Increasing resolution and instability for linear inverse scattering problems
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912261546704896 |
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| author | Kow, Pu-Zhao Salo, Mikko Zou, Sen |
| author_facet | Kow, Pu-Zhao Salo, Mikko Zou, Sen |
| contents | In this work we study the increasing resolution of linear inverse scattering problems at a large fixed frequency. We consider the problem of recovering the density of a Herglotz wave function, and the linearized inverse scattering problem for a potential. It is shown that the number of features that can be stably recovered (stable region) becomes larger as the frequency increases, whereas one has strong instability for the rest of the features (unstable region). To show this rigorously, we prove that the singular values of the forward operator stay roughly constant in the stable region and decay exponentially in the unstable region. The arguments are based on structural properties of the problems and they involve the Courant min-max principle for singular values, quantitative Agmon-Hörmander estimates, and a Schwartz kernel computation based on the coarea formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Increasing resolution and instability for linear inverse scattering problems Kow, Pu-Zhao Salo, Mikko Zou, Sen Analysis of PDEs 35P15, 35R25, 35R30, 35J05, 35J15 In this work we study the increasing resolution of linear inverse scattering problems at a large fixed frequency. We consider the problem of recovering the density of a Herglotz wave function, and the linearized inverse scattering problem for a potential. It is shown that the number of features that can be stably recovered (stable region) becomes larger as the frequency increases, whereas one has strong instability for the rest of the features (unstable region). To show this rigorously, we prove that the singular values of the forward operator stay roughly constant in the stable region and decay exponentially in the unstable region. The arguments are based on structural properties of the problems and they involve the Courant min-max principle for singular values, quantitative Agmon-Hörmander estimates, and a Schwartz kernel computation based on the coarea formula. |
| title | Increasing resolution and instability for linear inverse scattering problems |
| topic | Analysis of PDEs 35P15, 35R25, 35R30, 35J05, 35J15 |
| url | https://arxiv.org/abs/2404.18482 |