Removing small wavenumber constraints in Side B of the Probe Method
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914464495828992 |
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| author | Ikehata, Masaru |
| author_facet | Ikehata, Masaru |
| contents | The Probe Method is an analytical reconstruction scheme for inverse obstacle problems utilizing the Dirichlet-to-Neumann map associated with the governing partial differential equation. It consists of two distinct parts: Side A and Side B. Both are based on the indicator sequence which is calculated from the Dirichlet-to-Neumann map acting on "needle-like" specialized solution of the governing equation for the background medium, whose energy is concentrated on an arbitrary given needle inside. In Side A, the limit of the indicator sequence-referred to as the indicator function-is computed before the needles touch the obstacle, and the boundary is identified as the point where this function first blows up. In contrast, Side B states the blow-up of the indicator sequence after the needles have come into contact with the obstacle. For the Helmholtz equation, the validity of Side B has long required a small wavenumber constraint. This paper finally removes this long-standing restriction, establishing the method's applicability for broader cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_21124 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Removing small wavenumber constraints in Side B of the Probe Method Ikehata, Masaru Analysis of PDEs 35R30, 35E05, 35J05 The Probe Method is an analytical reconstruction scheme for inverse obstacle problems utilizing the Dirichlet-to-Neumann map associated with the governing partial differential equation. It consists of two distinct parts: Side A and Side B. Both are based on the indicator sequence which is calculated from the Dirichlet-to-Neumann map acting on "needle-like" specialized solution of the governing equation for the background medium, whose energy is concentrated on an arbitrary given needle inside. In Side A, the limit of the indicator sequence-referred to as the indicator function-is computed before the needles touch the obstacle, and the boundary is identified as the point where this function first blows up. In contrast, Side B states the blow-up of the indicator sequence after the needles have come into contact with the obstacle. For the Helmholtz equation, the validity of Side B has long required a small wavenumber constraint. This paper finally removes this long-standing restriction, establishing the method's applicability for broader cases. |
| title | Removing small wavenumber constraints in Side B of the Probe Method |
| topic | Analysis of PDEs 35R30, 35E05, 35J05 |
| url | https://arxiv.org/abs/2603.21124 |