Mesh adaptivity in optimal control of elliptic variational inequalities with point-tracking of the state

Brett, Charles; Elliott, Charles M.; Hintermüller, Michael; Löbhard, Caroline

doi:10.4171/ifb/332

Cited by 15 publications

(6 citation statements)

References 49 publications

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“…We deal with the second challenge by developing a goal-oriented error estimator based on the dual-weighted residual approach, cf., e.g., [14]. This allows us to implement an adaptive mesh refinement strategy, which acknowledges the error contributions of the primal residuals, the dual residuals and the mismatch in the complementarity terms, to reduce the computational effort.…”

Section: Strong Stationaritymentioning

confidence: 99%

Simulation and Control of a Nonsmooth Cahn-Hilliard Navier-Stokes System

Gräßle,

Hintermüller,

Hinze

et al. 2019

Preprint

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We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn-Hilliard Navier-Stokes system involving a nonsmooth energy potential. We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C-and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator. In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace highfidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.

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Section: Strong Stationaritymentioning

confidence: 99%

Simulation and Control of a Nonsmooth Cahn-Hilliard Navier-Stokes System

Gräßle,

Hintermüller,

Hinze

et al. 2019

Preprint

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“…This is because one reconstructs a conductivity based on measurements of the voltage over small regions, which could be approximated by measurements at points. In the paper Brett et al (2013) (written by ourselves) the point fidelity term is used for the optimal control of elliptic variational inequalities. The difficulty of the nonlinear control-to-state operator means that an a posteriori error estimator is derived but a priori error estimates are not considered.…”

Section: Introductionmentioning

confidence: 99%

Optimal control of elliptic PDEs at points

2015

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We consider an elliptic optimal control problem where the objective functional contains evaluations of the state at a finite number of points. In particular, we use a fidelity term that encourages the state to take certain values at these points, which means our problem is related to ones with state constraints at points. The analysis and numerical analysis differs from when the fidelity is in the L 2 norm because we need the state space to embed into the space of continuous functions. In this paper we discretise the problem using two different piecewise linear finite element methods. For each discretisation we use two different approaches to prove a priori L 2 error estimates for the control. We discuss the differences between these methods and approaches and present numerical results that agree with our analytical results.

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“…Problem (1.5)-(1.7) finds relevance in numerous applications where the observations are carried out at specific locations. For instance, in the so-called calibration problem with American options [1], in the optimal control of selective cooling of steel [60], in the active control of sound [7,48] and in the active control of vibrations [26,35]; see also [9,10,29,36,53] for other applications. The point observation terms in the cost (1.5), tend to enforce the state y to have the fixed value y z at the point z. Consequently, (1.5)-(1.7) can be understood as a penalty version of a PDE constrained optimization problem where the state is constrained at a collection of points.…”

mentioning

confidence: 99%

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confidence: 99%

Some applications of weighted norm inequalities to the error analysis of PDE-constrained optimization problems

Antil

Otárola

Salgado

2017

IMA Journal of Numerical Analysis

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The purpose of this work is to illustrate how the theory of Muckenhoupt weights, Muckenhoupt weighted Sobolev spaces and the corresponding weighted norm inequalities can be used in the analysis and discretization of PDE constrained optimization problems. We consider: a linear quadratic constrained optimization problem where the state solves a nonuniformly elliptic equation; a problem where the cost involves pointwise observations of the state and one where the state has singular sources, e.g. point masses. For all three examples we propose and analyze numerical schemes and provide error estimates in two and three dimensions. While some of these problems might have been considered before in the literature, our approach allows for a simpler, Hilbert space-based, analysis and discretization and further generalizations.

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Mesh adaptivity in optimal control of elliptic variational inequalities with point-tracking of the state

Cited by 15 publications

References 49 publications

Simulation and Control of a Nonsmooth Cahn-Hilliard Navier-Stokes System

Simulation and Control of a Nonsmooth Cahn-Hilliard Navier-Stokes System

Optimal control of elliptic PDEs at points

Some applications of weighted norm inequalities to the error analysis of PDE-constrained optimization problems

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