Inverse initial data reconstruction for Maxwell's equations via time-dimensional reduction method

TL;DR

This paper introduces a novel time-dimensional reduction method for reconstructing the initial electric field in Maxwell's equations from boundary data, employing Legendre polynomial-exponential basis projection and quasi-reversibility, with proven convergence and robust numerical validation.

Contribution

It develops a new approach combining time-basis projection and quasi-reversibility for inverse Maxwell problems, avoiding magnetic data and ensuring convergence.

Findings

Accurate initial electric field reconstruction with 10% data noise.

Method effectively handles inhomogeneous, anisotropic media.

Theoretical convergence guarantees the method's reliability.

Abstract

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 $\Omega \subset \mathbb{R}^3$, 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.