Inverse problem for the discrete Maxwell equations in a bounded paving

TL;DR

We address the Calderón-type inverse problem for the discrete anisotropic Maxwell operator on a bounded paving $Ω \subset \mathbf Z^3$, characterized by $H^{D_{\rm a}} = D_{\rm a}H_0$ with diagonal $D_{\rm a}$ and background $(\boldsymbol{\varepsilon}_0,\boldsymbol{\mu}_0)$. The analysis reduces to the discrete system $H_0u=Vu$ with a complex diagonal potential $V$ and uses plane-slice constructions to link interior coefficients to boundary data via the Dirichlet-to-Neumann map $Λ(D_{\rm a})$. A modified DN operator for partial data and cone-based propagation enable a backward reconstruction of $D_{\rm a}$ from $Λ(D_{\rm a})$, with existence for non-spectral $\lambda$. This work extends Calderón-type results to discrete Maxwell equations on lattices and provides a concrete reconstruction algorithm for recovering material parameters from boundary measurements.

Abstract

We consider the discrete anisotropic Maxwell operator DaH0 on a bounded paving $Ω$ $\subset$ Z3 , where H0 denotes discrete isotropic Maxwell operator and Da a diagonal operator of multiplication containing information about the anisotropy of the medium inside $Ω$. Letting a complex number $λ$ __ = 0 such the Dirichlet-to-Neumann operator $Λ$(Da) associated with the system DaH0 u = $λ$u on $Ω$ admits a unique solution, we show that knowing $Λ$(Da) is sufficient to determine Da by a reconstruction procedure for Da.