Superconvergence Properties of Optimal Control Problems

Meyer, Christian; Rösch, Arnd

doi:10.1137/s0363012903431608

Cited by 231 publications

(188 citation statements)

References 6 publications

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“…The first row of (6) yields δy L 2 ≤ Lδλ L 2 + r L 2 and thus (8). From the second row of (6) we conclude that…”

Section: Newton Linearization and Local Normsmentioning

confidence: 84%

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Superlinear convergence of the control reduced interior point method for PDE constrained optimization

Schiela

Weiser

2007

Comput Optim Appl

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A thorough convergence analysis of the Control Reduced Interior Point Method in function space is performed. This recently proposed method is a primal interior point pathfollowing scheme with the special feature, that the control variable is eliminated from the optimality system. Apart from global linear convergence we show, that this method converges locally almost quadratically, if the optimal solution satisfies a certain non-degeneracy condition. In numerical experiments we observe, that a prototype implementation of our method behaves as predicted by our theoretical results. AMS MSC 2000: 90C51, 65N15, 49M15

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“…The first row of (6) yields δy L 2 ≤ Lδλ L 2 + r L 2 and thus (8). From the second row of (6) we conclude that…”

Section: Newton Linearization and Local Normsmentioning

confidence: 84%

“…Distantly related work includes Rösch [8], who obtains similar accuracy on coarse meshes by a postprocessing step, and Hintermüller and Kunisch [3], who perform a regularization homotopy in the context of primal-dual active set methods.…”

Section: Introductionmentioning

confidence: 99%

Superlinear convergence of the control reduced interior point method for PDE constrained optimization

Schiela

Weiser

2007

Comput Optim Appl

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“…There are many papers devoted to the superconvergence analysis for integral equations (see, e.g., [5], [17], [18], and [27]) and to the optimal control problems governed by PDEs (see, e.g., [7], [10], [16], [21], [24], and [28]). The proofs of Theorems 4.1 and 4.2 use techniques similar those in references just cited.…”

Section: Supconvergence Analysismentioning

confidence: 99%

“…Superconvergence has been investigated for the finite element method of the optimal control problem governed by the partial differential equation, in, e.g., [7], [10], [16], [21], [24], and [28]. It has also been discussed for the Galerkin approximation of integral equations, see, e.g., [5], [17], and [18].…”

Section: Introductionmentioning

confidence: 99%

Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations

Yan

2009

Appl Math

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“…5. In certain situations, the need for refinement can be alleviated by a special postprocessing procedure [12], which results in an approximation order of O(h 2 ).…”

Section: Introductionmentioning

confidence: 99%

A control reduced primal interior point method for a class of control constrained optimal control problems

2007

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A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparatively coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples. AMS MSC 2000: 90C51, 65N15, 49M15

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Superconvergence Properties of Optimal Control Problems

Cited by 231 publications

References 6 publications

Superlinear convergence of the control reduced interior point method for PDE constrained optimization

Superlinear convergence of the control reduced interior point method for PDE constrained optimization

Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations

A control reduced primal interior point method for a class of control constrained optimal control problems

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