Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914713222250496 |
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| author | Aïcha, Ibtissem Ben Hu, Guanghui Si, Suliang |
| author_facet | Aïcha, Ibtissem Ben Hu, Guanghui Si, Suliang |
| contents | We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are givenby the far field patterns $u^\infity(\hat{x},k)$ for all observation directions in some neighborhood of a fixed direction $\hat{x}$ and for all wave numbers k belonging to a finite interval $(0,K)$. In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval $K>1$. In three dimensions we establish stability estimates of the $L^2$-norm and $H^{-1}$-norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of Hölder type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_08450 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies Aïcha, Ibtissem Ben Hu, Guanghui Si, Suliang Analysis of PDEs 35R30, 78A46 We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are givenby the far field patterns $u^\infity(\hat{x},k)$ for all observation directions in some neighborhood of a fixed direction $\hat{x}$ and for all wave numbers k belonging to a finite interval $(0,K)$. In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval $K>1$. In three dimensions we establish stability estimates of the $L^2$-norm and $H^{-1}$-norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of Hölder type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction. |
| title | Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies |
| topic | Analysis of PDEs 35R30, 78A46 |
| url | https://arxiv.org/abs/2403.08450 |