A Posteriori Error Estimates of Recovery Type for Distributed Convex Optimal Control Problems

Li, Ruo; Liu, Wenbin; Yan, Ningning

doi:10.1007/s10915-007-9147-7

Cited by 127 publications

(147 citation statements)

References 18 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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“…Residual-type a posteriori error estimators for control constrained problems have been developed and analyzed in [13,14,18,20,23,26,27]. State constrained optimal control problems are more difficult to handle than control constrained ones, since the Lagrange multiplier for the state constraints typically lives in a measure space.…”

Section: Solv E =⇒ Est Im At E =⇒ M Ark =⇒ Ref In Ementioning

confidence: 99%

Goal Oriented Mesh Adaptivity for Mixed Control-State Constrained Elliptic Optimal Control Problems

Hintermüller

Hoppe

2009

Computational Methods in Applied Sciences

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“…It is well-known that under the assumption (2.2) the distributed optimal control problem (2.1a)-(2.1c) admits a unique solution (y, u) ∈ H 1 0 (Ω) × L 2 (Ω) (cf., e.g., [15,[21][22][23]) which is characterized by the existence of a co-state (adjoint state) p ∈ H 1 0 (Ω) and a Lagrange multiplier for the inequality constraint (adjoint control)…”

Section: The Distributed Elliptic Control Problemmentioning

confidence: 99%

“…residual-type a posteriori error estimators in the control constrained case have been derived and analyzed in [20,23,24]. In contrast to the approach used in [20,23,24], the error analysis in this paper pertains to the error in the state, the adjoint state, the control, and the adjoint control and incorporates oscillations in terms of the data of the problem.…”

mentioning

confidence: 99%

“…In contrast to the approach used in [20,23,24], the error analysis in this paper pertains to the error in the state, the adjoint state, the control, and the adjoint control and incorporates oscillations in terms of the data of the problem. The data oscillations may significantly contribute to the error and thus have to be considered in the adaptive refinement process.…”

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confidence: 99%

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Ana posteriorierror analysis of adaptive finite element methods for distributed elliptic control problems with control constraints

et al. 2007

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Abstract. We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method.

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A Posteriori Error Estimates of Recovery Type for Distributed Convex Optimal Control Problems

Cited by 127 publications

References 18 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

Goal Oriented Mesh Adaptivity for Mixed Control-State Constrained Elliptic Optimal Control Problems

Ana posteriorierror analysis of adaptive finite element methods for distributed elliptic control problems with control constraints

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