6.4 A Further Approach to the ROM
6.4.2 Numerical Results to the Homogeneous Equation
The following experiment is conducted with a grid size of (Nx, Nt) = (200,400) and the reference parameter ˆµ= (6.5·10−4,2.5·104,1), which refers to the FOM solution ˆy. For the ˆy-ROM, a POD basis of size`= 2 was computed from the snapshot differencey(t, µ) =e y(t, µ)−y(t,ˆ µ), while itˆ was computed fromy(t, µ) in the standard ROM approach for comparison. Both approaches are compared by solving the ROM with ˆy and without ˆy. Table 6.4 shows the results of the ˆy-ROM compared to the standard ROM and the FOM for different parameters.
Parameter Model merr mdis msol εm εy εd
µ1= (6·10−2,3·104,1)
FOM 2.22·10−16 0.292461 0.207539
ROM 1.19·10−2 0.292815 0.195269 1.22·10−2 8.22·10−4 −3.53·10−4 ˆ
y-ROM −9.05·10−3 0.291795 0.217257 −9.71·10−3 1.30·10−3 6.65·10−4
µ2= (5·10−4,1.4206·104,1.1)
FOM 1.11·10−16 0.273283 0.226717
ROM 1.44·10−2 0.273375 0.212197 1.45·10−2 1.23·10−3 −9.16·10−5 ˆ
y-ROM −1.36·10−2 0.273195 0.240426 −1.37·10−2 2.51·10−3 8.86·10−5
µ3= (3.21·10−3,2.58·104,0.9)
FOM 5.55·10−17 0.370917 0.129083
ROM 5.30·10−4 0.370916 0.128554 5.28·10−4 2.30·10−6 1.51·10−6 ˆ
y-ROM −9.10·10−5 0.370912 0.129179 −9.60·10−5 1.92·10−7 4.99·10−6
Table 6.4: Comparison of the standard ROM and the ˆy-ROM solving the homogeneous equation.
The ansatz with ˆy is applicable and lead to a more precise approximation considering the final solid mass – consequently, the errorεmis smaller in the ˆy-ROM model. Besides,merrresulted to be smaller as well. Considering the final dissolved mass, orεd respectively, the measures resemble each other – except forµ2, the result of the standard ROM is a little bit better, being in the range from 10−4 to 106. The resulting snapshot errorεy resulted to be higher for ˆy-ROM, except for µ3, where a more accurate result was achieved.
The results show that the ˆy approximation depends crucially on the parameters – both the reference parameter ˆµand the parameters of the ˆy-ROM as well as of the POD basis size. Using a POD basis of higher size resulted in a better performance of the ROM in all cases with the contrary effect on the results of the ˆyapproximation in comparison.
The ˆy decomposition was applied in the following to the equation with aggregation as well, which is outlined briefly in the following subsection.
6.4.3 Numerical Results to the Non-Homogeneous Equation
Inserting the decomposition into the ROM aggregation equation, we get
˙
y`(t) =A`(t, µ)y`(t) +g`(t) +ψTW·
Bbagg(ξ, t)−Dbagg(ξ, t)
| {z }
h`(ξ,t)
, (6.4.7)
where
Bbagg(ξ, t) :=1 2
Z ξ λ=ξmin
β(ξ−λ, λ, t) ˆy(ξ−λ, t) +ψ(ξ−λ, t)y`(t)
· y(λ, t) +ˆ ψ(λ, t)y`(t) dλ
and
Here, all under-braced terms can be pre-computed, which is computationally expensive, but doesn’t affect our ROM computation time. For all terms, where ˆy appears, moreover an interpolation is done.
FOM 1.11·10−16 0.273916 0.226084
ROM 5.80·10−2 0.274395 0.167517 5.85·10−2 2.68·10−3 −4.78·10−4 ˆ
y-ROM 2.12·10−2 0.273919 0.204788 2.12·10−2 1.03·10−2 −2.86·10−6
µ3= (3.21·10−3,2.58·104,0.9)
FOM −1.11·10−16 0.370725 0.129275
ROM 1.34·10−3 0.370723 0.127929 1.34·10−3 4.02·10−6 2.07·10−6 ˆ
y-ROM −8.50·10−3 0.370543 0.137961 −8.68·10−3 2.18·10−3 1.81·10−4
Table 6.5: Comparison of the standard ROM and the ˆy-ROM solving the aggregation equation.
Table 6.5 shows the results for different parameters, using the same setting as in the homogeneous case. The results are comparable to the homogeneous case: For the parametersµ1 andµ2, better results where achieved with ˆy-ROM considering merr,εmandεd. Regarding εy, the error forµ1is better, but forµ2 worse than using the standard ROM approach. Again,µ3 shows how parameter dependent the results are, better results where achieved with the standard ROM here.
Since the model is very parameter sensitive, it is difficult to say a priori whether the ˆy-decomposition will work better than the standard ROM or not. Nonetheless, the final solid mass can be computed more precise in many cases, if a slightly higher error inεy can be condoned – the same holds for the final dissolved mass in most cases.
Conclusion
In this thesis we investigated in the application of POD based model reduction to population balance equations of particulate processes. The model was provided by our collaborators at the Research Center Pharmaceutical Engineering in Graz, however, it was modified as per our convenience and is outlined in detail in this work as well as its numerical realization is presented. The dynamic model consists of non-local PIDEs and describes crystallization processes in a model reactor.
The POD method was successfully applied to the full order model and the reduced order model enables significant advantages in runtime. Numerical examples underline the efficiency of the method applied in this work. Further, multiple snapshot sets where considered for a greedy POD method, which enables us to further determine the number of used basis vectors by a given specification, since there are no general rules yet.
Additionally, a greedy method was conducted over a whole parameter grid, where a POD basis was successfully constructed iteratively by identifying the worst approximations in the grid until a desired tolerance was reached. Finally, a further ROM approach was realized by a decomposition with a reference parameter in order to put additional information into the ROM in advance, which worked well considering the final solid mass of the simulations.
It was found that the reduced order modelling can be applied successfully, however, the results are very parameter dependent. A sensitivity analysis would help here to optimize the POD basis for certain parameter sets. Apart from that, parameter estimation techniques could be applied to the equations in order to localize reference parameters.
Nevertheless, an extension of the model to more dimensions seems natural, as it is done in praxis as well. Other discretization methods, like finite element methods could be applied. We only considered growth and aggregation, an extension of the equations including nucleation and breakage as they occur in real world experiments would be a good complement to the inclusion of several dimensions for the interior coordinate.
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Hiermit bestätige ich, Kevin Sieg, dass die vorliegende Masterarbeit ausschließlich unter Verwendung der angegebenen Quellen durch mich ohne Hilfsmittel erstellt wurde.
Kevin Sieg
Konstanz, den 24. Oktober 2014