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

Antil, Harbir; Otárola, Enrique; Salgado, Abner J.

doi:10.1093/imanum/drx018

Cited by 19 publications

(42 citation statements)

References 64 publications

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“…We recall the finite element approximation of the control problem with point sources detailed in [7]. In doing so, we consider T = {T } to be a conforming partition of Ω into simplices T with size h T = diam(T ) and define h T = max T ∈T h T .…”

Section: Finite Element Discretizationmentioning

confidence: 99%

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Ana posteriorierror analysis for an optimal control problem with point sources

et al. 2018

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We propose and analyze a reliable and efficient a posteriori error estimator for a controlconstrained linear-quadratic optimal control problem involving Dirac measures; the control variable corresponds to the amplitude of forces modeled as point sources. The proposed a posteriori error estimator is defined as the sum of two contributions, which are associated with the state and adjoint equations. The estimator associated with the state equation is based on Muckenhoupt weighted Sobolev spaces, while the one associated with the adjoint is in the maximum norm and allows for unbounded right hand sides. The analysis is valid for two and three-dimensional domains. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform. AN OPTIMAL CONTROL PROBLEM WITH POINT SOURCES3 polytope, we prove the global reliability and local efficiency of our proposed error estimator. The analysis is delicate since it involves the interaction of L ∞ (Ω), R l and weighted Sobolev spaces, combined with having to deal with the first-order necessary and sufficient optimality condition that characterizes the optimal controlū. It is important to comment that this work exploits the ideas developed in [4] for the a posteriori error analysis of the so-called pointwise tracking optimal control problem. Although the mathematical techniques are similar, the a posteriori error analysis of our control problem does not follow directly from [4]; it requires its own analysis. This is mainly due to the following reasons:• The optimal control variableū belongs to R l , while the one of the problem studied in [4] belongs to L 2 (Ω).This in a sense simplifies the analysis. For instance, as opposed to [4,30], we can obtain local efficiency estimates that do not require convexity of Ω. Nevertheless, it comes with its own set of complications. In particular, the low regularity of the state equation. • The adjoint problem is a Poisson equation with a forcing term y − y d , which does not belong to L ∞ (Ω).Consequently, we must consider a pointwise error indicator that accounts for unbounded right hand sides.Since we were not able to locate one in the literature, in section 4, we propose such an error indicator and provide its analysis on the basis of [9,15]. Notice that, thanks to the structure of the control problem of [4], such an estimator was not needed there. The outline of this paper is as follows. In section 2, we introduce the notation and functional framework we shall work with. Section 3 contains the description of our control problem and reviews the a priori error analysis developed in [7]. In section 4, we propose and analyze a pointwise a posteriori error estimator for the Laplacian that allows for unbounded right hand sides. Combining this estimator and another one based on Muckenhoupt weighted Sobolev spaces, in section 5 we devise an a posteriori error estimator for our optimal control problem. We show in sections 5.2 and 5.3, its ...

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Section: Finite Element Discretizationmentioning

confidence: 99%

“…The following a priori error analysis follows from [7]: Let ǫ > 0 and Ω 1 be such that D ⋐ Ω 1 ⋐ Ω. Assume that for every q ∈ (2, ∞), y d ∈ L q (Ω), Ω is convex, and the mesh T is quasiuniform with mesh size h T .…”

Section: Finite Element Discretizationmentioning

confidence: 99%

Ana posteriorierror analysis for an optimal control problem with point sources

et al. 2018

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“…The pointwise tracking optimal control problem for the Poisson equation has been considered in a number of works [8,9,12,14]. In [8], the authors operate under the framework of Muckenhoupt weighted Sobolev spaces [43] and circumvent the difficulties associated with the underlying adjoint equation: a Poisson equation with a linear combination of Dirac deltas as a forcing term. Weighted Sobolev spaces allow for working under a Hilbert space-based framework in comparison to the non-Hilbertian setting of [9,12,14].…”

Section: Introductionmentioning

confidence: 99%

An Adaptive FEM for the Pointwise Tracking Optimal Control Problem of the Stokes Equations

Allendes¹,

Fuica

Otárola³

et al. 2019

SIAM J. Sci. Comput.

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We propose and analyze a reliable and efficient a posteriori error estimator for the pointwise tracking optimal control problem of the Stokes equations. This linear-quadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the velocity field that solves the state equations. This leads to an adjoint problem with a linear combination of Dirac measures as a forcing term and whose solution exhibits reduced regularity properties. We also consider constraints on the control variable. The proposed a posteriori error estimator can be decomposed as the sum of four contributions: three contributions related to the discretization of the state and adjoint equations, and another contribution that accounts for the discretization of the control variable. On the basis of the devised a posteriori error estimator, we design a simple adaptive strategy that illustrates our theory and exhibits a competitive performance. L 2 (ω,G) 1 2 .

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“…(1. 4) As already pointed out in [7], the pointwise tracking optimal control problem (1.1)-(1.4) is relevant in several applications where the state observations are carried out at specific locations. For instance, the calibration problem with American options [1], selective cooling of steel [60], and many others.…”

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confidence: 98%

“…Therefore, the analysis of the finite element method applied to the pointwise tracking optimal control problem is not standard. An a priori error analysis has been recently provided in [7,12,14]. In [7], the authors operate under the framework of Muckenhoupt weighted Sobolev spaces analyzed in [49] and thus circumvent the difficulties associated with the adjoint equation (1.4).…”

mentioning

confidence: 99%

Adaptive finite element methods for an optimal control problem involving Dirac measures

et al. 2017

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The purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so-called pointwise tracking optimal control problem. This linearquadratic optimal control problem entails the minimization of a cost functional that involves point evaluations of the state, thus leading to an adjoint problem with Dirac measures on the right hand side; control constraints are also considered. The proposed error estimator relies on a posteriori error estimates in the maximum norm for the state and in Muckenhoupt weighted Sobolev spaces for the adjoint state. We present an analysis that is valid for two and three-dimensional domains. We conclude by presenting several numerical experiments which reveal the competitive performance of adaptive methods based on the devised error estimator.

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

Cited by 19 publications

References 64 publications

Ana posteriorierror analysis for an optimal control problem with point sources

Ana posteriorierror analysis for an optimal control problem with point sources

An Adaptive FEM for the Pointwise Tracking Optimal Control Problem of the Stokes Equations

Adaptive finite element methods for an optimal control problem involving Dirac measures

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