Convergence Analysis of an Adaptive Finite Element Method for Distributed Control Problems with Control Constraints

Gaevskaya, A.; Iliash, Y.; Kieweg, Michael; Hoppe, Ronald H. W.

doi:10.1007/978-3-7643-7721-2_3

Cited by 27 publications

(27 citation statements)

References 27 publications

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“…On the other hand, considerably less work has been done with regard to optimal control problems for PDEs. The so-called goal oriented dual weighted approach has been applied in the unconstrained case in [4,5] and to control constrained problems in [20,42], whereas residual-type a posteriori error estimators for control constrained problems have been derived and analyzed in [16,17,21,25,28,30,31]. Unlike the control constrained case, pointwise state constrained optimal control problems are much more difficult to handle due to the fact that the Lagrange multiplier for the state constraints lives in a measure space (see, e.g., [8,9,23,39]).…”

Section: Introductionmentioning

confidence: 99%

Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems

Hoppe

Kieweg

2008

Comput Optim Appl

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Abstract. Mixed control-state constraints are used as a relaxation of originally state constrained optimal control problems for partial differential equations to avoid the intrinsic difficulties arising from measure-valued multipliers in the case of pure state constraints. In particular, numerical solution techniques known from the pure control constrained case such as active set strategies and interior-point methods can be used in an appropriately modified way. However, the residual-type a posteriori error estimators developed for the pure control constrained case can not be applied directly. It is the essence of this paper to show that instead one has to resort to that type of estimators known from the pure state constrained case. Up to data oscillations and consistency error terms, they provide efficient and reliable estimates for the discretization errors in the state, a regularized adjoint state, and the control. A documentation of numerical results is given to illustrate the performance of the estimators.

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Section: Introductionmentioning

confidence: 99%

Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems

Hoppe

Kieweg

2008

Comput Optim Appl

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“…We are not aware of any previous studies in this direction. On the other hand, both residual-type a posteriori error estimators and dual weighted residuals for P1 conforming finite element approximations of control constrained H 1 (Ω)-elliptic optimal control problems have been developed in [17,19,27,29] and [18,41].…”

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confidence: 99%

Adaptive edge element approximation of H(curl)-elliptic optimal control problems with control constraints

Hoppe

Yousept

2014

Bit Numer Math

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Abstract. A three-dimensional H(curl)-elliptic optimal control problem with distributed control and pointwise constraints on the control is considered. We present a residual-type a posteriori error analysis with respect to a curl-conforming edge element approximation of the optimal control problem. Here, the lowest order edge elements of Nédélec's first family are used for the discretization of the state and the control with respect to an adaptively generated family of simplicial triangulations of the computational domain. In particular, the a posteriori error estimator consists of element and face residuals associated with the state equation and the adjoint state equation. The main results are the reliability of the estimator and its efficiency up to oscillations in terms of the data of the problem. In the last part of the paper, numerical results are included which illustrate the performance of the adaptive approach.Key words. Optimal control of PDEs, H(curl)-elliptic problems, curl-conforming edge elements, residual a posteriori error estimator, reliability and efficiency.AMS subject classifications. Primary: 65K10; Secondary: 49M05,65N30,65N50,78M101. Introduction. This paper is devoted to an a posteriori error analysis of adaptive edge element methods for control constrained distributed optimal control of H(curl)-elliptic problems in R 3 . Adaptive edge element methods for H(curl)-elliptic boundary value problems on the basis of residual-type a posteriori error estimators have been initiated in [7,8,31] and later on considered in [11,35]. A convergence analysis has been provided in [24]. For nonstandard discretizations such as Discontinuous Galerkin methods, we refer to [12,25]. In case of the time-harmonic Maxwell equations, convergence and quasi-optimality of adaptive edge element approximations have been established in [42,46] in the spirit of the results obtained in [13] for linear second order elliptic boundary value problems.

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“…A posteriori error estimates of "energy-norm"-type for problems with control and/or state constraints have intensively been studied in the literature, see, e.g., Gaevskaya et al [32], Hoppe & Kieweg [45], Hintermueller & et al [41] and Liu et al [52].…”

Section: Control and State Constraintsmentioning

confidence: 99%

Adaptive finite element discretization in PDE‐based optimization

Rannacher

Vexler

2010

GAMM-Mitteilungen

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Key words PDE-based optimization, finite element method, goal-oriented adaptivity MSC (2000) 49K20, 49M15, 49M15, 65M50, 65N50This article surveys recent developments in the adaptive numerical solution of optimal control problems governed by partial differential equations (PDE). By the Euler-Lagrange formalism the optimization problem is reformulated as a saddle-point problem (KKT system) that is discretized by a Galerkin finite element method (FEM). Following the Dual Weighted Residual (DWR) approach the accuracy of the approximation is controlled by residual-based a posteriori error estimates. This opens the way toward systematic complexity reduction in the solution of PDE-based optimal control problems occurring in science and engineering.

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Convergence Analysis of an Adaptive Finite Element Method for Distributed Control Problems with Control Constraints

Cited by 27 publications

References 27 publications

Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems

Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems

Adaptive edge element approximation of H(curl)-elliptic optimal control problems with control constraints

Adaptive finite element discretization in PDE‐based optimization

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