Research Topic

Multilevel Methods for PDE-constrained Optimization with State Constraints

An adaptive multilevel SQP method with reduced order models is designed for optimal control problems with constraints on the state and the control. We apply this novel theory to optimization problems governed by the unsteady Navier-Stokes equations. Main contributions on this topic are listed below. We distinguish between results concerning a general optimal control problem and a problem with the Navier-Stokes equations.

- We extend the adaptive multilevel SQP method of Ziems and Ulbrich [8] by reduced order models (ROM) for control constrained problems, leading to the adaptive SQP_ROM method. More precisely, the inexactness of the ROM is controlled comparing the predicted reduction of the SQP subproblems with a posteriori error estimators of the ROM. At the same time, the FEM grid is refined adaptively as in [8]. We show first-order convergence results.
- The adaptive SQP_ROM method is extended to solve optimal control problems with constraints on the state and control based on [2], [3]. That means that the adaptive SQP_ROM method is coupled with the Moreau-Yosida regularization. We prove first-order convergence results for a general problem.
- Assuming a second-order sufficient condition and that the iterates of our algorithm are close to an optimal control,  the convergence of a subsequence of the iterates to this optimal control is proven.

- The functional-analytical foundation is provided in order to cope with optimal control problems governed by the Navier-Stokes equations and constraints on the state. Especially, a unique solution of an adjoint system of the Navier-Stokes equations involving a Borel measure is shown to exist.
- Basic second-order sufficient conditions and a second-order sufficient condition with strongly active sets, c.f. [7], are derived for problems with the Navier-Stokes equations and constraints on the state.
- On the basis of the two penalization approaches Moreau-Yosida regularization and virtual control concept, the adaptive SQP_ROM method for constraints on the state is applied to problems with the Navier-Stokes equations. For that purpose, we investigate the adaptive SQP_ROM method in a Banach space setting. In the control constrained case, a new criticality measure is used in order to tackle this setting.
- We present numerical results for a flow around a cylinder with state and gradient constraints based on a posteriori error estimators and a goal-oriented error estimate for the ROM. In this regard, practical implementations of the adaptive SQP_ROM algorithm are considered, too.




[1] E. Arian, M. Fahl, and E. W. Sachs. Trust-region proper orthogonal decomposition for flow control. Technical report, DTIC Document, 2000.

[2] S. Bott. Adaptive Multilevel SQP Method for State Constrained Optimization with PDEs. Diplom thesis, TU Darmstadt, 2012.

[3] S. Bott, D. Clever, J. Lang, S. Ulbrich, J. C. Ziems, and D. Schröder. On a fully adaptive SQP method for PDAE-constrained optimal control problems with control and state constraints. In G. Leugering, P. Benner, S. Engell, A. Griewank, H. Harbrecht, M. Hinze, R. Rannacher, and S. Ulbrich, editors, Trends in PDE constrained optimization, volume 165 of International Series of Numerical Mathematics, pages 85-108. Springer International Publishing, 2014.

[4] J. Ghiglieri and S. Ulbrich. Optimal flow control based on POD and MPC and an application to the cancellation of Tollmien-Schlichting waves. Optim. Methods Softw., 29(5):1042-1074, 2014.

[5] K. Krumbiegel, I. Neitzel, and A. Rösch. Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints. Comput. Optim. Appl., 52(1):181-207, 2012.

[6] C. Meyer and I. Yousept. State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions. SIAM J. Control Optim. , 48(2):734–755, 2009.

[7] F. Tröltzsch and D. Wachsmuth. Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations. ESAIM Control Optim. Calc. Var., 12(1):93–119, 2006.

[8] J. C. Ziems and S. Ulbrich. Adaptive Multilevel Generalized SQP-Methods for PDE-constrained Optimization. Preprint, TU Darmstadt, 2011.


Key Research Area

Optimal control with constraints on the state, Navier-Stokes equations, reduced order methods, multilevel SQP-method


Stefanie Bott


Dolivostr. 15

D-64293 Darmstadt



+49 6151 16 - 23446


+49 6151 16 - 24404




bott (at) mathem...

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