A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction
In this article we introduce a new dimensional reduction approach which is based on the application of reduced basis (RB) techniques in the hierarchical model reduction (HMR) framework. Considering problems that exhibit a dominant spatial direction, the idea of HMR is to perform a Galerkin projectio...
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| Format: | Article |
| Language: | en_US |
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Society for Industrial and Applied Mathematics
2014
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| Online Access: | http://hdl.handle.net/1721.1/88243 https://orcid.org/0000-0003-4245-6586 |
| _version_ | 1826195237630902272 |
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| author | Ohlberger, Mario Smetana, Kathrin |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Ohlberger, Mario Smetana, Kathrin |
| author_sort | Ohlberger, Mario |
| collection | MIT |
| description | In this article we introduce a new dimensional reduction approach which is based on the application of reduced basis (RB) techniques in the hierarchical model reduction (HMR) framework. Considering problems that exhibit a dominant spatial direction, the idea of HMR is to perform a Galerkin projection onto a reduced space, which combines the full solution space in the dominant direction with a reduction space in the transverse direction. The latter is spanned by modal orthonormal basis functions. While so far the basis functions in the HMR approach have been chosen a priori [S. Perotto, A. Ern, and A. Veneziani, Multiscale Model. Simul., 8 (2010), pp. 1102--1127], for instance, as Legendre or trigonometric polynomials, in this work a highly nonlinear approximation is employed for the construction of the reduction space. To this end we first derive a lower dimensional parametrized problem in the transverse direction from the full problem where the parameters reflect the influence from the unknown solution in the dominant direction. Exploiting the good approximation properties of RB methods, we then construct a reduction space by applying a proper orthogonal decomposition to a set of snapshots of the parametrized partial differential equation. For an efficient construction of the snapshot set we apply adaptive refinement in parameter space based on an a posteriori error estimate that is also derived in this article. We introduce our method for general elliptic problems such as advection-diffusion equations in two space dimensions. Numerical experiments demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full dimensional problem and the computational efficiency of our new adaptive approach. |
| first_indexed | 2024-09-23T10:09:53Z |
| format | Article |
| id | mit-1721.1/88243 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T10:09:53Z |
| publishDate | 2014 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | mit-1721.1/882432022-09-30T19:16:30Z A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction Ohlberger, Mario Smetana, Kathrin Massachusetts Institute of Technology. Department of Mechanical Engineering Smetana, Kathrin In this article we introduce a new dimensional reduction approach which is based on the application of reduced basis (RB) techniques in the hierarchical model reduction (HMR) framework. Considering problems that exhibit a dominant spatial direction, the idea of HMR is to perform a Galerkin projection onto a reduced space, which combines the full solution space in the dominant direction with a reduction space in the transverse direction. The latter is spanned by modal orthonormal basis functions. While so far the basis functions in the HMR approach have been chosen a priori [S. Perotto, A. Ern, and A. Veneziani, Multiscale Model. Simul., 8 (2010), pp. 1102--1127], for instance, as Legendre or trigonometric polynomials, in this work a highly nonlinear approximation is employed for the construction of the reduction space. To this end we first derive a lower dimensional parametrized problem in the transverse direction from the full problem where the parameters reflect the influence from the unknown solution in the dominant direction. Exploiting the good approximation properties of RB methods, we then construct a reduction space by applying a proper orthogonal decomposition to a set of snapshots of the parametrized partial differential equation. For an efficient construction of the snapshot set we apply adaptive refinement in parameter space based on an a posteriori error estimate that is also derived in this article. We introduce our method for general elliptic problems such as advection-diffusion equations in two space dimensions. Numerical experiments demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full dimensional problem and the computational efficiency of our new adaptive approach. 2014-07-10T13:37:57Z 2014-07-10T13:37:57Z 2014-04 2014-01 Article http://purl.org/eprint/type/JournalArticle 1064-8275 1095-7197 http://hdl.handle.net/1721.1/88243 Ohlberger, Mario, and Kathrin Smetana. “A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction.” SIAM Journal on Scientific Computing 36, no. 2 (January 2014): A714–A736. © 2014, Society for Industrial and Applied Mathematics. https://orcid.org/0000-0003-4245-6586 en_US http://dx.doi.org/10.1137/130939122 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 and Applied Mathematics Society for Industrial and Applied Mathematics |
| spellingShingle | Ohlberger, Mario Smetana, Kathrin A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction |
| title | A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction |
| title_full | A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction |
| title_fullStr | A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction |
| title_full_unstemmed | A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction |
| title_short | A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction |
| title_sort | dimensional reduction approach based on the application of reduced basis methods in the framework of hierarchical model reduction |
| url | http://hdl.handle.net/1721.1/88243 https://orcid.org/0000-0003-4245-6586 |
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