Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation
We present a technique for the rapid and reliable evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN span...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
2004
|
| Subjects: | |
| Online Access: | http://hdl.handle.net/1721.1/7375 |
| _version_ | 1811096803700899840 |
|---|---|
| author | Nguyen, N. C. Liu, Guirong Patera, Anthony T. |
| author_facet | Nguyen, N. C. Liu, Guirong Patera, Anthony T. |
| author_sort | Nguyen, N. C. |
| collection | MIT |
| description | We present a technique for the rapid and reliable
evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation
process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse
computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and
error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense)
system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is
decreased. |
| first_indexed | 2024-09-23T16:49:16Z |
| format | Article |
| id | mit-1721.1/7375 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:49:16Z |
| publishDate | 2004 |
| record_format | dspace |
| spelling | mit-1721.1/73752019-04-10T16:47:38Z Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation Nguyen, N. C. Liu, Guirong Patera, Anthony T. Linear elliptic equations Reduced-basis method Reduced-basis approximation A posteriori error estimation Parameter estimation Inverse computational method Possibility region We present a technique for the rapid and reliable evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense) system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is decreased. Singapore-MIT Alliance (SMA) 2004-12-10T14:42:50Z 2004-12-10T14:42:50Z 2005-01 Article http://hdl.handle.net/1721.1/7375 en High Performance Computation for Engineered Systems (HPCES); 1117342 bytes application/pdf application/pdf |
| spellingShingle | Linear elliptic equations Reduced-basis method Reduced-basis approximation A posteriori error estimation Parameter estimation Inverse computational method Possibility region Nguyen, N. C. Liu, Guirong Patera, Anthony T. Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation |
| title | Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation |
| title_full | Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation |
| title_fullStr | Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation |
| title_full_unstemmed | Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation |
| title_short | Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation |
| title_sort | certified rapid solution of parametrized linear elliptic equations application to parameter estimation |
| topic | Linear elliptic equations Reduced-basis method Reduced-basis approximation A posteriori error estimation Parameter estimation Inverse computational method Possibility region |
| url | http://hdl.handle.net/1721.1/7375 |
| work_keys_str_mv | AT nguyennc certifiedrapidsolutionofparametrizedlinearellipticequationsapplicationtoparameterestimation AT liuguirong certifiedrapidsolutionofparametrizedlinearellipticequationsapplicationtoparameterestimation AT pateraanthonyt certifiedrapidsolutionofparametrizedlinearellipticequationsapplicationtoparameterestimation |