Adaptive finite element approximation for a class of parameter estimation problems

Chang, Yanzhen; Yang, Dinghui; Zhang, Zhijuan

doi:10.1016/j.amc.2013.12.141

Cited by 4 publications

(3 citation statements)

References 13 publications

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“…and define the discrete admissible set U ad (T ) := U(T ) ∩ U ad . The state and adjoint state variables, associated to a locally optimal control, are discretized by using the finite element space V(T ) defined in (7). With this setting at hand, the fully discrete scheme reads as follows: Find min J(y T , u T ) subject to the discrete state equation (18) for all v T ∈ V(T ) and the discrete constraints u T ∈ U ad (T ).…”

Section: The Fully Discrete Schemementioning

confidence: 99%

“…In this work, the authors develop a posteriori error estimators for two fully discrete approximation schemes of ( 2)-( 4) and obtain global reliability estimates [19,Theorems 4.1 and 4.3] and global efficiency results [19,Lemmas 4.2 and 4.3]. We also mention the work [7], where a posteriori error estimates for a parabolic version of (2)-(4) have been analyzed; an efficiency analysis, however, was not provided. We conclude this paragraph by mentioning the work [24], where the authors provide, on the basis of a pos-teriori error estimators, upper bounds for discretization errors with respect to a cost functional and with respect to a given quantity of interest; the latter being an arbitrary functional depending on the control and the state variables.…”

Section: Introductionmentioning

confidence: 99%

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A posteriori error estimates for an optimal control problem with a bilinear state equation

Fuica¹,

Otárola²

2022

Preprint

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We propose and analyze a posteriori error estimators for an optimal control problem that involves an elliptic partial differential equation as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We consider two different strategies to approximate optimal variables: a fully discrete scheme in which the admissible control set is discretized with piecewise constant functions and a semi-discrete scheme where the admissible control set is not discretized; the latter scheme being based on the so-called variational discretization approach. We design, for each solution technique, an a posteriori error estimator and show, in two and three dimensional Lipschitz polygonal/polyhedral domains (not necessarily convex), that the proposed error estimator is reliable and efficient. We design, based on the devised estimators, adaptive strategies that deliver optimal experimental rates of convergence for the performed numerical examples. Keywords optimal control problems • bilinear equations • finite elements • a posteriori error estimates • adaptive finite element methods Mathematics Subject Classification (2000) 49M25 • 65N15 • 65N30 • 65N50

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Section: The Fully Discrete Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for an optimal control problem with a bilinear state equation

Fuica¹,

Otárola²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Although adaptive mesh refinement has been found to be very useful in computational optimal control problems [12][13][14][15][16][17], but there is less work on adaptive finite element methods for bilinear optimal control problems. In [18], the optimal control problem for two-dimensional bilinear parabolic equations has been studied. The a posteriori error estimation results are verified using the adaptive finite element method.…”

Section: Introductionmentioning

confidence: 99%

School Mathematics Textbooks in China

Wang¹

2015

East China Normal University Scientific Reports

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Let µ and ν be two probability measures on R d , where µ(dx) = e −V (x) dx R d e −V (x) dx for some V ∈ C 1 (R d ). Explicit sufficient conditions on V and ν are presented such that µ * ν satisfies the log-Sobolev, Poincaré and super Poincaré inequalities. In particular, if V (x) = λ|x| 2 for some λ > 0 and ν(e λθ|•| 2 ) < ∞ for some θ > 1, then µ * ν satisfies the log-Sobolev inequality. This improves and extends the recent results on the log-Sobolev inequality derived in [20] for convolutions of the Gaussian measure and compactly supported probability measures. On the other hand, it is well known that the log-Sobolev inequality for µ * ν implies ν(e ε|•| 2 ) < ∞ for some ε > 0. RésuméDes conditions explicites suffisantes sur V et ν sont présentées telles que µ * ν satisfait des inégalités de Sobolev logarithmique, de Poincaré et de super-Poincaré. En particulier, si V (x) = λ|x| 2 pour quelque λ > 0 et ν(e λθ|•| 2 ) < ∞ avec θ > 1, alors µ * ν satisfait l'inégalité de Sobolev logarithmique. Cela améliore et étend des résultats récents sur l'inégalité de Sobolev logarithmique obtenus dans [20] pour des convolutions de la mesure de Gauss et des mesures de probabilité à support compact. D'autre part, il est bien connu que l'inégalité de Sobolev logarithmique pour µ * ν implique ν(e ε|•| 2 ) < ∞ pour quelque ε > 0.

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A posteriori error estimates for optimal control problems constrained by convection-diffusion equations

Rui

Zhou

2015

Front. Math. China

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Adaptive finite element approximation for a class of parameter estimation problems

Cited by 4 publications

References 13 publications

A posteriori error estimates for an optimal control problem with a bilinear state equation

A posteriori error estimates for an optimal control problem with a bilinear state equation

School Mathematics Textbooks in China

A posteriori error estimates for optimal control problems constrained by convection-diffusion equations

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