A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization
Parameter optimization problems constrained by partial differential equations (PDEs) appear in many science and engineering applications. Solving these optimization problems may require a prohibitively large number of computationally expensive PDE solves, especially if the dimension of the design sp...
| Main Authors: | , , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Published: |
Society for Industrial & Applied Mathematics (SIAM)
2018
|
| Online Access: | http://hdl.handle.net/1721.1/116912 https://orcid.org/0000-0001-6713-3746 https://orcid.org/0000-0003-2156-9338 |
| _version_ | 1826212908709707776 |
|---|---|
| author | Grepl, Martin Veroy, Karen Qian, Elizabeth Y. Willcox, Karen E |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Grepl, Martin Veroy, Karen Qian, Elizabeth Y. Willcox, Karen E |
| author_sort | Grepl, Martin |
| collection | MIT |
| description | Parameter optimization problems constrained by partial differential equations (PDEs) appear in many science and engineering applications. Solving these optimization problems may require a prohibitively large number of computationally expensive PDE solves, especially if the dimension of the design space is large. It is therefore advantageous to replace expensive high-dimensional PDE solvers (e.g., finite element) with lower-dimensional surrogate models. In this paper, the reduced basis (RB) model reduction method is used in conjunction with a trust region optimization framework to accelerate PDE-constrained parameter optimization. Novel a posteriori error bounds on the RB cost and cost gradient for quadratic cost functionals (e.g., least squares) are presented and used to guarantee convergence to the optimum of the high-fidelity model. The proposed certified RB trust region approach uses high-fidelity solves to update the RB model only if the approximation is no longer sufficiently accurate, reducing the number of full-fidelity solves required. We consider problems governed by elliptic and parabolic PDEs and present numerical results for a thermal fin model problem in which we are able to reduce the number of full solves necessary for the optimization by up to 86%. Key words: model reduction, optimization, trust region methods, partial differential equations, reduced basis methods, error bounds, parametrized systems |
| first_indexed | 2024-09-23T15:40:04Z |
| format | Article |
| id | mit-1721.1/116912 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T15:40:04Z |
| publishDate | 2018 |
| publisher | Society for Industrial & Applied Mathematics (SIAM) |
| record_format | dspace |
| spelling | mit-1721.1/1169122022-09-29T15:22:32Z A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization Grepl, Martin Veroy, Karen Qian, Elizabeth Y. Willcox, Karen E Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Qian, Elizabeth Y. Willcox, Karen E Parameter optimization problems constrained by partial differential equations (PDEs) appear in many science and engineering applications. Solving these optimization problems may require a prohibitively large number of computationally expensive PDE solves, especially if the dimension of the design space is large. It is therefore advantageous to replace expensive high-dimensional PDE solvers (e.g., finite element) with lower-dimensional surrogate models. In this paper, the reduced basis (RB) model reduction method is used in conjunction with a trust region optimization framework to accelerate PDE-constrained parameter optimization. Novel a posteriori error bounds on the RB cost and cost gradient for quadratic cost functionals (e.g., least squares) are presented and used to guarantee convergence to the optimum of the high-fidelity model. The proposed certified RB trust region approach uses high-fidelity solves to update the RB model only if the approximation is no longer sufficiently accurate, reducing the number of full-fidelity solves required. We consider problems governed by elliptic and parabolic PDEs and present numerical results for a thermal fin model problem in which we are able to reduce the number of full solves necessary for the optimization by up to 86%. Key words: model reduction, optimization, trust region methods, partial differential equations, reduced basis methods, error bounds, parametrized systems Fulbright U.S. Student Program National Science Foundation (U.S.). Graduate Research Fellowship Program Hertz Foundation United States. Department of Energy. Office of Advanced Scientific Computing Research (Award DEFG02-08ER2585) United States. Department of Energy. Office of Advanced Scientific Computing Research (Award DE-SC0009297) 2018-07-11T18:40:57Z 2018-07-11T18:40:57Z 2017-10 2017-02 2018-04-17T16:59:52Z Article http://purl.org/eprint/type/JournalArticle 1064-8275 1095-7197 http://hdl.handle.net/1721.1/116912 Qian, Elizabeth, et al. “A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization.” SIAM Journal on Scientific Computing, vol. 39, no. 5, Jan. 2017, pp. S434–60. https://orcid.org/0000-0001-6713-3746 https://orcid.org/0000-0003-2156-9338 http://dx.doi.org/10.1137/16M1081981 SIAM Journal on Scientific Computing Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Society for Industrial & Applied Mathematics (SIAM) SIAM |
| spellingShingle | Grepl, Martin Veroy, Karen Qian, Elizabeth Y. Willcox, Karen E A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization |
| title | A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization |
| title_full | A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization |
| title_fullStr | A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization |
| title_full_unstemmed | A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization |
| title_short | A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization |
| title_sort | certified trust region reduced basis approach to pde constrained optimization |
| url | http://hdl.handle.net/1721.1/116912 https://orcid.org/0000-0001-6713-3746 https://orcid.org/0000-0003-2156-9338 |
| work_keys_str_mv | AT greplmartin acertifiedtrustregionreducedbasisapproachtopdeconstrainedoptimization AT veroykaren acertifiedtrustregionreducedbasisapproachtopdeconstrainedoptimization AT qianelizabethy acertifiedtrustregionreducedbasisapproachtopdeconstrainedoptimization AT willcoxkarene acertifiedtrustregionreducedbasisapproachtopdeconstrainedoptimization AT greplmartin certifiedtrustregionreducedbasisapproachtopdeconstrainedoptimization AT veroykaren certifiedtrustregionreducedbasisapproachtopdeconstrainedoptimization AT qianelizabethy certifiedtrustregionreducedbasisapproachtopdeconstrainedoptimization AT willcoxkarene certifiedtrustregionreducedbasisapproachtopdeconstrainedoptimization |