take 1 hour (for Output Oriented) or 3 hours (for System Oriented) for each solve. The inefficiency of these last two methods is due to the fact that we further complicate the SVD of the snapshot set. This is especially the case for the System Oriented POD, where a full SVD solve is required for each parameter set, and the Snapshot Set must be always available. This quickly blows up the usage of RAM (which was the case), and the system is forced to use the swap space of the hard drive, seriously impacting the performance of the computation (even if the machine was using a solid-sate drive). For the case of the Output oriented ROM, this only impacts the SVD computation, but it does so in a severe way, destroying the performance of the method.
Table 3.7: Wall clock times for the HDM, SVD computation and the several ROMs for the Static Advection problem. The ROMs using the Output or System Oriented take slightly more than one hour to compute.
HDM Solve : 19.07 s
ROM order Galerkin LMS Others
2 0.69 s 0.72 s >1 hour
4 1.05 s 1.16 s -
6 1.32 s 1.57 s -
8 1.87 s 2.05 s -
10 2.39 s 2.70 s -
∆S 1 5 10
SVD time 164.9 s 33.8 s 17.9 s
0 5 10 15 20
10−7 10−5 10−3 10−1
Model Order
RelativeError
ROM error for∆S=1 with Standard POD
eout
et e⊥ epriori⊥
e//
Figure 3.16: ROM error for∆S=1 with Standard POD for the Static Advection problem.
2 4 6 8 10
10−1 100
Model Order
RelativeError
ROM error for∆S=5 with Least Mean Squares
Output Total
Figure 3.17: ROM error for ∆S=5 with Least Mean Squares Reduction Standard ROM for the Static Advection problem.
2 4 6 8 10
10−7 10−6 10−5 10−4 10−3 10−2 10−1 100
Model Order
RelativeError
ROM error for∆S=5 with Standard ROM
eout
et
e⊥ epriori⊥
e//
Figure 3.18: ROM error for ∆S=5 with Stan- dard POD for the Static Advection problem.
2 2.5 3 3.5 4 100
101 102 103
Model Order
RelativeError
ROM error for∆S=5 with System Induced POD Output
Total
Figure 3.19: ROM error for∆S=5 with System Induced POD for the Static Advection problem.
2 4 6 8 10
10−142 10−104 10−66 10−28 1010
Model Order
RelativeError
ROM error for∆S=10 with Output Oriented POD
eout
et e⊥ epriori⊥
e//
Figure 3.20: ROM error for∆S=10 with Output Oriented POD for the Static Advection prob- lem.
2 4 6 8 10
10−9 10−7 10−5 10−3 10−1 101
Model Order
RelativeError
ROM error for∆S=10 with Standard POD
eout
et
e⊥ epriori⊥
e//
Figure 3.21: ROM error for∆S=10 with Stan- dard POD for the Static Advection problem.
2 4 6 8 10
100 100.5
Model Order
RelativeError
ROM error for∆S=10 with Least Mean Squares Output
Total
Figure 3.22: ROM error for ∆S=10 with Least Mean Squares Reduction Standard ROM for the Static Advection problem.
(a) The Static Diffusion problem. (b) HDM output.[74]
Figure 3.23: Static Diffusion Problem in [74]. In our model there is no heat flux coming from the left.
Since our model has different coefficients, the responses should be slightly different.