Variational source condition for Ill-posed backward nonlinear Maxwell's equations

Chen, De-Han; Yousept, Irwin LSF

This paper analyzes the Tikhonov regularization for ill-posed backward nonlinear Maxwell's equations. We propose a variational source condition (VSC), leading to power-type convergence rates for the Tikhonov regularization. By means of the complex interpolation theory, the proposed VSC is verified under a piecewise H⁸-type Sobolev regularity assumption on the exact initial data. The second part of the paper is focused on the sensitivity analysis for the nonlinear forward Maxwell system and the first-order optimality conditions for the Tikhonov regularization. As a final result, we prove a strong convergence result for the corresponding adjoint state with power-type convergence rates based on VSC.