# A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD

Eileen Kammann; Fredi Tröltzsch; Stefan Volkwein

- Volume: 47, Issue: 2, page 555-581
- ISSN: 0764-583X

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topKammann, Eileen, Tröltzsch, Fredi, and Volkwein, Stefan. "A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.2 (2013): 555-581. <http://eudml.org/doc/273256>.

@article{Kammann2013,

abstract = {We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a local solution of the reduced problem to a local solution of the original problem is estimated.},

author = {Kammann, Eileen, Tröltzsch, Fredi, Volkwein, Stefan},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {optimal control; semilinear partial differential equations; error estimation; proper orthogonal decomposition; semilinear parabolic equations},

language = {eng},

number = {2},

pages = {555-581},

publisher = {EDP-Sciences},

title = {A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD},

url = {http://eudml.org/doc/273256},

volume = {47},

year = {2013},

}

TY - JOUR

AU - Kammann, Eileen

AU - Tröltzsch, Fredi

AU - Volkwein, Stefan

TI - A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2013

PB - EDP-Sciences

VL - 47

IS - 2

SP - 555

EP - 581

AB - We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a local solution of the reduced problem to a local solution of the original problem is estimated.

LA - eng

KW - optimal control; semilinear partial differential equations; error estimation; proper orthogonal decomposition; semilinear parabolic equations

UR - http://eudml.org/doc/273256

ER -

## References

top- [1] A.C. Antoulas, Approximation of large-scale dynamical systems, Advances in Design and Control. Society for Industrial and Applied Mathematics SIAM, Philadelphia, PA (2005). With a foreword by Jan C. Willems. Zbl1158.93001MR2155615
- [2] N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl.23 (2002) 201–229. Zbl1033.65044MR1937089
- [3] M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An ‘empirical interpolation’ method : application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris339 (2004) 667–672. Zbl1061.65118MR2103208
- [4] P. Benner and E.S. Quintana-Ortí, Model reduction based on spectral projection methods, in Reduction of Large-Scale Systems, edited by P. Benner, V. Mehrmann, D.C. Sorensen, Lect. Notes Comput. Sci. Eng. 45 (2005) 5–48. Zbl1106.93015MR2503778
- [5] E. Casas, J.C. De los Reyes and F. Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM : J. Optim. 19 (2008) 616–643. Zbl1161.49019MR2425032
- [6] E. Casas and F. Tröltzsch, First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations. SIAM : J. Control Optim. 48 (2009) 688–718. Zbl1194.49025MR2486089
- [7] S. Chaturantabut and D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM : J. Sci. Comput. 32 (2010) 2737–2764. Zbl1217.65169MR2684735
- [8] A.L. Dontchev, W.W. Hager, A.B. Poore and B. Yang, Optimality, stability, and convergence in nonlinear control. Appl. Math. Optim.31 (1995) 297–326. Zbl0821.49022MR1316261
- [9] M.A. Grepl and M. Kärcher, Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. C. R. Acad. Sci. Paris, Ser. I 349 (2011) 873–877. Zbl1232.49039MR2835894
- [10] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints. Springer-Verlag, Berlin. Math. Model. Theory Appl. 23 (2009). Zbl1167.49001MR2516528
- [11] M. Hinze and S. Volkwein, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl.39 (2008) 319–345. Zbl1191.49040MR2396870
- [12] P. Holmes, J.L. Lumley and G. Berkooz, Turbulence, coherent structures, dynamical systems and symmetry. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (1996). Zbl0923.76002MR1422658
- [13] A.D. Ioffe, Necessary and sufficient conditions for a local minimum 3 : Second order conditions and augmented duality. SIAM : J. Control Optim. 17 (1979) 266–288. Zbl0417.49029MR525027
- [14] K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008). Zbl1156.49002MR2441683
- [15] M. Kahlbacher and S. Volkwein, POD aposteriori error based inexact SQP method for bilinear elliptic optimal control problems. ESAIM : M2AN 46 (2012) 491–511. Zbl1272.49059
- [16] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math.90 (2001) 117–148. Zbl1005.65112MR1868765
- [17] O. Lass and S. Volkwein, POD Galerkin schemes for nonlinear elliptic-parabolic systems (2011). Submitted. Zbl1272.49060
- [18] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin (1971). Zbl0203.09001MR271512
- [19] K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control–constrained optimal control problems. Appl. Math. Optim.8 (1981) 69–95. Zbl0479.49017MR646505
- [20] K. Malanowski, Two-norm approach in stability and sensitivity analysis of optimization and optimal control problems. Adv. Math. Sci. Appl.2 (1993) 397–443. Zbl0791.49015MR1239267
- [21] K. Malanowski, Ch. Büskens and H. Maurer, Convergence of approximations to nonlinear optimal control problems, in Mathematical Programming with Data Perturbations, edited by Marcel-Dekker, Inc. Lect. Notes Pure Appl. Math. 195 (1997) 253–284. Zbl0883.49025MR1472274
- [22] I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math.120 (2012) 345–386. Zbl1245.65074MR2874969
- [23] A. Rösch and D. Wachsmuth, Numerical verification of optimality conditions. SIAM J. Control Optim.47 (2008) 2557–2581. Zbl1171.49018MR2448474
- [24] A. Rösch and D. Wachsmuth, How to check numerically the sufficient optimality conditions for infinite-dimensional optimization problems, in Optimal control of coupled systems of partial differential equations, Int. Ser. Numer. Math. 158 (2009) 297–317. Zbl1197.49022MR2588562
- [25] E.W. Sachs and M. Schu, A priori error estimates for reduced order models in finance. ESAIM : M2AN 47 (2013) 449–469. Zbl1268.91182MR3021694
- [26] K. Schittkowski, Numerical solution of a time–optimal parabolic boundary-value control problem. JOTA27 (1979) 271–290. Zbl0372.49014MR529864
- [27] A. Studinger and S. Volkwein, Numerical analysis of POD a posteriori error estimation for optimal control (2012). Zbl1275.49050
- [28] T. Tonn, K. Urban and S. Volkwein, Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear quadratic optimal control problem. Mathematical and Computer Modelling of Dynamical Systems, Math. Comput. Modell. Dyn. Syst. 17 (2011) 355–369. Zbl1302.49045MR2823468
- [29] F. Tröltzsch, Optimal Control of Partial Differential Equations. American Math. Society, Providence, Theor. Methods Appl. 112 (2010). Zbl1195.49001
- [30] F. Tröltzsch and S. Volkwein, POD a posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl.44 (2009) 83–115. Zbl1189.49050MR2556846
- [31] S. Volkwein, Optimal control of a phase-field model using proper orthogonal decomposition. ZAMM Z. Angew. Math. Mech.81 (2001) 83–97. Zbl1007.49019MR1818724
- [32] S. Volkwein, Model Reduction using Proper Orthogonal Decomposition. Lecture notes, Institute of Mathematics and Statistics, University of Konstanz (2011).

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