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It is demonstrated in this paper that the propagation of the electric wave field in a heterogeneous medium in 3D can sometimes be governed well by a single scalar second order PDE, which is derived from Maxwell’s equations. The corresponding component of the electric field dominates two other components. This justifies some past results of the second author with coauthors about numerical solutions of coefficient inverse problems with experimental electromagnetic data. In addition, since it is simpler to work in applications with a single scalar second order PDE rather than with the complete Maxwell system, the result of this paper might be useful to researchers working on applied issues of the propagation of electromagnetic waves in inhomogeneous media.

The problem of continuation of a solution of electrodynamic equations from the time-like half-plane S = { x ∈ R 3 ∣ x 3 = 0 } S=\{x\in\mathbb{R}^{3}\mid x_{3}=0\} inside the half-space R + 3 = { x ∈ R 3 ∣ x 3 > 0 } \mathbb{R}^{3}_{+}=\{x\in\mathbb{R}^{3}\mid x_{3}>0\} is considered. A regularization method for a solution of this problem with approximate data is proposed, and the convergence of this method for the class of functions that are analytic with respect to space variables is stated.

We consider two phaseless inverse problems for elliptic equation. The statements of these problems differ from have considered. Namely, instead of given information about modulus of scattering waves, we consider the information related to modulus of full fields, which consist of sums of incident and scattering fields. These full fields are the interference fields generated by point sources. We introduce a set of auxiliary point sources for solving the inverse problems and demonstrate that the corresponding data allow us to solve the inverse problems in a way similar to the case of measurements of scattering waves.

A long standing problem is completely solved here for the first time. This problem was posed by K. Chadan and P. C. Sabatier in their classical book “Inverse Problems in Quantum Scattering Theory”, Springer, New York, 1977. The inverse scattering problem of the reconstruction of the unknown potential with a compact support in the three-dimensional Schrödinger equation is considered. Only the modulus of the scattering complex-valued wave field is known, whereas the phase is unknown. It is shown that the unknown potential can be reconstructed via the inverse Radon transform. This solution has potential applications in imaging of nanostructures.

Imaging of nano structures is necessary for the quality control in their manufacturing. In the case when X-rays probe the medium, only the modulus of the complex valued scattered wave field can be measured. The phase cannot be measured. In the case of the Born approximation, we obtain an explicit reconstruction formula for the unknown coefficient from the phaseless scattering data.

A mammography inverse problem for the frequency domain diffusion equation is considered when the cancer region is small. The extrapolated boundary condition is used for the compressed breast and the problem is linearized. Explicit identification formulas are given for two different cases, namely, the case using measurement of one-point reflection and one-point transmission, and the case using measurement of two-point transmission or reflection. Numerical results for the identification of a small cancer ball are presented.