The algorithm.The reduced-order approximations yh,`and ph,` for the solutions to (1b) and (3) depend on the chosen control ubc ∈Uad and on the number ` of POD basis functions. Hence, the approximations can be improved by utilizing a POD basis generated from a better controlubc∈Uad(i.e. by computing a new POD basis) and/or by increasing the number`(i.e. by using more POD basis functions).
We pursue the approach of successively adapting the reduced-order models in the course of optimization just by increasing the number`or, if this does not lead to a sufficient accuracy, by computing a new POD basis from the current control iterate (and then adapting the number`).
Our TR-POD algorithm belongs to the class of trust region methods that use a quadratic model function with inexact gradient information. For an introduction to trust region methods we refer to [25, 27] whereas extensive details on trust region methods are given in [9]. Assume we are in iterationkof Algorithm 1 with current iterateukand with a given accurate reduced order model. In line 6 the cost functional Jˆhis replaced by the quadratic model
mh,`k (d) =Jˆh(uk) +h∇Jˆh,`(uk),diU+1
2hHkh,`d,diU, (31) whereHkh,`denotes the POD Galerkin approximation of the reduced Hessian ˆJ00(uk).
In line 7 a trial stepdkis computed by approximately solving the so-called trust re-gion subproblem with trust rere-gion radius∆k. Here, we compute a truncated Newton stepHkh,`dk=−∇Jˆh,`(uk)with the CG-Steihaug algorithm; see [33, p. 171]. Control constraints are handled by using active set projection according to [27, Section 5.4].
The CG-Steihaug algorithm provides sufficient model decrease since the obtained step is at least as good as the so-called Cauchy step. Lines 9 to 18 coincide with a standard trust region method. In these lines it is decided whether or not to accept the trial pointuk+dkas next iterate and the trust region radius gets adjusted. This deci-sion is based upon the quotientρkin line 8 which compares the reductionpredkof the model function to the reductionaredkof the cost functional and which provides a measure for the approximation quality of the model function. In caseρk≥η1(e.g.
η1=0.2), the trial point is accepted. If evenρk≥η2(e.g.η2=0.8), the
approxi-Algorithm 1(TR-POD method)
Require: An inital trust region radius∆0>0, an initial controlu0, a maximal number`maxof POD basis functions, constantsη1,η2,γ1,γ2,γ3,ζCCsatisfying 0<η1≤η2<1, 0<γ1≤ γ2<1≤γ3, 0<ζCC<1−η2, safeguardσCE≥1;
1: Compute initial POD basis of length`maxand choose`≤`maxsuch that (32) is satisfied;
2: Setk=0;
3: ifk>0then
4: adapt reduced order model using Algorithm 2;
5: end if
6: Build up the model functionmh,`k ;
7: Determine an approximate solutiondk∈Uto
minmh,`k (d) s.t. uk+d∈Uad,kdkU≤∆k; 8: Compute the quotient
ρk=aredpredk
k = J(uˆ k)−J(uˆ k+dk)
mh,`k (0)−mh,`k (dk); 9: ifρk≥η2then
10: Setuk+1=uk+dkand∆k+1∈[∆k,γ3∆k];
11: Setk=k+1 and go to line 4;
12: else ifη1≤ρk<η2then
13: Setuk+1=uk+dkand∆k+1∈[γ2∆(k),∆k];
14: Setk=k+1 and go to line 4;
15: else ifρk<η1then
16: Setuk+1=ukand∆k+1∈[γ1∆k,γ2∆k];
17: Setk=k+1 and go to line 7;
18: end if
mation quality of the model function is as good that the trust region radius can be increased for the next iteration. Otherwise the radius should be kept or decreased.
In caseρk<η1, the quadratic model is a poor approximation on the trust region and the trial point is rejected. The trust region subproblem has to be solved again for a smaller trust region radius. Note that the model function and hence the reduced order model is kept in this case.
In the upper part of Algorithm 1 gradient accuracy is guaranteed by adapting the reduced order model in a sufficient manner. The adaption is based upon the relative error condition
egr(uk)≤ζCCk∇Jˆh,`(uk)kU with 0<ζCC≤1−η2, (32) which was introduced by Carter in [8] and which leads to global convergence. At the beginning of Algorithm 1 we compute an initial POD basis of length`max. Conse-quently, the high-dimensional solutionsyhandphare given, so that the computation of the high-dimensional gradient∇Jˆh(uk)is extremely cheap. Therefore, the Carter condition (32) can be directly tested to determine the required number `of POD basis functions. In all other iterations the reduced order model gets adapted at the beginning in line 4 using Algorithm 2. In Algorithm 2 the a-posteriori error estimate
∆apo` (u)from Proposition 3 is computed instead of the gradient erroregr(uk)as long as the high-dimensional adjoint state phis not given. In order to compensate for
Algorithm 2(Update strategy) 1: if(33) not fulfilled for`=`maxthen
2: Eventually set new`max, compute new POD basis of length`maxand choose`≤`maxsuch that (32) is fulfilled;
3: else
4: if(33) not fulfilled for`then
5: Choose`≤`maxsuch that (33) is satisfied;
6: else
7: Keep reduced order model;
8: end if 9: end if
overestimation we set a safeguard factorσCE≥1 and test for 1
σCE∆apo` (u)≤ζCCk∇Jˆh,`(uk)kU. (33) Concretely, this means that inequality (33) is first tested for the maximum number
`maxof POD basis functions in order to see if gradient accuracy can be reached with the current POD basis. If this is the case, the POD basis is kept and the number of required POD basis functions is determined using∆apo` (u)together with condition (33). If not, a new POD basis of `max is computed. Hence, the high-dimensional adjoint statephis given and the high-dimensional gradient∇Jˆh(uk)is computed for determining the required number of POD basis functions with the Carter condition (32).
In line 8 of Algorithm 1 we have to compute the high-dimensional stateyh(uk+dk).
If the trial pointuk+1=uk+dkis afterwards accepted, the associated stateyh(uk+1) is already computed. Consequently, we somehow loose the speedup factor shown in Table 1. This can be avoided by utilizting the approach presented in [36]. Here, the trust-region subproblem (cf. line 7 of Algorithm 1) is replaced by
minmh,`k (d) s.t. uk+d∈Uadand ˆ∆apo` (uk+d)≤εrelJ Jˆh,`(uk+d) with a chosenεrelJ >0, where ˆ∆apo` (u)is an a-posteriori error estimate for the dif-ference|Jˆh(u)−Jˆh,`(u)|. Now, we do not have to evaluate ˆJhatuk+d, but only the a-posteriori error estimate and ˆJh,`. Thus, the high-dimensional stateyh(uk+d)is not reqired.
Numerical experiments.The initial state is set toy◦(x)≡0 forx∈Ω. For optimiza-tion we use the initial controlu0≡0.5,η1=0.2,η2=0.8,γ1=0.5,γ2=1,γ3=1.5, ζCC=1−η2=0.2 and∆0=3. In Table 2 we compare the standard TR-POD algo-rithm (i.e. without∆apo` and computation of the gradient error) to the TR-POD algo-rithm utilizing the a-posteriori error estimate∆apo` as described above withσCE=16.
The choice ofσCEis justified since the smallest overestimation (∆apo` (uk)/egr(uk)) is 16.1 as can be seen in Table 2. Note that in case of TR-POD with∆apo` (u)a POD
TR-POD TR-POD with∆apo` Quality`max Expansion Quality`max Expansion IterkBasis k∇eˆgr(uk)
Jh,`(uk)kU
egr(uk)
k∇Jˆh,`(uk)kU ` ∆
apo` (uk) egr(uk)
∆apo` (uk)
egr(uk),k∇eˆgr(uk)
Jh,`(uk)kU(*) `
0 Init 5.2·10−3 3 5.2·10−3(*) 3
1 3.9·10−2 8.8·10−2 3 16.2 21.3 3
2 New 2.2·10−1 1.8·10−1 4 21.9 1.8·10−1(*) 4
3 8.1·10−2 2.0·10−1 9 16.7 16.1 10
4 1.0·10−1 1.7·10−1 13 29.2 29.2 14
5 New 3.2·10−1 6.0·10−2 8 21.5 7.0·10−2(*) 8
Jˆh(u)¯ 1.559·10−1 1.559·10−1
Time [s] 35.7 36.8
Table 2 Standard TR-POD compared to TR-POD with the a-posteriori error estimate∆apo` .
basis with number`is accepted in case of∆apo` (uk)/k∇Jˆh,`(uk)kU≤σCEζCC=3.2.
For the maximum number of POD basis functions we initially choose `max=10 because at the beginning only few POD basis functions usually suffice to satisfy the Carter condition. In iterationk=2 it is then set to`max=14. The results are very satisfactory although the overestimation is not perfectly uniform. This is why in iteration k=3 andk=4 one additional POD basis is chosen compared to the standard TR-POD run. The total computation time of TR-POD with APE is even a bit above the time required by the standard TR-POD method. The decay of the error between the FE and the POD reduced gradient together with the a-posteriori error estimate∆apo` (u)is plotted in Figure 4. We observe that the a-posteriori error
5 10 15 20 25 30
ℓ 10-10
10-5 100
Iterationk= 3: Gradient error k∇Jˆh(u)−∇Jˆℓ(u)kU APE
5 10 15 20 25 30
ℓ 10-10
10-5 100
Iterationk= 4: Gradient error k∇Jˆh(u)−∇Jˆℓ(u)kU APE
Fig. 4 Decay of gradient error and gradient estimator in TR-POD run, at the beginning of the iteration where the POD basis for`maxis tested.
estimate is larger than the real error. Moreover, the a-posteriori error estimates stag-nates provided the accuracy of the underlying FE and implicit Euler discretization is reached; cf. [19]. The change of the first POD basis function within the TR-POD method can be seen from Figure 5.
0.9 1 0.95
1 1
ψ1(x1,x2) 1.05
x2 0.5
x1 0.5 1.1
0 0
-1 1 -0.5
1 ψ2(x1,x2) 0
x2 0.5
x1 0.5 0.5
0 0
0.4 1 0.6
1 ψ1(x1,x2)0.8
x2 0.5
x1 0.5 1
0 0
-0.8 1 -0.6
1 ψ2(x1,x2)-0.4
x2 0.5
x1 0.5 -0.2
0 0
Fig. 5 Adaption of POD basis functions. Upper row: first two POD basis functions computed from u0. Lower row: first two POD basis functions computed fromu5.
Remark 6.In [4, 13, 40, 41] a TR-POD algorithm with the non-quadratic model functionmk(d) =Jˆh,`(uk+d)is presented. In contrast to our quadratic model func-tion sufficient model decrease is then not a-priori given but has to be guaranteed via the so-called step determination algorithm. For the given example, this algorithm
needs in average 1.6 seconds (per TR iteration). ♦
References
1. R.A. Adams.Sobolev Spaces. Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.
2. K. Afanasiev and M. Hinze. Adaptive control of a wake flow using proper orthogonal decom-position.Lecture Notes in Pure and Applied Mathematics, 216:317-332, 2001.
3. H.W. Alt.Lineare Funktionalanalysis. Springer-Verlag Berlin, 2006.
4. E. Arian, M. Fahl and E.W. Sachs. Trust-region proper orthogonal decomposition for flow control. Technical Report 2000-25, ICASE, 2000.
5. P. Astrid, S. Weiland, K. Willcox, and T. Backx. Missing point estimation in models described by proper orthogonal decomposition. IEEE Transactions on Automatic Control, 53:2237-2251, 2008.
6. J.A. Atwell, J.T. Borggaard, and B.B. King. Reduced-order controllers for Burgers’ equa-tion with a nonlinear observer. International Journal of Applied Mathematics and Computer Science, 11:1311-1330, 2001.
7. H.T. Banks, M.L. Joyner, B. Winchesky, and W.P. Winfree. Nondestructive evaluation using a reduced-order computational methodology.Inverse Problems, 16:1-17, 2000.
8. R.G. Carter. On the global convergence of trust region algorithms using inexact gradient information.SIAM J. on Numerical Analysis, 28:251-265, 1991.
9. A.R. Conn, N.I.M. Gould, and P.L. Toint.Trust-region methods. SIAM, 2000.
10. R. Dautray and J.-L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5: Evolution Problems I. Springer-Verlag, Berlin, 2000.
11. M. Dihlmann and B. Haasdonk. Certified nonlinear parameter optimization with reduced basis surrogate models.Proc. Appl. Math. Mech., 13:3-6, 2013.
12. L.C. Evans. Partial Differential Equations. American Math. Society, Providence, Rhode Island, 2002.
13. M. Fahl. Trust-Region Methods for Flow Control Based on Reduced Order Modelling. PhD thesis, University of Trier, 2000.
14. C. Gr¨aßle. POD based inexact SQP methods for optimal control problems gov-erned by a semilinear heat equation. Diploma Thesis, University of Konstanz, 2014.
http://nbn-resolving.de/urn:nbn:de:bsz:352-0-265093
15. C. Gr¨aßle, M. Gubisch, S. Metzdorf, S. Rogg and S. Volkwein. POD basis up-dates for nonlinear PDE control. at - Automatisierungstechnik, to appear, 2017.
https://kops.uni-konstanz.de/handle/123456789/34979
16. M.A. Grepl and M. K¨archer. A certified reduced basis method for parametrized elliptic optimal control problems.ESAIM: COCV, Volume 20, 2, 416-441, 2014.
17. M.A. Grepl, Y. Maday, N.C. Nguyen and A.T. Patera. Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis, 41:575-605, 2007.
18. E. Grimm, M. Gubisch, and S. Volkwein. A-posteriori error analysis and optimality-system POD for constrained optimal control.Lecture Notes in Computational Science and Engineer-ing, 105:297-317, 2015.
19. M. Gubisch, I. Neitzel, and S. Volkwein. A-posteriori error estimation of discrete POD mod-els for PDE-constrained optimal controls. Submitted, 2016. Preprint available: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-324122
20. M. Gubisch and S. Volkwein. Proper orthogonal decomposition for linear-quadratic optimal control. To appear in: P. Benner, A. Cohen, M. Ohlberger, and K. Willcox (eds.): Model Re-duction and Approximation: Theory and Algorithms. SIAM, Philadelphia, PA, 20167. Preprint available: http://nbn-resolving.de/urn:nbn:de:bsz:352-250378
21. B. Haasdonk. Convergence rates of the POD-Greedy method. ESAIM: Mathematical Mod-elling and Numerical Analysis, 47:859-873, 2013.
22. B. Haasdonk and M. Ohlberger. Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Mathematical Modelling and Numerical Analysis, 42:277-302, 2008.
23. C. Himpe and M. Ohlberger. Cross-gramian based combined state and parameter reduction.
Mathematical Problems in Engineering, 2014:1-13, 2014.
24. P. Holmes, J.L. Lumley, G. Berkooz, and C.W. Rowley. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics, Cambridge Uni-versity Press, second edition, 2012.
25. M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich.Optimization with PDE constraints. Math-ematical Modelling: Theory and Applications, Springer Series, 2009.
26. K. Ito and S.S. Ravindran. A reduced basis method for control problems governed by PDEs.
In W. Desch, F. Kappel, and K. Kunisch, eds., Control and Estimation of Distributed Parame-ter Systems. Proceedings of the InParame-ternational Conference in Vorau, 1996, Birkh¨auser-Verlag, Basel, 126:153-168, 1998.
27. C.T. Kelley. Iterative Methods for Optimization. Frontiers in Applied Mathematics, SIAM, Philadelphia, PA, 1999.
28. K. Kunisch and S. Volkwein. Galerkin proper orthogonal decomposition methods for parabolic problems.Numerische Mathematik, 90:117-148, 2001.
29. K. Kunisch and S. Volkwein. Proper orthogonal decomposition for optimality systems.
ESAIM: Mathematical Modelling and Numerical Analysis, 42:1-23, 2008.
30. S. Lall, J.E. Marsden, and S. Glavaski. A subspace approach to balanced truncation for model reduction of nonlinear control systems. International Journal of Robust and Nonlinear Con-trol, 12:519-535, 2002.
31. F. Negri, G. Rozza, A. Manzoni, and A. Quarteroni. Reduced basis method for parametrized elliptic optimal control problems. SIAM Journal on Scientific Computing, 35:A2316-A2340, 2013.
32. J. Nocedal and S.J. Wright.Numerical Optimization. Springer-Verlag, New York, 2006.
33. M. Ohlberger and M. Sch¨afer. Error control based model reduction for parameter optimization of elliptic homogenization problems. Proceedings of the 1st IFAC Workshop on Control of Systems Governed by Partial Differential Equations, pp. 251-256, IFAC Papers Online, 2013.
34. A.T. Patera and G. Rozza. Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT Pappalardo Graduate Monographs in Mechanical Engineering, 2006.
35. E. Qian, M. Grapl, K.Veroy, and K. Willcox. A certified trust region reduced basis approach to PDE-constrained optimization. Submitted, 2016.
36. M. Rathinam and L. Petzold. Dynamic iteration using reduced order models: a method for simulation of large scale modular systems. SIAM Journal on Numerical Analysis, 40:1446-1474, 2002.
37. S. Rogg. Trust region POD for optimal boundary control of a semi-linear heat equation.. Diploma Thesis, University of Konstanz, 2014.
http://nbn-resolving.de/urn:nbn:de:bsz:352-0-255879
38. C.W. Rowley. Balanced model reduction via the proper orthogonal decompositionModel re-duction for fluids, using balanced proper orthogonal decomposition.International Journal on Bifurcation and Chaos, 15:997-1013, 2005.
39. E.W. Sachs, M. Schneider, and M. Schu. Adaptive trust-region POD methods in PIDE-constrained optimization.Numerische Mathematik, 165:327-342, 2014.
40. M. Schu. Adaptive Trust-Region POD Methods and Their Applications in Finance. Ph.D thesis, University of Trier, 2012.
41. J.R. Singler. New POD error expressions, error bounds, and asymptotic results for reduced order models of parabolic PDEs. SIAM Journal on Numerical Analysis, 52:852-87, 2014.
42. F. Tr¨oltzsch. Optimal Control of Partial Differential Equations: Theory, Methods and Appli-cations. AMS American Mathematical Society, 2nd ed., 2010.
43. S. Volkwein. Distributed control problems for the Burgers equation. Computational Opti-mization and Applications, 18:115-140, 2001.
44. S. Volkwein. Optimality system POD and a-posteriori error analysis for linear-quadratic prob-lems.Control and Cybernetics, 40:1109-1125, 2011.
45. S. Volkwein.Proper Orthogonal Decomposition for PDE Constrained Optimal Control Prob-lems. In preparation, 2017.
46. K. Willcox and J. Peraire. Balanced model reduction via the proper orthogonal decomposition.
American Institute of Aeronautics and Astronautics, 40:2323-2330, 2002.
47. Y. Yue and K. Meerbergen. Accelerating PDE-constrained optimization by model order re-duction with error control.SIAM J. Optimization, 23:1344-1370, 2013.