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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2506.23559 |
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| _version_ | 1866912869668356096 |
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| author | Busch, Leonard Lassas, Matti Oksanen, Lauri Salo, Mikko |
| author_facet | Busch, Leonard Lassas, Matti Oksanen, Lauri Salo, Mikko |
| contents | For a time-independent potential $q\in L^\infty$, consider the source-to-solution operator that maps a source $f$ to the solution $u=u(t,x)$ of $(\Box+q)u=f$ in Euclidean space with an obstacle, where we impose on $u$ vanishing Cauchy data at $t=0$ and vanishing Dirichlet data at the boundary of the obstacle. We study the inverse problem of recovering the potential $q$ from this source-to-solution map restricted to some measurement domain. By giving an example where measurements take place in some subset and the support of $q$ lies in the `shadow region' of the obstacle, we show that recovery of $q$ is exponentially unstable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23559 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Exponential Instability of an Inverse Problem for the Wave Equation Busch, Leonard Lassas, Matti Oksanen, Lauri Salo, Mikko Analysis of PDEs For a time-independent potential $q\in L^\infty$, consider the source-to-solution operator that maps a source $f$ to the solution $u=u(t,x)$ of $(\Box+q)u=f$ in Euclidean space with an obstacle, where we impose on $u$ vanishing Cauchy data at $t=0$ and vanishing Dirichlet data at the boundary of the obstacle. We study the inverse problem of recovering the potential $q$ from this source-to-solution map restricted to some measurement domain. By giving an example where measurements take place in some subset and the support of $q$ lies in the `shadow region' of the obstacle, we show that recovery of $q$ is exponentially unstable. |
| title | On Exponential Instability of an Inverse Problem for the Wave Equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.23559 |