Functional Regression for State Prediction Using Linear PDE Models and Observations
Partial differential equations (PDEs) are commonly used to model a wide variety of physical phenomena. A PDE model of a physical problem is typically described by conservation laws, constitutive laws, material properties, boundary conditions, boundary data, and geometry. In most practical applicatio...
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| Format: | Article |
| Language: | en_US |
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Society for Industrial and Applied Mathematics (SIAM)
2016
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| Online Access: | http://hdl.handle.net/1721.1/103578 https://orcid.org/0000-0002-8556-685X https://orcid.org/0000-0002-1733-5363 |
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| author | Nguyen, Ngoc Cuong Men, Han Freund, Robert Michael Peraire, Jaime |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Nguyen, Ngoc Cuong Men, Han Freund, Robert Michael Peraire, Jaime |
| author_sort | Nguyen, Ngoc Cuong |
| collection | MIT |
| description | Partial differential equations (PDEs) are commonly used to model a wide variety of physical phenomena. A PDE model of a physical problem is typically described by conservation laws, constitutive laws, material properties, boundary conditions, boundary data, and geometry. In most practical applications, however, the PDE model is only an approximation to the real physical problem due to both (i) the deliberate mathematical simplification of the model to keep it tractable and (ii) the inherent uncertainty of the physical parameters. In such cases, the PDE model may not produce a good prediction of the true state of the underlying physical problem. In this paper, we introduce a functional regression method that incorporates observations into a deterministic linear PDE model to improve its prediction of the true state. Our method is devised as follows. First, we augment the PDE model with a random Gaussian functional which serves to represent various sources of uncertainty in the model. We next derive a linear regression model for the Gaussian functional by utilizing observations and adjoint states. This allows us to determine the posterior distribution of the Gaussian functional and the posterior distribution for our estimate of the true state. Furthermore, we consider the problem of experimental design in this setting, wherein we develop an algorithm for designing experiments to efficiently reduce the variance of our state estimate. We provide several examples from the heat conduction, the convection-diffusion equation, and the reduced wave equation, all of which demonstrate the performance of the proposed methodology. |
| first_indexed | 2024-09-23T16:34:03Z |
| format | Article |
| id | mit-1721.1/103578 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:34:03Z |
| publishDate | 2016 |
| publisher | Society for Industrial and Applied Mathematics (SIAM) |
| record_format | dspace |
| spelling | mit-1721.1/1035782022-10-02T08:18:41Z Functional Regression for State Prediction Using Linear PDE Models and Observations Nguyen, Ngoc Cuong Men, Han Freund, Robert Michael Peraire, Jaime Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Sloan School of Management Nguyen, Ngoc Cuong Men, Han Freund, Robert Michael Peraire, Jaime Partial differential equations (PDEs) are commonly used to model a wide variety of physical phenomena. A PDE model of a physical problem is typically described by conservation laws, constitutive laws, material properties, boundary conditions, boundary data, and geometry. In most practical applications, however, the PDE model is only an approximation to the real physical problem due to both (i) the deliberate mathematical simplification of the model to keep it tractable and (ii) the inherent uncertainty of the physical parameters. In such cases, the PDE model may not produce a good prediction of the true state of the underlying physical problem. In this paper, we introduce a functional regression method that incorporates observations into a deterministic linear PDE model to improve its prediction of the true state. Our method is devised as follows. First, we augment the PDE model with a random Gaussian functional which serves to represent various sources of uncertainty in the model. We next derive a linear regression model for the Gaussian functional by utilizing observations and adjoint states. This allows us to determine the posterior distribution of the Gaussian functional and the posterior distribution for our estimate of the true state. Furthermore, we consider the problem of experimental design in this setting, wherein we develop an algorithm for designing experiments to efficiently reduce the variance of our state estimate. We provide several examples from the heat conduction, the convection-diffusion equation, and the reduced wave equation, all of which demonstrate the performance of the proposed methodology. United States. Air Force Office of Scientific Research (AFOSR grant FA9550-15-1-0276) 2016-07-12T18:43:37Z 2016-07-12T18:43:37Z 2016-03 2015-11 Article http://purl.org/eprint/type/JournalArticle 1064-8275 1095-7197 http://hdl.handle.net/1721.1/103578 Nguyen, N. C., H. Men, R. M. Freund, and J. Peraire. “Functional Regression for State Prediction Using Linear PDE Models and Observations.” SIAM Journal on Scientific Computing 38, no. 2 (January 2016): B247–B271. © 2016, Society for Industrial and Applied Mathematics. https://orcid.org/0000-0002-8556-685X https://orcid.org/0000-0002-1733-5363 en_US http://dx.doi.org/10.1137/14100275x 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) SIAM |
| spellingShingle | Nguyen, Ngoc Cuong Men, Han Freund, Robert Michael Peraire, Jaime Functional Regression for State Prediction Using Linear PDE Models and Observations |
| title | Functional Regression for State Prediction Using Linear PDE Models and Observations |
| title_full | Functional Regression for State Prediction Using Linear PDE Models and Observations |
| title_fullStr | Functional Regression for State Prediction Using Linear PDE Models and Observations |
| title_full_unstemmed | Functional Regression for State Prediction Using Linear PDE Models and Observations |
| title_short | Functional Regression for State Prediction Using Linear PDE Models and Observations |
| title_sort | functional regression for state prediction using linear pde models and observations |
| url | http://hdl.handle.net/1721.1/103578 https://orcid.org/0000-0002-8556-685X https://orcid.org/0000-0002-1733-5363 |
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