Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem
To calculate the parameters k in a stationary model Aku=ffrom measurements yˆof the solution u, i.e. to solve the inverse problem, usually a regularized minimization problem is solved and its solution kˆis considered as estimate of the true parameters. We focus on a situation which occurs e.g. for t...
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| Format: | Article |
| Language: | English |
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Elsevier
2022-06-01
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| Series: | Partial Differential Equations in Applied Mathematics |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S266681812200064X |
| _version_ | 1811343550537793536 |
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| author | Jochen Merker Aleš Matas |
| author_facet | Jochen Merker Aleš Matas |
| author_sort | Jochen Merker |
| collection | DOAJ |
| description | To calculate the parameters k in a stationary model Aku=ffrom measurements yˆof the solution u, i.e. to solve the inverse problem, usually a regularized minimization problem is solved and its solution kˆis considered as estimate of the true parameters. We focus on a situation which occurs e.g. for the coefficient functions k of a linear elliptic operator Aku=−divk∇uon a domain Ω⊂Rdand the total variation as regularization term. As functions k∈BV(Ω)of bounded variation can be discontinuous, existence of strong solutions u of Aku=fcannot be guaranteed and existence of minimizers cannot be obtained by standard methods. In this article, we prove solvability and stability for a general regularized minimization problem under weak assumptions, which particularly hold in case of BV(Ω)as parameter space due to a higher integrability result. |
| first_indexed | 2024-04-13T19:32:16Z |
| format | Article |
| id | doaj.art-4d81c4286ef14f97ab02b0c4b00247ad |
| institution | Directory Open Access Journal |
| issn | 2666-8181 |
| language | English |
| last_indexed | 2024-04-13T19:32:16Z |
| publishDate | 2022-06-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Partial Differential Equations in Applied Mathematics |
| spelling | doaj.art-4d81c4286ef14f97ab02b0c4b00247ad2022-12-22T02:33:09ZengElsevierPartial Differential Equations in Applied Mathematics2666-81812022-06-015100384Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problemJochen Merker0Aleš Matas1HTWK Leipzig, Leipzig, Germany; Corresponding author.Pilsen, CzechiaTo calculate the parameters k in a stationary model Aku=ffrom measurements yˆof the solution u, i.e. to solve the inverse problem, usually a regularized minimization problem is solved and its solution kˆis considered as estimate of the true parameters. We focus on a situation which occurs e.g. for the coefficient functions k of a linear elliptic operator Aku=−divk∇uon a domain Ω⊂Rdand the total variation as regularization term. As functions k∈BV(Ω)of bounded variation can be discontinuous, existence of strong solutions u of Aku=fcannot be guaranteed and existence of minimizers cannot be obtained by standard methods. In this article, we prove solvability and stability for a general regularized minimization problem under weak assumptions, which particularly hold in case of BV(Ω)as parameter space due to a higher integrability result.http://www.sciencedirect.com/science/article/pii/S266681812200064XLinear elliptic PDEDiscontinuous coefficientsOptimal controlInverse problemTotal variation minimizationHigher integrability |
| spellingShingle | Jochen Merker Aleš Matas Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem Partial Differential Equations in Applied Mathematics Linear elliptic PDE Discontinuous coefficients Optimal control Inverse problem Total variation minimization Higher integrability |
| title | Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem |
| title_full | Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem |
| title_fullStr | Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem |
| title_full_unstemmed | Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem |
| title_short | Estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem |
| title_sort | estimation of discontinuous parameters in linear elliptic equations by a regularized inverse problem |
| topic | Linear elliptic PDE Discontinuous coefficients Optimal control Inverse problem Total variation minimization Higher integrability |
| url | http://www.sciencedirect.com/science/article/pii/S266681812200064X |
| work_keys_str_mv | AT jochenmerker estimationofdiscontinuousparametersinlinearellipticequationsbyaregularizedinverseproblem AT alesmatas estimationofdiscontinuousparametersinlinearellipticequationsbyaregularizedinverseproblem |