Details
| Original language | English |
|---|---|
| Pages (from-to) | 3001-3026 |
| Number of pages | 26 |
| Journal | Computers and Mathematics with Applications |
| Volume | 79 |
| Issue number | 10 |
| Early online date | 25 Jan 2020 |
| Publication status | Published - 15 May 2020 |
Abstract
In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to a semi-linear monotone PDE and 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 with the help of several numerical examples.
Keywords
- Dual-weighted residuals, Finite elements, Multigoal-oriented a posteriori error estimation, Optimal control, Regularized p-Laplacian
ASJC Scopus subject areas
- Mathematics(all)
- Modelling and Simulation
- Computer Science(all)
- Computational Theory and Mathematics
- Mathematics(all)
- Computational Mathematics
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In: Computers and Mathematics with Applications, Vol. 79, No. 10, 15.05.2020, p. 3001-3026.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Multigoal-oriented optimal control problems with nonlinear PDE constraints
AU - Endtmayer, B.
AU - Langer, U.
AU - Neitzel, I.
AU - Wick, T.
AU - Wollner, W.
N1 - Funding Information: This work has been supported by the Austrian Science Fund (FWF) under the grant P 29181 ‘Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multiphysics Problems’ and the DFG - SPP 1962 ‘Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization’ within the project ‘Optimizing Fracture Propagation Using a Phase-Field Approach’ under the project number . Furthermore the authors thank Fredi Tröltzsch, Huidong Yang and Behzad Azmi for helpful discussions. Additionally we would like to thank the reviewers for their careful readings and suggestions, which improved the paper.
PY - 2020/5/15
Y1 - 2020/5/15
N2 - In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to a semi-linear monotone PDE and 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 with the help of several numerical examples.
AB - In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to a semi-linear monotone PDE and 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 with the help of several numerical examples.
KW - Dual-weighted residuals
KW - Finite elements
KW - Multigoal-oriented a posteriori error estimation
KW - Optimal control
KW - Regularized p-Laplacian
UR - http://www.scopus.com/inward/record.url?scp=85078461211&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1903.02799
DO - 10.48550/arXiv.1903.02799
M3 - Article
AN - SCOPUS:85078461211
VL - 79
SP - 3001
EP - 3026
JO - Computers and Mathematics with Applications
JF - Computers and Mathematics with Applications
SN - 0898-1221
IS - 10
ER -