Inverse initial data reconstruction for Maxwell's equations via time-dimensional reduction method
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911022762164224 |
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| author | Le, Thuy T. Van, Cong B. Dang, Trong D. Nguyen, Loc H. |
| author_facet | Le, Thuy T. Van, Cong B. Dang, Trong D. Nguyen, Loc H. |
| contents | We study an inverse problem for the time-dependent Maxwell system in an inhomogeneous and anisotropic medium. The objective is to recover the initial electric field $\mathbf{E}_0$ in a bounded domain $Ω\subset \mathbb{R}^3$, using boundary measurements of the electric field and its normal derivative over a finite time interval. Informed by practical constraints, we adopt an under-determined formulation of Maxwell's equations that avoids the need for initial magnetic field data and charge density information. To address this inverse problem, we develop a time-dimension reduction approach by projecting the electric field onto a finite-dimensional Legendre polynomial-exponential basis in time. This reformulates the original space-time problem into a sequence of spatial systems for the projection coefficients. The reconstruction is carried out using the quasi-reversibility method within a minimum-norm framework, which accommodates the inherent non-uniqueness of the under-determined setting. We prove a convergence theorem that ensures the quasi-reversibility solution approximates the true solution as the noise and regularization parameters vanish. Numerical experiments in a fully three-dimensional setting validate the method's performance. The reconstructed initial electric field remains accurate even with $10\%$ noise in the data, demonstrating the robustness and applicability of the proposed approach to realistic inverse electromagnetic problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_20777 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inverse initial data reconstruction for Maxwell's equations via time-dimensional reduction method Le, Thuy T. Van, Cong B. Dang, Trong D. Nguyen, Loc H. Numerical Analysis We study an inverse problem for the time-dependent Maxwell system in an inhomogeneous and anisotropic medium. The objective is to recover the initial electric field $\mathbf{E}_0$ in a bounded domain $Ω\subset \mathbb{R}^3$, using boundary measurements of the electric field and its normal derivative over a finite time interval. Informed by practical constraints, we adopt an under-determined formulation of Maxwell's equations that avoids the need for initial magnetic field data and charge density information. To address this inverse problem, we develop a time-dimension reduction approach by projecting the electric field onto a finite-dimensional Legendre polynomial-exponential basis in time. This reformulates the original space-time problem into a sequence of spatial systems for the projection coefficients. The reconstruction is carried out using the quasi-reversibility method within a minimum-norm framework, which accommodates the inherent non-uniqueness of the under-determined setting. We prove a convergence theorem that ensures the quasi-reversibility solution approximates the true solution as the noise and regularization parameters vanish. Numerical experiments in a fully three-dimensional setting validate the method's performance. The reconstructed initial electric field remains accurate even with $10\%$ noise in the data, demonstrating the robustness and applicability of the proposed approach to realistic inverse electromagnetic problems. |
| title | Inverse initial data reconstruction for Maxwell's equations via time-dimensional reduction method |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2506.20777 |