In this contribution we propose and rigorously analyze new variants of adaptive Trust-Region methods for parameter optimization with PDE constraints and bilateral parameter constraints. The approach employs successively enriched Reduced Basis surrogate models that are constructed during the outer optimization loop and used as model function for the Trust-Region method. Each Trust-Region sub-problem is solved with the projected BFGS method. Moreover, we propose a non-conforming dual (NCD) approach to improve the standard RB approximation of the optimality system. Rigorous improved a posteriori error bounds are derived and used to prove convergence of the resulting NCD-corrected adaptive Trust-Region Reduced Basis algorithm. Numerical experiments demonstrate that this approach enables to reduce the computational demand for large scale or multi-scale PDE constrained optimization problems significantly.
Keywords: PDE constrained optimization, Trust-Region method, error analysis, Reduced Basis method, model order reduction, parametrized systems, large scale problems
@article{M2AN_2021__55_3_1239_0,
author = {Keil, Tim and Mechelli, Luca and Ohlberger, Mario and Schindler, Felix and Volkwein, Stefan},
title = {A non-conforming dual approach for adaptive {Trust-Region} reduced basis approximation of {PDE-constrained} parameter optimization},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1239--1269},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {3},
doi = {10.1051/m2an/2021019},
mrnumber = {4269464},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021019/}
}
TY - JOUR AU - Keil, Tim AU - Mechelli, Luca AU - Ohlberger, Mario AU - Schindler, Felix AU - Volkwein, Stefan TI - A non-conforming dual approach for adaptive Trust-Region reduced basis approximation of PDE-constrained parameter optimization JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 1239 EP - 1269 VL - 55 IS - 3 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021019/ DO - 10.1051/m2an/2021019 LA - en ID - M2AN_2021__55_3_1239_0 ER -
%0 Journal Article %A Keil, Tim %A Mechelli, Luca %A Ohlberger, Mario %A Schindler, Felix %A Volkwein, Stefan %T A non-conforming dual approach for adaptive Trust-Region reduced basis approximation of PDE-constrained parameter optimization %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 1239-1269 %V 55 %N 3 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021019/ %R 10.1051/m2an/2021019 %G en %F M2AN_2021__55_3_1239_0
Keil, Tim; Mechelli, Luca; Ohlberger, Mario; Schindler, Felix; Volkwein, Stefan. A non-conforming dual approach for adaptive Trust-Region reduced basis approximation of PDE-constrained parameter optimization. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 3, pp. 1239-1269. doi: 10.1051/m2an/2021019
[1] , and , Trust-region proper orthogonal decomposition for flow control. Technical Report 2000-25, ICASE (2000). http://www.dtic.mil/docs/citations/ADA377382.
[2] , , and , An “empirical interpolation’’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339 (2004) 667–672. | MR | Zbl | DOI
[3] , and , Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J. Control Optim. 39 (2000) 113–132. | MR | Zbl | DOI
[4] and , A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints. Comput. Optim. Appl. 44 (2009) 3–25. | MR | Zbl | DOI
[5] , , and , editors. Model Reduction and Approximation. In: Vol. 15 of Computational Science & Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2017). Theory and algorithms. | Zbl
[6] , , , and , editors. Model Reduction of Parametrized Systems. In: Vol. 17 of Modeling, Simulation and Applications. Springer-Verlag GmbH (2017). | MR
[7] and , Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models. J. Comput. Phys. 227 (2008) 7813–7840. | MR | Zbl | DOI
[8] , and , Reduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDE. In: To Appear in New Trends in PDE Constrained Optimization (2020). | MR
[9] , , , , and , Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. | MR | Zbl | DOI
[10] , , and , A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations. In: 11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014 (2014) 4094–4102.
[11] , , and , ArbiLoMod, a simulation technique designed for arbitrary local modifications. SIAM J. Sci. Comput. 39 (2017) A1435–A1465. | MR | DOI
[12] and , Second order optimality conditions and their role in PDE control. Jahresber. Dtsch. Math. Ver. 117 (2015) 3–44. | MR | DOI
[13] and , Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32 (2010) 2737–2764. | MR | Zbl | DOI
[14] , and , A dual-weighted trust-region adaptive POD 4-D Var applied to a finite-volume shallow water equations model on the sphere. Int. J. Numer. Methods Fluids 68 (2012) 377–402. | MR | Zbl | DOI
[15] , , and , Generalized Multilevel SQP-methods for PDAE-constrained Optimization Based on Space-Time Adaptive PDAE Solvers. Springer Basel, Basel (2012) 51–74. | MR
[16] , and , Trust Region Methods. MOS-SIAM Series on OptimizationSociety for Industrial and Applied Mathematics (2000). | MR | Zbl
[17] , Reduced basis method and error estimation for parametrized optimal control problems with control constraints. J. Sci. Comput. 50 (2012) 287–305. | MR | Zbl | DOI
[18] and , Certified PDE-constrained parameter optimization using reduced basis surrogate models for evolution problems. Comput. Optim. App. 60 (2015) 753–787. | MR | DOI
[19] , and , Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34 (2012) A937–A969. | MR | Zbl | DOI
[20] , and , An “” certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput. 32 (2010) 3170–3200. | MR | Zbl | DOI
[21] , and , A reduced basis Landweber method for nonlinear inverse problems. Inverse Prob. 32 (2016) 035001. | MR | DOI
[22] and , Gmsh: a 3-d finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Methods Eng. 79 (2009) 1309–1331. | MR | Zbl | DOI
[23] , , , and , POD basis updates for nonlinear PDE control. at – Automatisierungstechnik 65 (2017) 298–307. | DOI
[24] and , Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. C. R. Math. Acad. Sci. Paris 349 (2011) 873–877. | MR | Zbl | DOI
[25] and , Proper orthogonal decomposition for linear-quadratic optimal control, edited by , , and . In: Model Reduction and Approximation: Theory and Algorithms. SIAM, Philadelphia, PA (2017) 3–63. | MR | DOI
[26] , Convergence rates of the POD-greedy method. ESAIM: M2AN 47 (2013) 859–873. | MR | Zbl | Numdam | DOI
[27] , Reduced basis methods for parametrized PDEs: a tutorial introduction for stationary and instationary problems. Model Reduction Approximation: Theory Algorithms 15 (2017) 65. | MR | DOI
[28] , and , A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space. Math. Comput. Model. Dyn. Syst. 17 (2011) 423–442. | MR | Zbl | DOI
[29] and , Analysis of inexact trust-region SQP algorithms. SIAM J. Optim. 12 (2002) 283–302. | MR | Zbl | DOI
[30] , and , Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics. Springer International Publishing (2016). | MR | DOI
[31] and , Cross-gramian-based combined state and parameter reduction for large-scale control systems. Math. Probl. Eng. 13 (2014) 843869. | MR
[32] and , Data-driven combined state and parameter reduction for inverse problems. Adv. Comput. Math. 41 (2015) 1343–1364. | MR | DOI
[33] , and , Weak-duality based adaptive finite element methods for PDE-constrained optimization with pointwise gradient state-constraints. J. Comput. Math. 30 (2012) 101–123. | MR | Zbl | DOI
[34] and , Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. App. 39 (2008) 319–345. | MR | Zbl | DOI
[35] , , and , Optimization with PDE Constraints. Springer, Netherlands (2009). | MR | Zbl
[36] , , and , A successive constraint linear optimization method for lower bounds of parametric coercivity and inf–sup stability constants. C. R. Math. 345 (2007) 473–478. | MR | Zbl | DOI
[37] , and , A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD. ESAIM: M2AN 47 (2013) 555–581. | MR | Zbl | Numdam | DOI
[38] , , and , Certified reduced basis methods for parametrized elliptic optimal control problems with distributed controls. J. Sci. Comput. 75 (2018) 276–307. | MR | DOI
[39] , , , and , Software for NCD corrected TR-RB approach for PDE constrained optimization. (2020). DOI: 10.5281/zenodo.3897230.
[40] , Iterative Methods for Optimization. Society for Industrial and Applied Mathematics (1999). | MR | Zbl | DOI
[41] , and , Parameter and state model reduction for large-scale statistical inverse problems. SIAM J. Sci. Comput. 32 (2010) 2523–2542. | MR | Zbl | DOI
[42] and , A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math. 15 (2001) 285–309. | MR | Zbl | DOI
[43] , and , pyMOR – Generic Algorithms and Interfaces for Model Order Reduction. SIAM J. Sci. Comput. 38 (2016) S194–S216. | MR | DOI
[44] , , and , Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci. Comput. 35 (2013) A2316–A2340. | MR | Zbl | DOI
[45] and , Numerical Optimization, 2nd edition. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006). | MR | Zbl
[46] and , Error control for the localized reduced basis multiscale method with adaptive on-line enrichment. SIAM J. Sci. Comput. 37 (2015) A2865–A2895. | MR | DOI
[47] and , Non-conforming localized model reduction with online enrichment: towards optimal complexity in PDE constrained optimization. In: Vol. 200 ofFinite Volumes for Complex Applications VIII– Hyperbolic, Elliptic and Parabolic Problems. Springer Proc. Math. Stat. Springer, Cham (2017) 357–365. | MR | DOI
[48] , and , Localized model reduction in PDE constrained optimization. Int. Ser. Numer. Math. 169 (2018) 143–163. | MR | DOI
[49] and , Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optim. Eng. 8 (2007) 43–65. | MR | Zbl | DOI
[50] , , and , A certified trust region reduced basis approach to PDE-constrained optimization. SIAM J. Sci. Comput. 39 (2017) S434–S460. | MR | Zbl | DOI
[51] , and , Reduced Basis Methods for Partial Differential Equations, 1st edition. La Matematica per il 3+2. Springer International Publishing (2016). | MR | Zbl
[52] , On the adaptive discretization of PDE-based optimization problems. In: PDE Constrained Optimization. Springer (2006).
[53] , and , Trust-region POD using a-posteriori error estimation for semilinear parabolic optimal control problems. Konstanzer Schriften in Mathematik No. 359 (2017)http://nbn-resolving.de/urn:nbn:de:bsz:352-0-401106.
[54] and , A-posteriori error estimates for optimal control problems with state and control constraints. Numer. Math. 120 (2012) 733–762. | MR | Zbl | DOI
[55] , and , Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15 (2007) 1. | DOI
[56] and , A-priori error estimates for reduced order models in finance. ESAIM: M2AN 47 (2013) 449–469. | MR | Numdam | DOI
[57] and , Accelerating optimization of parametric linear systems by model order reduction. SIAM J. Optim. 23 (2013) 1344–1370. | MR | Zbl | DOI
[58] and , Progressive construction of a parametric reduced-order model for PDE-constrained optimization. Int. J. Numer. Methods Eng. 102 (2015) 1111–1135. | MR | Zbl | DOI
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