Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems
We present a method for approximating the solution of a parameterized, nonlinear dynamical system using an affine combination of solutions computed at other points in the input parameter space. The coefficients of the affine combination are computed with a nonlinear least squares procedure that mini...
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| Format: | Article |
| Language: | en_US |
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Society for Industrial and Applied Mathematics
2013
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| Online Access: | http://hdl.handle.net/1721.1/77927 https://orcid.org/0000-0001-9669-2563 |
| _version_ | 1811097527710121984 |
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| author | Constantine, Paul G. Wang, Qiqi |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Constantine, Paul G. Wang, Qiqi |
| author_sort | Constantine, Paul G. |
| collection | MIT |
| description | We present a method for approximating the solution of a parameterized, nonlinear dynamical system using an affine combination of solutions computed at other points in the input parameter space. The coefficients of the affine combination are computed with a nonlinear least squares procedure that minimizes the residual of the governing equations. The approximation properties of this residual minimizing scheme are comparable to existing reduced basis and POD-Galerkin model reduction methods, but its implementation requires only independent evaluations of the nonlinear forcing function. It is particularly appropriate when one wishes to approximate the states at a few points in time without time marching from the initial conditions. We prove some interesting characteristics of the scheme, including an interpolatory property, and we present heuristics for mitigating the effects of the ill-conditioning and reducing the overall cost of the method. We apply the method to representative numerical examples from kinetics---a three-state system with one parameter controlling the stiffness---and conductive heat transfer---a nonlinear parabolic PDE with a random field model for the thermal conductivity. |
| first_indexed | 2024-09-23T17:00:48Z |
| format | Article |
| id | mit-1721.1/77927 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T17:00:48Z |
| publishDate | 2013 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | mit-1721.1/779272022-09-29T23:04:28Z Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems Constantine, Paul G. Wang, Qiqi Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Wang, Qiqi We present a method for approximating the solution of a parameterized, nonlinear dynamical system using an affine combination of solutions computed at other points in the input parameter space. The coefficients of the affine combination are computed with a nonlinear least squares procedure that minimizes the residual of the governing equations. The approximation properties of this residual minimizing scheme are comparable to existing reduced basis and POD-Galerkin model reduction methods, but its implementation requires only independent evaluations of the nonlinear forcing function. It is particularly appropriate when one wishes to approximate the states at a few points in time without time marching from the initial conditions. We prove some interesting characteristics of the scheme, including an interpolatory property, and we present heuristics for mitigating the effects of the ill-conditioning and reducing the overall cost of the method. We apply the method to representative numerical examples from kinetics---a three-state system with one parameter controlling the stiffness---and conductive heat transfer---a nonlinear parabolic PDE with a random field model for the thermal conductivity. 2013-03-15T19:43:19Z 2013-03-15T19:43:19Z 2012-07 2010-12 Article http://purl.org/eprint/type/JournalArticle 1064-8275 1095-7197 http://hdl.handle.net/1721.1/77927 Constantine, Paul G., and Qiqi Wang. “Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems.” SIAM Journal on Scientific Computing 34.4 (2012): A2118–A2144. © 2012, Society for Industrial and Applied Mathematics https://orcid.org/0000-0001-9669-2563 en_US http://dx.doi.org/10.1137/100816717 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 SIAM |
| spellingShingle | Constantine, Paul G. Wang, Qiqi Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems |
| title | Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems |
| title_full | Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems |
| title_fullStr | Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems |
| title_full_unstemmed | Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems |
| title_short | Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems |
| title_sort | residual minimizing model interpolation for parameterized nonlinear dynamical systems |
| url | http://hdl.handle.net/1721.1/77927 https://orcid.org/0000-0001-9669-2563 |
| work_keys_str_mv | AT constantinepaulg residualminimizingmodelinterpolationforparameterizednonlineardynamicalsystems AT wangqiqi residualminimizingmodelinterpolationforparameterizednonlineardynamicalsystems |