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Inverse initial data reconstruction for Maxwell’s equations via time splitting method
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DOI：10.1016/j.jcp.2026.114896.png)
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Abstract
En 中文
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 E0 in a bounded domain Ω⊂R3, 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 splitting 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.
Keywords:
Time-domain Maxwell equations
inverse problem
initial condition recovery
time splitting
Legendre polynomial-exponential basis
minimum-norm solution
convergence analysis
AMS subject classification:
35R30
35L50
35Q61
65M32
78A25,
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