Two-Side a Posteriori Error Estimates for the Dual-Weighted Residual Method

Endtmayer, Bernhard; Langer, Ulrich; Wick, Thomas

doi:10.1137/18m1227275

Cited by 34 publications

(73 citation statements)

References 70 publications

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“…Remark 3.7. From numerical experiments for the regularized p-Laplacian computed in [27], we can deduce that R (3) can be neglected on sufficiently refined meshes.…”

Section: Error Representation For the Reduced Systemmentioning

confidence: 99%

“…The replacement is justified if a strengthened saturation assumption is fulfilled as shown in [27] for both the nonlinear state equation and the goal functionals.…”

Section: The Parts Of the Error Estimatormentioning

confidence: 99%

“…However, we would have to solve the adjoint problem N times, leading to high computational cost. There are several ways to tackle this problem as for example discussed in [33,32,42,1] and more recently in [35,28,25,26,27].…”

Section: Extension To Multiple Goal Functionalsmentioning

confidence: 99%

“…The error functionalĨ E is constructed in a way, that avoids error cancellation between two or more functionals. For a more detailed discussion, we refer the reader to [25,27]. However, we can use a higher order polynomial approximation to approximate this quantity, as done in [33,25,27], where consequences of the replacement are discussed in [27].…”

Section: Extension To Multiple Goal Functionalsmentioning

confidence: 99%

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Multigoal-oriented optimal control problems with nonlinear PDE constraints

Endtmayer

Langer

Neitzel³

et al. 2020

Computers & Mathematics with Applications

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In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to the regularized p-Laplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest.All quantities are combined to one, and then the dual-weighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated via several numerical examples. numerical example of a quasi-linear PDE constraint, and multiple goal-oriented a posteriori error estimation.In the following, we briefly refer to studies that treat parts of the three topics. Optimal control problems (specifically, a priori estimates and optimality conditions) with quasi-linear (as the p-Laplacian can be classified) elliptic PDE constraints were considered in [16,18,15]. More recently, the extension to optimal control with parabolic PDEs was discussed in [9] and [14].Optimal control problems with (single) goal functionals were investigated in [6,40,5,50,52,43]. The p-Laplacian and a posteriori error estimates were considered in [36,12,20,13], and, more specifically, for goal functional evaluations, we refer to [34,44,25]. To estimate goal functionals, we adopt the dual-weighted residual (DWR) method [7,8] in which an adjoint problem is solved to obtain (local) sensitivity measures that are used for mesh refinement. As is well-known, using a gradientbased approach for the numerical solution of optimal control problems, the same adjoint problem as for the DWR error estimator can be employed. For this reason, it is natural to combine gradient-based optimization with adjoint-based error estimation.We are specifically interested in an extended DWR version in which the discretization and (linear/nonlinear) iteration error are balanced [39,44,37]. As localization technique we employ integration by parts as done in [8] or, for residual based error estimates, in [49]. The extension of [44] to multiple goal functionals was recently undertaken in [25].Three major aims constitute the main contents of this paper: first, the design of a framework for goal-oriented error estimation for optimal control subject to a nonlinear PDE and balancing the discretization and nonlinear iteration error (Section 3). From the optimization point of view, we carefully revisit the important elements for the DWR estimator for optimization problems. The main result in this respect is the a posteriori error representation for the reduced optimal control system for an abstract problem formulation. The second aim is the extension to the simultaneous control of multiple goal functionals (Section 4). As a third goal, based on our theoretical developments, we carefully design an adaptive soluti...

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“…Remark 3.7. From numerical experiments for the regularized p-Laplacian computed in [27], we can deduce that R (3) can be neglected on sufficiently refined meshes.…”

Section: Error Representation For the Reduced Systemmentioning

confidence: 99%

“…The replacement is justified if a strengthened saturation assumption is fulfilled as shown in [27] for both the nonlinear state equation and the goal functionals.…”

Section: The Parts Of the Error Estimatormentioning

confidence: 99%

Section: Extension To Multiple Goal Functionalsmentioning

confidence: 99%

Section: Extension To Multiple Goal Functionalsmentioning

confidence: 99%

See 2 more Smart Citations

Multigoal-oriented optimal control problems with nonlinear PDE constraints

Endtmayer

Langer

Neitzel³

et al. 2020

Computers & Mathematics with Applications

Self Cite

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“…Since |J (u) − J (ũ)| = 0 may happen, the lower bound estimate remains a questionable task and was only recently achieved in [6] using a so-called saturation assumption. Findings as discussed in this book are useful for two research directions of equal importance.…”

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confidence: 99%

Sergey I. Repin, Stefan A. Sauter: “Accuracy of Mathematical Models”

Wick

2020

Jahresber. Dtsch. Math. Ver.

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The book by Repin and Sauter is an inspiring piece of work. The key question is centered around 'How accurate is a mathematical model?' by deriving model error estimates arising in different approaches for treating partial differential equations and variational problems of elliptic type. Accuracy is nothing else than measuring distances, in metric spaces or normed vector spaces, between (unknown) exact solutions and their (numerical) approximations in terms of error estimates. There is a rich theory on complete metric spaces and Banach spaces which forms the basis for some well-posedness results and numerical analysis of partial differential equations and variational problems. While the theory for linear (elliptic) problems is fairly complete, nonlinear (time-dependent) differential equations and variational inequality systems are subject of ongoing research. Error estimates often include several sources such as model-, regularization-, homogenization-, discretization-, interpolation-, linear solver-, nonlinear solver-, quadrature-, and implementation (bugs) errors. But also the exact solution to compare with, might be error-prone from inaccurate experimental setups, uncertain data, and measurement errors. The important confession one has to make is the fact that such errors never can be avoided. Rather, the aim is to design error estimates that allow us to quantify their influence and to adjust those terms with the largest contributions. B T. Wick

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