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

Allendes, Alejandro; Otárola, Enrique; Rankin, Richard; Salgado, Abner J.

doi:10.1007/s00211-017-0867-9

Cited by 18 publications

(21 citation statements)

References 69 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The theory for linear and second-order elliptic boundary value problems is well-established. For an up to date survey on a posteriori error analysis for optimal control problems we refer the reader to [52][53][54]. The a posteriori error analysis for finite element approximations of constrained optimal control problems is currently under development; the main source of difficulty being its inherent nonlinear feature.…”

Section: An Ideal a Posteriori Error Estimatormentioning

confidence: 99%

“…Starting with the pioneering work [51], several authors have contributed to its advancement. For an up to date survey on a posteriori error analysis for optimal control problems we refer the reader to [52][53][54]. In contrast to these advances, the theory for optimal control problems involving a sparsity functional, as (1.2), is much less developed.…”

Section: An Ideal a Posteriori Error Estimatormentioning

confidence: 99%

See 1 more Smart Citation

An adaptive finite element method for the sparse optimal control of fractional diffusion

Otárola

2019

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

We propose and analyze an a posteriori error estimator for a partial differential equation (PDE)-constrained optimization problem involving a nondifferentiable cost functional, fractional diffusion, and control-constraints. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly PDE and propose an equivalent optimal control problem with a local state equation. For such an equivalent problem, we design an a posteriori error estimator which can be defined as the sum of four contributions: two contributions related to the approximation of the state and adjoint equations and two contributions that account for the discretization of the control variable and its associated subgradient. The contributions related to the discretization of the state and adjoint equations rely on anisotropic error estimators in weighted Sobolev spaces. We prove that the proposed a posteriori error estimator is locally efficient and, under suitable assumptions, reliable. We design an adaptive scheme that yields, for the examples that we perform, optimal experimental rates of convergence. KEYWORDSa posteriori error analysis, adaptive loop, anisotropic estimates, nonlocal operators, nonsmooth objectives, PDE-constrained optimization, sparse controls, spectral fractional Laplacian Enrique Otárola is partially supported by CONICYT through FONDECYT project 11180193.Numer Methods Partial Differential Eq. 2020;36:302-328. wileyonlinelibrary.com/journal/num

show abstract

Section: An Ideal a Posteriori Error Estimatormentioning

confidence: 99%

Section: An Ideal a Posteriori Error Estimatormentioning

confidence: 99%

An adaptive finite element method for the sparse optimal control of fractional diffusion

Otárola

2019

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

show abstract

“…The author of this work proposes a numerical scheme but its stability and convergence properties are not investigated. Another instance where a singular force like that of (1) may occur, see [3] and [9,11], is in a PDE constrained optimization problem where the state is governed by a standard Stokes problem, but the objective contains a point value of u. The idea in this problem is that one tries to optimize the flow profile so as to match the velocity at a certain point.…”

mentioning

confidence: 99%

A posteriori error estimates for the Stokes problem with singular sources

Allendes

Otárola

Salgado

2019

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

We propose a posteriori error estimators for classical low-order inf-sup stable and stabilized finite element approximations of the Stokes problem with singular sources in two and three dimensional Lipschitz, but not necessarily convex, polytopal domains. The designed error estimators are proven to be reliable and locally efficient. On the basis of these estimators we design a simple adaptive strategy that yields optimal rates of convergence for the numerical examples that we perform.

show abstract

“…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.…”

mentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

Ana posteriorierror analysis for an optimal control problem with point sources

et al. 2018

Self Cite

View full text Add to dashboard Cite

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 ...

show abstract

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

Cited by 18 publications

References 69 publications

An adaptive finite element method for the sparse optimal control of fractional diffusion

An adaptive finite element method for the sparse optimal control of fractional diffusion

A posteriori error estimates for the Stokes problem with singular sources

Ana posteriorierror analysis for an optimal control problem with point sources

Contact Info

Product

Resources

About