∆¯u:= ¯uPNCG−u¯PBFGS
` k∆¯ukU max
i (|∆¯ui|) 1 7.3399×10−3 3.4623×10−3 3 3.2948×10−2 1.0685×10−2 5 3.6749×10−2 1.1301×10−2 7 3.6934×10−2 1.1307×10−2 9 3.6898×10−2 1.1303×10−2 11 3.7671×10−2 1.1538×10−2 13 3.7674×10−2 1.1520×10−2 FE 1.1116×10−2 4.5083×10−3
Table 5.16: Difference∆¯u∈R25 of the computed optimal controls.
Active sets A(¯uh) andA(¯u`)
` FULLHS SENS BFGS
1 {1,2,3,4,5} {1,2,3,4,5} {1,2,3,4,5}
5 {2,3,4,5} {2,3,4,5} {2,3,4,5}
9 {2,3,4,5} {2,3,4,5} {2,3,4,5}
13 {2,3,4,5} {2,3,4,5} {2,3,4,5}
FE {2,3,4,5} {2,3,4,5} {2,3,4,5}
Table 5.17: Identified active sets at PBFGS solutions u¯h andu¯`.
the unconstrained case. Anyway, we are still confronted with the fact, that setting up the Hessian matrix explicetly for the a-posteriori error computation outperforms the temporal advantage gained by the RO optimization.
5.3 Time-dependent (unconstrained) controls
For time-dependent controls u ∈ Uad defined in (1.31) the situation is more ellaborate, since we have to consider also a time-discretization for each control parameter ui(tk), i = 1, . . . , Nu, k = 1, . . . , Nt, i.e., the dimension of the Hessian matrix is now up to (Nu·Nt)×(Nu·Nt), depending on the number of active constraints as we have seen in the previous section! Obviously, this can become very quickly extremly large and the capacities of standard computation platforms and workstations might no longer suffice. Therefore we restrict our setting to a smaller number Nu = 4 of time-dependent control parameter and perform the optimization on coarser spatial as well as time discretization, now for regular FE triangulation of maximum edge size∆x= 3.8×10−2 with Nx= 729mesh points and a time discretization ∆t= 1×10−2 with Nt= 101time points.
Of course, we accept in this regard less accuracy in our numerical solutions (compare Section 2.1.4). As before, the BFGS approximation is initialised by the identity matrix I= diag(1, . . . ,1)
FULLHS SENS BFGS Active sets Mat.Dim.
` η HF; ¯uh,u¯`
η HS; ¯uh,u¯`
η B; ¯uh,u¯`
A(¯uh)≡A(¯u`) n×n
1 n/a n/a n/a 6= n/a
5 2.7421×10−5 2.7462×10−5 2.4375×10−2 {2,3,4,5} 21×21 9 4.7495×10−6 4.7608×10−6 2.3899×10−2 {2,3,4,5} 21×21 13 1.0547×10−5 1.0560×10−5 8.0377×10−3 {2,3,4,5} 21×21
Table 5.18: Matrix deviation at PBFGS solutionsu¯h andu¯`.
FULLHS – SENS BFGS – FULLHS BFGS – SENS Active set Mat.Dim.
` η HF,HS; ¯u`
η B,HF; ¯u`
η B,HS; ¯u`
A(¯u`) n×n 1 5.9604×10−5 8.999×10−1 8.999×10−1 {1,2,3,4,5} 20×20 5 5.5393×10−5 8.999×10−1 8.999×10−1 {2,3,4,5} 21×21 9 5.5386×10−5 8.999×10−1 8.999×10−1 {2,3,4,5} 21×21 13 5.5390×10−5 8.999×10−1 8.999×10−1 {2,3,4,5} 21×21 FE 5.5390×10−5 8.999×10−1 8.999×10−1 {2,3,4,5} 21×21
Table 5.19: Matrix deviation at PBFGS solutions u¯`.
of corresponding dimension (Nu ·Nt)×(Nu ·Nt). The spatial domain and the cuboid shape functions are presented in Figures 5.9 and 5.10. The eigenvalue decay and corresponding energy ratio for the POD basis computation on the coarser spatial grid for ubc ≡ 1 ∈ RNu·Nt can be found in Figure 5.10 and Table 5.22. The EIM basis range reduced to ℘= 16.
0 2 4 6 8 10 12 14 16 18 20
Numberi
10-12 10-10 10-8 10-6 10-4 10-2 100 102 104
λi
Eigenvalues
Figure 5.10: Eigenvalue decay fori= 1, . . . ,20.
` E(`)
1 7.991 614×10−1 3 9.964 854×10−1 5 9.998 697×10−1 7 9.999 906×10−1 9 9.999 982×10−1 11 9.999 998×10−1 13 9.999 999×10−1 Table 5.22: Energy ratio for POD.
Due to the coarser grid, also the average FE and RO computation times needed in a single optimization iteration step to solve the (linearized) state and adjoint equations (P.SE), (P.AE), (P.LSE) and (P.LAE) reduced, see Table 5.23. Of course, reducing a system that is of lower
5.3 Time-dependent (unconstrained) controls
0 0.2 0.4 0.6 0.8 1
x1 0
0.2 0.4 0.6 0.8 1
x2
Spatial domainΩ
Ω1 Ω2
Ω3 Ω4
0 1 0.2 0.4
1
bi(x1,x2) 0.6
Shape functionsbi
x2 0.8
0.5
x1 1
0.5 0 0
Figure 5.9: Spatial domain and shape functions.
dimension from the beginning, we cannot expect the same effectivitiy. But still the numerical benefit of the ROM approach is clearly evident, even when the maximum speed-up in the solution time decreased to a factor around17. The same holds true for the amount of speed-up generated by emplyoing EIM, which can now be stated by a factor between 1.2 and 1.8. For the numerical
∅Time [s] per iteration step
` (P.SE) (P.AE) (P.LSE) (P.LAE) 1 0.111 0.035 0.031 0.039 5 0.079 0.027 0.027 0.031 9 0.053 0.015 0.015 0.022 13 0.054 0.017 0.018 0.024 FE 0.664 0.176 0.265 0.198
Table 5.23: Average solver times for a single optimization iteration step.
anlyses of optimization and a-posteriori error estimation for time-dependent controls we confine our research without restriction of generality on the unconstrained caseUad =U =RNu×Nt with bounds set touai(tk) =−∞andubi(tk) = +∞fori= 1, . . . , Nuandk= 1, . . . , Nt. This allows us to formulate “worst case” estimations concerning the numerical effort, since otherwise the existence of active constraints reduces, accompanied with the matrix dimension, also the computational time, of course (we refer to Section 5.2). The optimization results for NCG and BFGS are presented in Tables 5.24 and 5.25. A comparison of the differences between the computed optimal controls can be found in Table 5.26. A visualization of the FE optimal control u¯h computed by NCG is presented in Figure 5.11. For the a-posteriori error estimation we again compare the matrices HF,HS,B∈R(Nu·Nt)×(Nu·Nt)obtained for FULLHS, SENS and BFGS and analyse their difference by the spectral norm introduced in (5.1) for solutionsu¯h andu¯` generated by the BFGS method.
The results are shown in Tables 5.27 and 5.28. The norm of the perturbation as well as the
` NIt J Time [s] k¯yh−y¯`k kp¯h−p¯`k k¯uh−u¯`k 1 4 7.2291 1.2 2.2235×10−1 1.0108 3.5146 3 4 7.7484 1.3 2.2901×10−2 9.5154×10−2 2.3660×10−1 5 4 7.7680 1.1 2.0581×10−2 5.4454×10−2 1.2615×10−1 7 4 7.7691 1.3 1.0611×10−2 5.2306×10−2 1.1819×10−1 9 4 7.7707 1.2 1.0317×10−2 5.2141×10−2 1.1672×10−1 11 4 7.7701 1.4 1.0096×10−2 5.2029×10−2 1.1784×10−1 13 4 7.7694 1.4 1.0081×10−2 5.1713×10−2 1.1789×10−1 FE 4 7.9857 13.7
Table 5.24: FE and RO optimization results for NCG.
` NIt J Time [s] k¯yh−y¯`k kp¯h−p¯`k k¯uh−u¯`k 1 10 7.2300 1.2 2.2370×10−1 1.0204 3.5080 3 11 7.7499 1.5 2.3141×10−2 9.4656×10−2 2.3450×10−1 5 10 7.7698 1.3 2.0802×10−2 5.3832×10−2 1.2759×10−1 7 10 7.7711 1.3 1.1182×10−2 5.1675×10−2 1.2260×10−1 9 10 7.7726 1.3 1.0907×10−2 5.1382×10−2 1.2104×10−1 11 10 7.7720 1.4 1.0745×10−2 5.1209×10−2 1.2242×10−1 13 10 7.7713 1.4 1.0678×10−2 5.1008×10−2 1.2230×10−1 FE 10 7.9874 8.9
Table 5.25: FE and RO optimization results for BFGS.
smallest eigenvalues computed by the different approaches FULLHS, SENS, BFGS and CGHS are presented in Table 5.29. The corresponding computation times for evaluating the norm of the perturbation and the computation of the smallest eigenvalues are shown in Tables 5.11 and 5.12.
Please note, that for CGHS the average computation time of 100 sec in Table 5.12 has to be considered with caution (and is therefore put in brackets). This is due to the observation, that the computational time varied for the same setting, but multiple computation runs as well as for different numbers`, in a range from20sec up to180sec, while all other methods FULLHS, SENS and BFGS kept stable. A reason for that might be found in the implementation of the function eigs, which utilizes randomly generated start vectors (see MATLAB help on eigs). So far we have no indication how to establish a suitable initial vector that guarantees short computation times. Therefore we suggest this issue for further investigation. Concluding, we present again the a-posteriori error estimation at optimal solutions u¯h,u¯` obtained by the BFGS method in Table 5.32.
Observation
Also for the time-dependent (unconstrained) case we adopt for the optimization procedure in an analoguous way the observations from the previous sections. As remarked at the beginning, due to the coarser discretization, we expect less accuracy in the approximation of the optimal state,
5.3 Time-dependent (unconstrained) controls PNCG – PBFGS
∆¯u:= ¯uPNCG−u¯PBFGS
` k∆¯ukU max
i,k (|∆¯ui(tk)|) 1 2.3784×10−1 2.6487×10−2 3 6.8090×10−1 1.5652×10−1 5 7.0480×10−1 1.2703×10−1 7 7.2843×10−1 1.2321×10−1 9 7.2652×10−1 1.3021×10−1 11 7.3154×10−1 1.2768×10−1 13 7.2600×10−1 1.2754×10−1 FE 6.5833×10−1 1.2032×10−1
Table 5.26: Difference∆¯u∈RNu×Nt of the computed optimal controls.
FULLHS SENS BFGS
` η HF; ¯uh,u¯`
η HS; ¯uh,u¯`
η B; ¯uh,u¯` 1 1.5328×10−1 1.5450×10−3 8.9084×10−1 3 3.9236×10−3 3.9316×10−5 3.0305×10−1 5 3.8047×10−3 3.8143×10−5 1.4285×10−1 7 4.0533×10−3 4.0637×10−5 1.1899×10−1 9 4.1875×10−3 4.1991×10−5 1.1891×10−1 11 4.1273×10−3 4.1378×10−5 1.2152×10−1 13 4.0600×10−3 4.0700×10−5 1.2146×10−1 Table 5.27: Matrix deviation at BFGS solutionsu¯h andu¯`.
adjoint and control y¯h,p¯h andu¯h by the RO approximations y¯`,p¯` andu¯`. Here, more important is to maintain the striven behaviour of the procedure, that for an increase in the number`of POD basis functions also the approximation quality improves (compare Tables 5.24 and 5.25). The stagnation of the further improvement for` >9is traced back on the observation, that the energy ratio E(`) is not significantly changing any more.
Again, the more interesting part is provided by the a-posteriori error estimation. Of course, we find also less accuracy in the comparison of the Hessian matrices and approximations obtained by FULLHS, SENS and BFGS at FE and RO solutions as well as between themselves, pointing out, that SENS seems here to be the most reliable approach (see Tables 5.27 and 5.28). Nervertheless, the smallest eigenvalues preserved for the different approaches allow in all cases a (very) good a-posteriori estimation of the error, again providing narrow upper bounds, even when we identify a slight underestimation for `= 3 in case of BFGS, compare Table 5.32 and Figure 5.12. Let us also mention here, that we are mainly interested in the order of the error, which was finally met.
Completing, we comment again on the computational times. Due to the coarser discretization grid the time for the evaluation of the norm of the perturbation function has reduced, since (P.SE)
0