Influence of the Convection Term

In our first experiment we want to analyse the influence of the convection term on the solutions of the bicriterial optimal control problem. For this reason we first run the algorithm with the convection constant c_b = 0, i.e. without any convection, and compare the results with the ones we get forc_b= 1. For this experiment we use h_x= 0.1.

In Figure 6.2 (a) the Pareto front of the problem withc_b = 0 can be seen. First of all, we observe that the algorithm provides us with a smooth approximation of the Pareto front, for which 57 Pareto optimal points P⁰, . . . , P⁵⁶ have been computed, i.e. N_P= 55.

The range of Pareto optimal points reaches fromP⁰= (0.0231,4.5894) toP⁵⁶= (0.6667,0), i.e.

it holds y^id ≈(0.0231,0) and y^nad ≈ (4.5894,0.6667). In particular, it holds minu∈U_adJˆ₁(u) ≈ 0.0231, in which the heating costs ˆJ₂ have to amount to 4.5894 to reach this value. Note that there is still a weight of 0.02 on the cost function ˆJ2in the computation of the first Pareto optimal point P⁰, so that these values are only approximations. However, the slope of the Pareto front at P⁰ is -50, so that a small improvement in ˆJ₁ already implies a huge increase of the heating costs ˆJ2. Therefore, we are satisfied with this approximation for our purposes. We continue the analysis by looking at some optimal controls. The Figures 6.3 (a),(b) and (c) show the optimal controls ¯u⁰, ¯u²⁷ and ¯u⁵⁵. In all controls we can see that all four heaters have the same heating strategy, at least up to computational inaccuracies. This is because of the symmetry of the problem due to constant diffusion in the whole domain and homogeneous Neumann boundary conditions on the entire boundary.

Furthermore, the same qualitative behaviour can be noticed for all three optimal controls and they just differ in their scales. The controls have their maximum in the beginning of the time

Time

Figure 6.3: Optimal controls for different values of cb

interval and decrease in a parabolic way until they reach zero at T = 1. This can be explained by the fact that we want the temperature in the room to follow a given temperature distribution during the whole time interval and to not only coincide with a given temperature distribution at the end time pointT = 1. Therefore, not heating in the beginning would lead to a deviation, which could only be corrected by a disproportionately high heating input. The effect of the optimal heating strategies can be seen in Figure 6.4 (a), where the L²-deviation between the temperature and the desired temperature in the whole time interval, i.e. the graph of the mapping t 7→ kSu(t,¯ ·) + ˆy(t,·)−y_d(t,·)k²_L2(Ω), is shown. In the beginning the temperature distribution approximates the desired temperature distribution quite accurately for the two optimal controls ¯u⁰ and ¯u²⁷. Only after some time we can actually see a deviation, which is of course bigger in the case of ¯u²⁷ than in ¯u⁰. As the heating input decreases to 0 for all three optimal controls, theL²-deviation is increasing in time.

By looking at Figure 6.5 (a), it can be seen that the achieved temperature is below the desired temperature atT = 1 for ¯u⁰. In fact, we observe that at each node and each time instance, the achieved temperature is lower or equal to the desired temperature. As no control is active on the upper boundary in the whole time domain, we can conclude that the cost function ˆJ₁ could be decreased by just heating more, and only the weight of α₂ = 0.02 on the heating costs is preventing this from happening in the optimization routine.

Now, we turn to analysing the results we get forc_b= 1 and comparing them with the previous results. Again, we first look at the Pareto front for this case in Figure 6.2 (b). We observe again a smooth approximation of the Pareto front, this time with 47 Pareto optimal points. By

Time

0 0.2 0.4 0.6 0.8 1

DifferenceinL2 (Ω)

0 0.5 1 1.5 2 2.5 3 3.5 4

¯ u⁰

¯ u²⁷

¯ u⁵⁵

(a)cb= 0

Time

0 0.2 0.4 0.6 0.8 1

DifferenceinL2 (Ω)

0 0.5 1 1.5 2 2.5 3 3.5 4

¯ u⁰

¯ u²²

¯ u⁴⁵

(b) cb= 1

Figure 6.4: Deviation from the desired temperature inL²(Ω)

looking atP⁰ = (0.0191,3.5876), we find that the desired temperature distribution can actually be reached better with less heating costs than in the case of cb = 0. We will explain this in a moment, when we look at the optimal controls. Of course, the end of the Pareto front is again at P⁴⁶ = (0.6667,0). Since we start with a constant initial temperature, not controlling just leaves the temperature constant, even if there is an air flow in the room. This leads to the approximate values minu∈U_adJˆ₁(u) ≈0.0191, y^id ≈(0.0191,0) and y^nad ≈(3.5876,0.6667) (again with a weight of 0.02 on the heating costs for computingP⁰).

By looking at the optimal controls ¯u⁰, ¯u²² and ¯u⁴⁵ in Figure 6.3 (d),(e) and (f), we can clearly see the influence of the convection term. Figure 6.1 illustrates that the air flow goes from the top left corner of the room to the right bottom corner. Consequently, heater two needs to heat the most in order to reach a uniquely distributed rise in temperature in the whole room because the warm air is transported from the second region into the other ones, mainly region three. This is also the reason why heater three has to heat the least, whereas the heaters one and four show similar heating strategies in between the other two heaters. This behaviour can be observed in all three optimal controls and they just differ from each other in the quantitative strength of heating, but not in the qualitative heating strategy. So this characteristic occurs for both cases cb = 0 and cb= 1.

In the optimal control ¯u⁰ one can see that the control of the second heater is active on the upper bound of the constraints in the beginning of the heating process. The consequence is that the temperature in this region is actually overshooting the desired temperature distribution in the beginning. In the further progress the excessive heat of this region is transported into the other regions by the air flow, so that heaters one, three and four actually have to heat way less than in the case cb = 0. In Figure 6.4 (b) one can see that this strategy leads to a slightly bigger deviation from the desired temperature distribution in the beginning compared to the case c_b= 0 because the temperature is too high in region two and too low in regions one, three and four. In the end, however, the strategy of ’overshooting’ pays off and the deviation from the desired temperature is smaller than in the casec_b = 0, while using less heating input. This leads

(a)cb= 0 (b) cb= 1

Figure 6.5: Temperature at T = 1 for the optimal control ¯u⁰ for different values ofc_b to the observed effect that in the problem with convection the desired temperature distribution can be reached slightly better than in the problem with only diffusion, but with way less heating costs.

Yet, we also observe in this case that the desired end temperature is not reached by using any of the optimal control inputs. Figure 6.5 (b) illustrates the temperature atT = 1 that is reached by using ¯u⁰. In contrast to the temperature distribution at T = 1 in the case c_b = 0, the temperature is not homogeneous, but has its maximum in the top left corner and decreases towards the right left corner. Again, this can be explained by the direction of the air flow in the room.

In a last step we want to compare the computation times of both problems. In total, the computation time for solving all Euclidean reference point problems withcb = 0 was 566 s and for the problem withc_b = 1 it was 722 s. Additionally, one has to take into consideration that in the case of c_b = 0, 55 Euclidean reference point problems have been solved in contrast to 45 Euclidean reference point problems in the case of cb = 1. So on average the computation time for one Euclidean reference point problem is approximately 10 s for c_b = 0 and 16 s for c_b = 1. This is an expected result because on the one hand, including a convection term adds dynamics to the optimization problem which are more difficult to handle, i.e. more Newton-CG iterations are needed than without the convection term. On the other hand, solving the state and adjoint equation gets more costly due to the fact that the arising linear equation systems become non-symmetric.

Figure 6.6 shows the computational time for each Euclidean reference point problem in both cases. To be able to show both plots on the same scale, although 56 Pareto optimal points are computed in the case ofcb = 0 and only 46 Pareto optimal points forcb = 1, we use the relative location on the Pareto front on thex-axis, i.e. P⁰=0 andb P^NP+1=1.b

While for c_b = 0 the plot does not show a strong pattern but only a slight decrease of the computational time with increasing number of the Pareto point, one can clearly see a pattern in the plot for c_b = 1. With the exception of two Pareto optimal points, the computation

Relative Location on Pareto Front

0 0.2 0.4 0.6 0.8 1

Time

0 10 20 30 40 50 60

cb = 0 cb = 1

Figure 6.6: Computation times forc_b = 0 andc_b= 1

time is monotonically decreasing and there are several steps which correspond to the decrease of needed Newton-CG iterations in the optimization routine. The reason why the number of needed iterations is decreasing while traversing the Pareto front is that the optimization problem gets smoother if the factor ˆJ₂(u)−z₂ increases in comparison to ˆJ₁(u)−z₁ because ∇²F_z(u) is strictly positive definite with coercivity constant ˆJ2(u)−z2, if ˆJ(u)−z >0 holds (see Lemma 3.38).