Adaptive Optimal Control of the Obstacle Problem

Meyer, Christian; Rademacher, Andreas; Wollner, Winnifried

doi:10.1137/140975863

Cited by 30 publications

(25 citation statements)

References 28 publications

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“…With a major assumption, similar to the assumption for instance in [22], we see that the differentiation is well posed.…”

Section: 2mentioning

confidence: 58%

“…e.g. [22,30]. However, we have to consider the different sign in the penalty term and the normal trace τ.…”

Section: Regularization and Discretizationmentioning

confidence: 99%

“…We cannot adopt the techniques presented for instance in [22] straight forwardly, since V = W in the contact problem at hand. We tackle this problem by analyzing auxiliary problems.…”

Section: Regularization and Discretizationmentioning

confidence: 99%

“…A more efficient error estimator for the regularization error is developed in [22]; a paper dedicated to optimal control problems subject to standard obstacle problems. Therein, the extrapolation is based on Taylor expansion with respect to the regularization parameter.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive optimal control of Signorini’s problem

Rademacher

Rosin

2018

Comput Optim Appl

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ABSTRACT. In this article, we present a-posteriori error estimations in context of optimal control of contact problems; in particular of Signorini's problem. Due to the contact side-condition, the solution operator of the underlying variational inequality is not differentiable, yet we want to apply Newton's method. Therefore, the non-smooth problem is regularized by penalization and afterwards discretized by finite elements. We derive optimality systems for the regularized formulation in the continuous as well as in the discrete case. This is done explicitly for Signorini's contact problem, which covers linear elasticity and linearized surface contact conditions. The latter creates the need for treating trace-operations carefully, especially in contrast to obstacle contact conditions, which exert in the domain. Based on the dual weighted residual method and these optimality systems, we deduce error representations for the regularization, discretization and numerical errors. Those representations are further developed into error estimators. The resulting error estimator for regularization error is defined only in the contact area. Therefore its computational cost is especially low for Signorini's contact problem. Finally, we utilize the estimators in an adaptive refinement strategy balancing regularization and discretization errors. Numerical results substantiate the theoretical findings. We present different examples concerning Signorini's problem in two and three dimensions.

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“…With a major assumption, similar to the assumption for instance in [22], we see that the differentiation is well posed.…”

Section: 2mentioning

confidence: 58%

“…e.g. [22,30]. However, we have to consider the different sign in the penalty term and the normal trace τ.…”

Section: Regularization and Discretizationmentioning

confidence: 99%

“…We cannot adopt the techniques presented for instance in [22] straight forwardly, since V = W in the contact problem at hand. We tackle this problem by analyzing auxiliary problems.…”

Section: Regularization and Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adaptive optimal control of Signorini’s problem

Rademacher

Rosin

2018

Comput Optim Appl

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“…The most frequently applied and useful approach is the conversion of VI constraints into the linear complementarity form. According to , VI constraints can be rewritten as follows:

- normal Δ y - f - u ⩽ 0, y - ψ ⩽ 0, (- normal Δ y - f - u) (y - ψ) = 0 .

Consequently, the obstacle optimal control problems and can be equivalently reformulated as

Minimize J [y, u] = \frac{1}{2} \int_{normal Ω} {(y - y_{d})}^{2} d boldx + \frac{μ}{2} \int_{normal Ω} u^{2} d boldx

\begin{matrix} right Over & left (y, u) \in H_{0}^{1} (normalΩ) \times L^{2} (normalΩ), & right \\ right Subject to & left - normalΔ y - f - u ⩽ 0, \end{matrix}

y - ψ ⩽ 0,

(- normal Δ y - f - u) (y - ψ) = 0 .

The aforementioned problem is an infinite dimensional mathematical programming with complementarity constraint (MPCC).…”

Section: Problem Statementmentioning

confidence: 99%

Direct pseudo-spectral method for optimal control of obstacle problem - an optimal control problem governed by elliptic variational inequality

Oshagh

Shamsi

2017

Math. Meth. Appl. Sci.

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In this paper, a computational technique based on the pseudo‐spectral method is presented for the solution of the optimal control problem constrained with elliptic variational inequality. In fact, our aim in this paper is to present a direct approach for this class of optimal control problems. By using the pseudo‐spectral method, the infinite dimensional mathematical programming with equilibrium constraint, which can be an equivalent form of the considered problem, is converted to a finite dimensional mathematical programming with complementarity constraint. Then, the finite dimensional problem can be solved by the well‐developed methods. Finally, numerical examples are presented to show the validity and efficiency of the technique. Copyright © 2017 John Wiley & Sons, Ltd.

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First Order Limiting Optimality Conditions in the Coefficient Control of an Obstacle Problem

Simon

Wollner

2023

Proc Appl Math and Mech

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We consider an optimal control problem governed by an obstacle problem. The novelty and focus of this work will be the introduction of a control variable into the coefficients of this optimal control problem. In particular, we discuss the effects of the multiplicative coupling between control and, derivatives of, the state. This prevents the use of standard weak convergence arguments in limit arguments. As it is known, H‐convergence needs to be utilized, e.g., to prove existence of a solution to the problem. To obtain optimality conditions, we present a regularization approach to circumvent the non‐differentiability of the solution operator of the obstacle problems. We then discuss the limiting optimality conditions obtained by the proposed regularization.

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Adaptive Optimal Control of the Obstacle Problem

Cited by 30 publications

References 28 publications

Adaptive optimal control of Signorini’s problem

Adaptive optimal control of Signorini’s problem

Direct pseudo-spectral method for optimal control of obstacle problem - an optimal control problem governed by elliptic variational inequality

First Order Limiting Optimality Conditions in the Coefficient Control of an Obstacle Problem

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