A numerical method of determining permittivity from the modulus of the electric intensity vector of an electromagnetic field

The system of equations of electrodynamics is considered for a nonmagnetic nonconducting medium. For this system, the problem is under study of determining the permittivity $\varepsilon$ from the given modulus of the electric intensity vector of the electromagnetic field that is the result of the interference of the two fields generated by some point sources of an extraneous current. It is assumed that permittivity is different from a given positive constant $\varepsilon_0$ only inside a compact domain $\Omega_0\subset\mathbb{R}^3$, whereas the modulus of the electric field intensity vector is given for all frequencies starting from a fixed frequency $\omega_0$ on the boundary $S$ of some domain $\Omega$ that includes $\Omega_0$. It is shown that this information allows us to reduce the original problem to the well-known inverse kinematic problem of determining the refraction index inside $\Omega$ by the travel time of the electromagnetic wave between the points of the boundary of this domain. The algorithm for numerical solution of the inverse problem is constructed, and the test calculations on the simulated data are presented.