4.2 Model order reduction via proper orthogonal decomposition
4.2.1 Homogeneous proper orthogonal decomposition
In (1.26) & (1.27) we introduced a splitting of the state solution y into a controlled homogeneous part and an uncontrolled inhomogeneous one,y=yu+ ˆy where
˙
yu(t) +Ayu(t) =Bu(t) inV0 f.a.a. t∈Θ, yu(0) = 0 inH,
˙ˆ
y(t) +Aˆy(t) =f(t) inV0 f.a.a. t∈Θ, y(0) =ˆ y◦ inH
which allows to define a linear control-to-state operator S : u → yu. This procedure turns out to be very useful in the context of POD model reduction: One observes that discontinuities of the initial value such as jumps or noise have a strong impact on the POD elements, but not on the trajectory of the state solution if the dissipation in the state equation dominates. Furthermore, due to the smoothing property of parabolic equations, both the POD elements and the snapshots y(t) for t > 0 take values in V; accordingly, the progress of the heat flow may be represented essentially by just one or two POD elements where the remaining ones are exclusively required to approximate an initial valuey◦ ∈/ V.
We consider the setting of Run 1 with pure control constraints and control cost parameter σu =1.0e-01. Fig. 4.10 shows the optimal statey¯and the decay of the model reduction errors gained by the classical ansatz, using the trajectory of y¯ directly for the POD basis generation, and of the modified one where the POD elements are generated bySu.¯ Of course, the classical representation y¯` ≈ y¯ is replaced by y¯` + ˆy ≈ y¯ or y¯` ≈ Su,¯ respectively, if the modified basis choice strategy is applied.
0 1
2 3 0
1 2
−2 0 2
timet state
directionx
y(t,x)
10 20 30 40 50
10−10 10−5 100
Reduced order model errors
Pod basis rankℓ
Error(ℓ)
Modified aposti bound Modified reduction error Classical aposti bound Classical reduction error
Figure 4.10:The state trajectory and the control error decay`7→ k¯u−u¯`kL2(Θ,Rm)of inhomogeneous PODx and homogeneous PODx. We observe that the errors decay faster for the homogeneous POD elements, but both procedures reduce the errors to zero if the basis rank`is chosen sufficiently large.
Next we compare the first POD basis elements of the two strategies, see Fig. 4.11. The classical ansatz generates POD functions which develop discontinuities at the jumps of y◦ while the modified POD elements are smooth.
0 0.5 1 1.5 2
−2
−1 0 1
2 first six POD elements
directionx
2 first six POD elements
directionx
Figure 4.11:The first six POD basis elements of the modified homogeneous basis generation strategy (left) and of the the classical inhomogeneous one (right).
Fig. 4.12 shows the rank-`approximation ofy(0)andy(1)for`= 1,2,3,4,5provided by inhomogeneous POD elements. Since the shape ofψ3is already dominated by the jumps iny◦, just the first two basis functions are available for the approximation ofy(t),t >0, while the remaining POD elements ensure an accurate approximation of the reduced initial value. On the other hand, with the unregular POD elements, one achieves a far better approximation of y◦ then with a Fourier series based on trigonometric functions, for instance, cp. [8], Ex. 7.10: The POD solution does not develop oscillations at the jumps ofy`(0).
reduced state value att= 1
directionx
Figure 4.12:The snapshots ofy`(0)andy`(1)for the classical POD ansatz with different POD ranks`.
Now we add noise on the initial value and repeat the investigations performed above.
Fig. 4.13 shows the optimal state solution with respect to the perturbed initial value and the reduced model errors. Here, the classical basis construction does not lead to an error decay on the level of the modified one; the error stagnates on a higher level instead although the optimal trajectory was used to generate the POD elements.
0
Figure 4.13: The state trajectory and the error decay of inhomogeneous PODxand homogeneous POD xin case of noise covering the initial value.
The shape of the first three classical POD elements is similar to the situation without noise. Nevertheless, already ψ2 contains oscillations of small amplitude; from the fourth basis function on, the noise dominates the shape of the element, see Fig. 4.14.
0 0.5 1 1.5 2
−2
−1 0 1
2 first six POD elements
directionx
2 first six POD elements
directionx
Figure 4.14:The first six POD basis elements of the modified homogeneous basis generation strategy (left) and of the the classical inhomogeneous one (right) with noise.
As before, the classical rank-five POD basis leads to a very good pointwise approximation not only of the state snapshots y(t) for t >0, but also of the chaotic initial value, see Fig. 4.15. However, the stong oscillations of the POD elements ψl,l > 3, progressively destabilizes the system matrices and the data projections of the classical reduced order model.
0 0.5 1 1.5 2
−1.5
−1
−0.5 0 0.5 1 1.5
reduced initial value
directionx yℓ ◦(x)
y◦(ψ5) y◦(ψ4) y◦(ψ3) y◦(ψ2) y◦(ψ1) y◦
0 0.5 1 1.5 2
−0.5 0 0.5
reduced state value att= 1
directionx yℓ(t=1,x)
y(ψ5;t= 1) y(ψ4;t= 1) y(ψ3;t= 1) y(ψ2;t= 1) y(ψ1;t= 1) y(t= 1)
Figure 4.15:The snapshots ofy`(0)andy`(1)for the classical POD ansatz with different POD ranks`covering noise.
Comparing the two methods, we see that the modified ansatz performs better in case of challenging initial conditions; nevertheless, also the classical basis generation strategy turns out to be rather stable versus perturbations in the initial value.
In all our tests, no notable difference appeared when different norms where used in the construction of the POD operator R(y), see Fig. 5.4 in [62] where the impact of the choices X = V and X =H was investigated. Further, the addition of time derivatives
˙
y or adjoint states p in the snapshot sample did not give a different decay behavior. In the a-priori error estimes, Sec. 3.1, and for the convergence analysis, Thm. 3.5 & Thm.
3.6, however, the choice of the appropriate norms is essential, of course.