<abstract xmlns="http://www.w3.org/1999/xhtml"><p>We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an “optimize-then-reduce” approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows.</p></abstract>

We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an “optimize-then-reduce” approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows.

A certified RB method for PDE-constrained parametric optimization problems

Communications in Applied and Industrial Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"A certified RB method for PDE-constrained parametric optimization problems"}

We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization...

A certified RB method for PDE-constrained parametric optimization problems

Published Online: Jun 15, 2019

Page range: 123 - 152

Received: Jun 14, 2018

Accepted: May 17, 2019

DOI: https://doi.org/10.2478/caim-2019-0017

Keywords
PDE-constrained optimization, Reduced basis method, A posteriori error estimation, Adjoint-based approach, Non-convex cost functionals

© 2019 Andrea Manzoni et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

A certified RB method for PDE-constrained parametric optimization problems

Published Online: Jun 15, 2019

Page range: 123 - 152

Received: Jun 14, 2018

Accepted: May 17, 2019

DOI: https://doi.org/10.2478/caim-2019-0017

KeywordsPDE-constrained optimization, Reduced basis method, A posteriori error estimation, Adjoint-based approach, Non-convex cost functionals

© 2019 Andrea Manzoni et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
PDE-constrained optimization, Reduced basis method, A posteriori error estimation, Adjoint-based approach, Non-convex cost functionals