Variational Source Condition For Ill-Posedbackward Nonlinear Maxwell'S Equations

This paper examines the mathematical analysis of an electromagnetic inverse problem governed by nonlinear evolutionary Maxwell's equations. The aim of the inverse problem is to recover electromagnetic fields at the past time by noisy measurement data at the present time. We consider the Tikhonov regularization method to cope with the ill-posedness of the governing backward nonlinear Maxwell's equations. By means of the semigroup theory, we study its convergence analysis and derive optimality conditions through a rigorous first-order analysis and adjoint calculus. The final part of the paper is focused on the convergence rate analysis of the Tikhonov regularization method under a variational source condition (VSC), which leads to power-type convergence rates. Employing the spectral theory, the complex interpolation theory and fractional Sobolev spaces, we validate the proposed VSC on account of an appropriate regularity assumption on the exact initial data and the material parameters.