Localized Reduced Basis Methods for PDE-constrained Parameter Optimization
Final Report Abstract
Within this very successful project we developed and fully analyzed a novel Reduced Basis Trust Region (RB-TR) model reduction framework for large scale and multiscale PDE-constrained optimization and inverse parameter identification problems. The framework comes with a full a-posteriori error estimation for the approximation errors in the underlying primal and dual problems as well as for the corresponding sensitivities, the approximated objective functional and its gradient with respect to the parameters. Moreover convergence of the overall iterative methods to accumulation points could be shown, including the situation of box constraints for the parameters. Concerning localization for large scale or multiscale forward problems, we investigated two approaches. The first approach is particularly suited for multiscale problems and is based on a novel reduced basis approximation of the local orthogonal decomposition (LOD) method, while the second approach, based on the localized reduced basis method in a discontinuous Galerkin setting is more general and therefore also applicable to wider classes of large scale problems. Moreover, we extended and applied the novel approaches also to parabolic forward operators, as well as elliptic-parabolic systems. Finally, we also addressed elliptic inverse parameter identification problems, where in addition to the primal and dual states, also the parameter space is infinite or very high dimensional and needs to be reduced. To this end we suggested a new strategy for combined parameter and space reduction where the adaptive enrichment of the reduced parameter space is naturally inherited from the Tikhonov regularization within an iteratively regularized Gauß-Newton method. The scientific output of this project is considerably large which is also documented by the number and quality of the resulting publications. We emphasize that the results of this project evidently show, how the research achievements profited from the complementary expertise that was brought in from the participating PIs.
Publications
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Software for adaptive projected Newton NCD corrected TR-RB approach for PDE constrained parameter optimization, 2020.
S. Banholzer; T. Keil; L. Mechelli; M. Ohlberger; F. Schindler & S. Volkwein
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Software for NCD corrected TR-RB approach for PDE constrained optimization
T. Keil; L. Mechelli; M. Ohlberger; F. Schindler & S. Volkwein
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A non-conforming dual approach for adaptive Trust-Region reduced basis approximation of PDE-constrained parameter optimization. ESAIM: Mathematical Modelling and Numerical Analysis, 55(3), 1239-1269.
Keil, Tim; Mechelli, Luca; Ohlberger, Mario; Schindler, Felix & Volkwein, Stefan
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Software for An online efficient two-scale reduced basis approach for the localized orthogonal decomposition
T. Keil & S. Rave
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Software for Model reduction for large scale systems
T. Keil & M. Ohlberger
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A Trust Region Reduced Basis Pascoletti-Serafini Algorithm for Multi-Objective PDE-Constrained Parameter Optimization. Mathematical and Computational Applications, 27(3), 39.
Banholzer, Stefan; Mechelli, Luca & Volkwein, Stefan
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Adaptive machine learning-based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery. Advances in Computational Mathematics, 48(6).
Keil, Tim; Kleikamp, Hendrik; Lorentzen, Rolf J.; Oguntola, Micheal B. & Ohlberger, Mario
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Adaptive Reduced Basis Methods for Multiscale Problems and Large-scale PDE- ¨ Constrained Optimization. PhD thesis, Universitat Munster, Germany, 2022.
T. Keil
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An adaptive projected Newton non-conforming dual approach for trust-region reduced basis approximation of PDE-constrained parameter optimization. Pure and Applied Functional Analysis, 7:1561-1596, 2022
S. Banholzer; T. Keil; L. Mechelli; M. Ohlberger; F. Schindler & S. Volkwein
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Model Reduction for Large Scale Systems. Lecture Notes in Computer Science, 16-28. Springer International Publishing.
Keil, Tim & Ohlberger, Mario
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Software for A relaxed localized trust-region reduced basis approach for optimization of multiscale problems
Keil, Tim
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Supplementary software for the PhD-thesis Adaptive reduced basis methods for multiscale problems and large-scale PDE-constrained optimization, 2022.
T. Keil
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Trust-Region RB Methods for PDE-Constrained Optimization and Optimal Input Design. IFAC-PapersOnLine, 55(26), 149-154.
Petrocchi, Andrea; Scharrer, Matthias K. & Volkwein, Stefan
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Adaptive parameter optimization for an elliptic-parabolic system using the reduced-basis method with hierarchical a-posteriori error analysis
B. Azmi; A. Petrocchi & S. Volkwein
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An Online Efficient Two-Scale Reduced Basis Approach for the Localized Orthogonal Decomposition. SIAM Journal on Scientific Computing, 45(4), A1491-A1518.
Keil, Tim & Rave, Stephan
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Software for Adaptive reduced basis trust region methods for inverse parameter identification problems, 2023.
M. Kartmann
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A relaxed localized trust-region reduced basis approach for optimization of multiscale problems. ESAIM: Mathematical Modelling and Numerical Analysis, 58(1), 79-105.
Keil, Tim & Ohlberger, Mario
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Adaptive Localized Reduced Basis Methods forLarge Scale PDE-Constrained Optimization. Lecture Notes in Computer Science, 108-116. Springer Nature Switzerland.
Keil, Tim; Ohlberger, Mario & Schindler, Felix
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Adaptive reduced basis trust region methods for parameter identification problems. Computational Science and Engineering, 1(1).
Kartmann, Michael; Keil, Tim; Ohlberger, Mario; Volkwein, Stefan & Kaltenbacher, Barbara
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Optimal Input Design for Large-Scale Inverse Problems using PDE-Constrained Optimization. PhD thesis, Universität Konstanz, Germany
A. Petrocchi
