Regularization Techniques for Estimating the Source in a Complete Parabolic Equation in $\mathbb{R}^n$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917744385982464 |
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| author | Umbricht, Guillermo Federico Rubio, Diana |
| author_facet | Umbricht, Guillermo Federico Rubio, Diana |
| contents | In this article, the problem of identifying the source term in transport processes given by a complete parabolic equation is studied mathematically from noisy measurements taken at an arbitrary fixed time. The problem is solved analytically with Fourier techniques and it is shown that this solution is not stable. Three single parameter families of regularization operators are proposed to dealt with the instability of the solution. Each of them is designed to compensate the factor that causes the instability of the inverse operator. Moreover, a rule of choice for the regularization parameter is included and a Hölder error bound type is obtained for each estimation. Numerical examples of different characteristics are presented to demonstrate the benefits of the proposed strategies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13094 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularization Techniques for Estimating the Source in a Complete Parabolic Equation in $\mathbb{R}^n$ Umbricht, Guillermo Federico Rubio, Diana Analysis of PDEs Mathematical Physics In this article, the problem of identifying the source term in transport processes given by a complete parabolic equation is studied mathematically from noisy measurements taken at an arbitrary fixed time. The problem is solved analytically with Fourier techniques and it is shown that this solution is not stable. Three single parameter families of regularization operators are proposed to dealt with the instability of the solution. Each of them is designed to compensate the factor that causes the instability of the inverse operator. Moreover, a rule of choice for the regularization parameter is included and a Hölder error bound type is obtained for each estimation. Numerical examples of different characteristics are presented to demonstrate the benefits of the proposed strategies. |
| title | Regularization Techniques for Estimating the Source in a Complete Parabolic Equation in $\mathbb{R}^n$ |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2404.13094 |