Nonlinear Model Reduction for Uncertainty Quantification in Large-Scale Inverse Problems
We present a model reduction approach to the solution of large-scale statistical inverse problems in a Bayesian inference setting. A key to the model reduction is an efficient representation of the non-linear terms in the reduced model. To achieve this, we present a formulation that employs masked p...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | en_US |
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John Wiley & Sons, Inc.
2011
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| Online Access: | http://hdl.handle.net/1721.1/61711 https://orcid.org/0000-0003-2156-9338 |
| _version_ | 1811075292116025344 |
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| author | Galbally, David Fidkowski, Krzysztof Willcox, Karen E. Ghattas, O. |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Galbally, David Fidkowski, Krzysztof Willcox, Karen E. Ghattas, O. |
| author_sort | Galbally, David |
| collection | MIT |
| description | We present a model reduction approach to the solution of large-scale statistical inverse problems in a Bayesian inference setting. A key to the model reduction is an efficient representation of the non-linear terms in the reduced model. To achieve this, we present a formulation that employs masked projection of the discrete equations; that is, we compute an approximation of the non-linear term using a select subset of interpolation points. Further, through this formulation we show similarities among the existing techniques of gappy proper orthogonal decomposition, missing point estimation, and empirical interpolation via coefficient-function approximation. The resulting model reduction methodology is applied to a highly non-linear combustion problem governed by an advection–diffusion-reaction partial differential equation (PDE). Our reduced model is used as a surrogate for a finite element discretization of the non-linear PDE within the Markov chain Monte Carlo sampling employed by the Bayesian inference approach. In two spatial dimensions, we show that this approach yields accurate results while reducing the computational cost by several orders of magnitude. For the full three-dimensional problem, a forward solve using a reduced model that has high fidelity over the input parameter space is more than two million times faster than the full-order finite element model, making tractable the solution of the statistical inverse problem that would otherwise require many years of CPU time. |
| first_indexed | 2024-09-23T10:03:45Z |
| format | Article |
| id | mit-1721.1/61711 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T10:03:45Z |
| publishDate | 2011 |
| publisher | John Wiley & Sons, Inc. |
| record_format | dspace |
| spelling | mit-1721.1/617112022-09-26T15:26:44Z Nonlinear Model Reduction for Uncertainty Quantification in Large-Scale Inverse Problems Galbally, David Fidkowski, Krzysztof Willcox, Karen E. Ghattas, O. Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Willcox, Karen E. Willcox, Karen E. We present a model reduction approach to the solution of large-scale statistical inverse problems in a Bayesian inference setting. A key to the model reduction is an efficient representation of the non-linear terms in the reduced model. To achieve this, we present a formulation that employs masked projection of the discrete equations; that is, we compute an approximation of the non-linear term using a select subset of interpolation points. Further, through this formulation we show similarities among the existing techniques of gappy proper orthogonal decomposition, missing point estimation, and empirical interpolation via coefficient-function approximation. The resulting model reduction methodology is applied to a highly non-linear combustion problem governed by an advection–diffusion-reaction partial differential equation (PDE). Our reduced model is used as a surrogate for a finite element discretization of the non-linear PDE within the Markov chain Monte Carlo sampling employed by the Bayesian inference approach. In two spatial dimensions, we show that this approach yields accurate results while reducing the computational cost by several orders of magnitude. For the full three-dimensional problem, a forward solve using a reduced model that has high fidelity over the input parameter space is more than two million times faster than the full-order finite element model, making tractable the solution of the statistical inverse problem that would otherwise require many years of CPU time. MIT-Singapore Alliance. Computational Engineering Programme United States. Air Force Office of Scientific Research (Contract Nos. FA9550-06-0271) National Science Foundation (U.S.) (Grant No. CNS-0540186) National Science Foundation (U.S.) (Grant No. CNS-0540372) Caja Madrid Foundation (Graduate Fellowship) 2011-03-17T12:04:42Z 2011-03-17T12:04:42Z 2009-09 2009-06 Article http://purl.org/eprint/type/JournalArticle 1097-0207 http://hdl.handle.net/1721.1/61711 Galbally, D., Fidkowski, K., Willcox, K. and Ghattas, O. (2010), Non-linear model reduction for uncertainty quantification in large-scale inverse problems. International Journal for Numerical Methods in Engineering, 81: 1581–1608. doi: 10.1002/nme.2746 https://orcid.org/0000-0003-2156-9338 en_US http://dx.doi.org/10.1002/nme.2746 International Journal for Numerical Methods in Engineering Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf John Wiley & Sons, Inc. Karen Willcox |
| spellingShingle | Galbally, David Fidkowski, Krzysztof Willcox, Karen E. Ghattas, O. Nonlinear Model Reduction for Uncertainty Quantification in Large-Scale Inverse Problems |
| title | Nonlinear Model Reduction for Uncertainty Quantification in Large-Scale Inverse Problems |
| title_full | Nonlinear Model Reduction for Uncertainty Quantification in Large-Scale Inverse Problems |
| title_fullStr | Nonlinear Model Reduction for Uncertainty Quantification in Large-Scale Inverse Problems |
| title_full_unstemmed | Nonlinear Model Reduction for Uncertainty Quantification in Large-Scale Inverse Problems |
| title_short | Nonlinear Model Reduction for Uncertainty Quantification in Large-Scale Inverse Problems |
| title_sort | nonlinear model reduction for uncertainty quantification in large scale inverse problems |
| url | http://hdl.handle.net/1721.1/61711 https://orcid.org/0000-0003-2156-9338 |
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