Inverse problem of electrodynamics for anisotropic medium: linear approximation

For electrodynamic equations with permittivity specified by a symmetric matrix $\varepsilon (x)=({{\varepsilon }_{{ij}}}(x),i,j=1,2,3)$, the inverse problem of determining this matrix from information on solutions of these equations is considered. It is assumed that the permittivity is a given positive constant ${\varepsilon}_{0}>0$ outside a bounded domain $\Omega \subset {{\mathbb{R}}^{3}}$, while, inside $\Omega $, it is an anisotropic quantity such that the differences ${{\varepsilon }_{{ij}}}(x)-{{\varepsilon }_{0}}{{\delta }_{{ij}}}=:{{\tilde {\varepsilon }}_{{ij}}}(x),$ $i,j=1,2,3,$ are small. Here, ${{\delta }_{{ij}}}$ is the Kronecker delta. The inverse problem is studied in the linear approximation. The structure of the solution to a linearized direct problem for the electrodynamic equations is investigated, and it is proved that all elements of the matrix $\tilde {\varepsilon }(x)={{\tilde {\varepsilon }}_{{ij}}}(x), i,j=1,2,3$, can be uniquely determined by special observation data. Moreover, the problem of recovering the diagonal components ${{\tilde {\varepsilon }}_{{ij}}}(x), i=1,2,3,$ leads to a usual X-ray tomography problem, so these components can be efficiently computed. The recovery of the other components leads to a more complicated algorithmic procedure.