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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.

For fixed \begin{document} $y \in \mathbb{R}^3$ \end{document} , we consider the equation \begin{document} $L u+k^2u = - δ(x-y), \>x \in \mathbb{R}^3$ \end{document} , where \begin{document} $L=\text{div}(n(x)^{-2}\nabla)+q(x)$ \end{document} , \begin{document} $k >0$ \end{document} is a frequency, \begin{document} $n(x)$ \end{document} is a refraction index and \begin{document} $q(x)$ \end{document} is a potential. Assuming that the refraction index \begin{document} $n(x)$ \end{document} is different from \begin{document} $1$ \end{document} only inside a bounded compact domain \begin{document} $Ω$ \end{document} with a smooth boundary \begin{document} $S$ \end{document} and the potential \begin{document} $q(x)$ \end{document} vanishes outside of the same domain, we study an inverse problem of finding both coefficients inside \begin{document} $Ω$ \end{document} from some given information on solutions of the elliptic equation. Namely, it is supposed that the point source located at point \begin{document} $y \in S$ \end{document} is a variable parameter of the problem. Then for the solution \begin{document} $u(x,y,k)$ \end{document} of the above equation satisfying the radiation condition, we assume to be given the following phaseless information \begin{document} $f(x,y,k)=|u(x,y,k)|^2$ \end{document} for all \begin{document} $x,y \in S$ \end{document} and for all \begin{document} $k≥ k_0>0$ \end{document} , where \begin{document} $k_0$ \end{document} is some constant. We prove that this phaseless information uniquely determines both coefficients \begin{document} $n(x)$ \end{document} and \begin{document} $q(x)$ \end{document} inside \begin{document} $Ω$ \end{document} .

Inverse problems for wave equations have been extensively studied in the last 50 years due to a large number of engineering and technological applications. The objective of this chapter is to present an analysis of the two basic inverse coefficient problems related to 1D damped wave equations m(x)utt + μ(x)ut = (r(x)ux)x and \(m(x) u_{tt}+\mu (x)u_{t}=\left (r(x)u_{x}\right )_{x}+q(x)u\), \(\Omega _{T}:=\{(x,t)\in \mathbb {R}^2\,:\,x\in (0,\ell ),\, t\in (0,T)\}\), when l > 0 and T > 0 are finite. Here and below μ(x) ≥ 0 is the damping coefficient. Specifically, we study the inverse problem of identifying the unknown principal coefficient r(x) from Dirichlet-to-Neumann operator, and the inverse problem of recovering the unknown potential q(x) from either from Neumann-to-Dirichlet or Dirichlet-to-Neumann operators.

Beams and plates are important parts of main engineering constructions such as aircraft wings, flexible robotic manipulators, large space structures and robots, marine risers and moving strips. Within the scope of the linear theory of elasticity, classical Euler-Bernoulli beam and Kirchhoff plate theories have been a cornerstone tools in engineering since the development of the Eiffel Tower and the Ferris wheel in the late nineteenth century. Recently, small size nanobeams and nanoplates are found also in nanotechnology, including emerging systems for medical diagnostics and nanoscale measurements such as the Atomic Force Microscopy (AFM) and Transverse Dynamic Force Microscope (TDFM). The direct and inverse problems for the simplest Euler-Bernoulli ρ(x)utt + (EI(x)uxx)xx = F(x, t) and Kirchhoff utt + D Δ2u = g(t)f(x), (x, t) ∈ ΩT := Ω × (0, T), \(\Omega \subset \mathbb {R}^2\) equations have been extensively studied in the last 50 years due to a large number of engineering and technological applications.

In this chapter we consider some inverse problems for the Maxwell equations. Here H = (H1, H2, H3) and E = (E1, E2, E3) are vectors of electric and magnetic strengths, e > 0, μ > 0 and σ are the permittivity, permeability and conductivity coefficients, respectively, which define electro-dynamical parameters of a medium. The function j = j(x, t) is the external current source which generates the electromagnetic waves.

Inverse source problems for evolution PDEs \(u_t=Au+F\), \(t\in (0,T_f]\), represent a well-known area in inverse problems theory and has many engineering applications.

This Chapter deals with inverse coefficient problems for linear second-order 1D parabolic equations. We establish, first, a relationship between solutions of direct problems for parabolic and hyperbolic equations.

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.

In this paper we discuss the unique determination of unknown spatial load $F(x)$ in the damped Euler-Bernoulli beam equation $\rho(x) u_{tt}+\mu u_{t}+(r(x)u_{xx})_{xx}=F(x)G(t)$ from final time measured output (displacement, $u_{T}(x):=u(x,T)$ or velocity, $\nu_{t,T}(x):=u_t(x,T)$). It is shown in [A. Hasanov Hasanoglu and V.G. Romanov, \emph{Introduction to Inverse Problems for Differential Equations}, Springer, New York, 2017] that the unique determination of $F(x)$ in the undamped wave equation $u_{tt}-(k(x)u_{x})_{x}=F(x)G(t)$ from final time output is not possible. This result is also valid for the undamped beam equation $\rho(x) u_{tt}+(r(x)u_{xx})_{xx}=F(x)G(t)$. We prove that in the presence of damping term $\mu u_{t}$, the spatial load can be uniquely determined by the final time output, in terms of the convergent Singular Value Expansion (SVE), as $F(x)=\sum_{n=1}^{\infty} u_{T,n}\psi_n(x)/\sigma_n$, under some acceptable conditions with respect to the final time $T>0$, the damping coefficient $\mu>0$ and the temporal load $G(t)>0$. As an alternative method we propose the Adjoint Problem Approach (APA) and derive an explicit gradient formula for the Fr\'{e}chet derivative of the Tikhonov functional $J(F)=\Vert u(\cdot,T;F)-u_T \Vert_{L^2(0,l)}^2$. Comparative analysis of numerical algorithms based on SVE and APA methods are provided for the harmonic loading $G(t)= \cos (\omega t)$, $\omega>0$, as a most common dynamic loading case. The results presented in this paper not only clearly demonstrate the key role of the damping term $\mu u_{t}$ the inverse problems arising in vibration and wave phenomena, but also allows us, firstly, to find admissible values of the final time $T>0$, at which a final time measured output can be extracted, and secondly, to reconstruct the unknown spatial load $F(x)$ in the damped Euler-Bernoulli beam equation from this measured output.

The inverse problem of recovering the potential q(x) in the damped wave equation m ( x ) u t t + μ ( x ) u t = r ( x ) u x x + q ( x ) u , (x, t) ∈ Ω T ≔ (0, ℓ) × (0, T) subject to the boundary conditions u(0, t) = ν(t), u(ℓ, t) = 0, from the Neumann boundary measured output f(t) ≔ r(0)u x (0, t), t ∈ (0, T] is studied. The approach proposed in this paper allows us to derive behavior of the direct problem solution in the subdomains defined by characteristics of the wave equation and along the characteristic lines, as well. Based on these results, a local existence theorem and the stability estimate are proved. The compactness and Lipschitz continuity of the Dirichlet-to-Neumann operator are derived. Fréchet differentiability of the Tikhonov functional is proved and an explicit gradient formula is derived by means of an appropriate adjoint problem. It is proved that this gradient is Lipschitz continuous.

The inverse problem of identifying the unknown spacewise dependent source F(x) in 1D wave equation is considered. Measured data are taken in the form g(t) := u(0; t). The relationship between that problem and Ground Penetrating Radar (GRR) data interpretation problem is shown. The non-iterative algorithm for reconstructing the unknown source F(x) is developed. The algorithm is based on the Fourier expansion of the source F(x) and the explicit representation of the direct problem solution via the function F(x). Then the minimization problem for discrete form of the Tikhonov functional is reduced to the linear algebraic system and solved numerically. Calculations show that the proposed algorithm allows to reconstruct the spacewise dependent source F(x) with enough accuracy for noise free and noisy data., 15 pages, 4 figures, 3 tables

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.

We use a new method to prove uniqueness theorem for a coefficient inverse scattering problem without the phase information for the 3-D Helmholtz equation. We consider the case when only the modulus of the scattered wave field is measured and the phase is not measured. The spatially distributed refractive index is the subject of the interest in this problem. Applications of this problem are in imaging of nanostructures and biological cells.

In the first part of this chapter we study two inverse source problems related to the second order hyperbolic equations \(u_{tt}-u_{xx}=\rho (x, t)g(t)\) and \(u_{tt}-u_{xx}=\rho (x, t)\varphi (x)\) for the quarter plane \(\mathbb {R}^2_+=\{(x, t)|\, x>0, t>0\}\), with Dirichlet type measured output data \(f(t):=u(x, t)\vert _{x=0}\).

Inverse problems arise in almost all areas of science and technology, in modeling of problems motivated by various physical and social processes.

Here \(H=(H_1,H_2,H_3)\) and \(E=( E_1,E_2,E_3)\) are vectors of electric and magnetic strengths, \(\varepsilon >0\), \(\mu >0\) and \(\sigma \) are the permittivity, permeability and conductivity coefficients, respectively, which define electro-dynamical parameters of a medium.

The problem of recovering the sound speed (or index of refraction) from travel time measurements is an important issue in determining the substructure of the Earth.

The main objective of this chapter is to present some necessary results of functional analysis, frequently used in study of inverse problems.

This chapter is an introduction to the basic inverse problems for elliptic equations. One class of these inverse problems arises when the Born approximation is used for scattering problem in quantum mechanics, acoustics or electrodynamics. In the first part of this chapter two inverse problems, the inverse scattering problem at a fixed energy and the inverse scattering problems at a fixed energy, are studied. The last problem is reduced to the tomography problem which is studied in the next chapter. In the second part of the chapter, the Dirichlet-to-Neumann operator is introduced. It is proved that this operator uniquely defines the potential q(x) in Δu(x) + q(x) = 0.

Inverse problems related to the transport equations arise in many areas of applied sciences and have various applications in medical imaging and tomography.

A numerical method possessing the approximate global convergence property is developed for a 3-D coefficient inverse problem for hyperbolic partial differential equations with backscattering data resulting from a single measurement. An important part of this technique is the quasireversibility method. An approximate global convergence theorem is proved. Results of two numerical experiments are presented.

The 3-d inverse scattering problem of the reconstruction of the unknown dielectric permittivity in the generalized Helmholtz equation is considered. The main difference with the conventional inverse scattering problems is that only the modulus of the scattering wave field is measured. The phase is not measured. The initializing wave field is the incident plane wave. On the other hand, in the previous recent works of the authors about the "phaseless topic" the case of the point source was considered [20,21,22]. Two reconstruction procedures are developed for a linearized case. However, the linearization is not the Born approximation. This means that, unlike the Born approximation, our linearization does not break down when the frequency tends to the infinity. Applications are in imaging of nanostructures and biological cells.

This is a continuation of two recent publications of the authors about reconstruction procedures for 3-d phaseless inverse scattering problems. The main novelty of this paper is that the Born approximation for the case of the wave-like equation is not considered. It is shown here that the phaseless inverse scattering problem for the 3-d wave-like equation in the frequency domain leads to the well known Inverse Kinematic Problem. Uniqueness theorem follows. Still, since the Inverse Kinematic Problem is very hard to solve, a linearization is applied. More precisely, geodesic lines are replaced with straight lines. As a result, an approximate explicit reconstruction formula is obtained via the inverse Radon transform. The second reconstruction method is via solving a problem of the integral geometry using integral equations of the Abel type.

We discuss here a method for investigating inverse problems for hyperbolic equations based on combining an asymptotic expansion of solutions to the equations in a neighborhood of a characteristic surface, the connections between coefficients of the expansion and the unknown coefficients of the equation and the estimates of a solution to the Cauchy problem in terms of the data on a time-like surface. We give some applications of this method to a number of inverse problems. Stability results to the inverse problems are given.

A simple and effective method is introduced to calculate the bending loss and phase enhancement of a bent planar waveguide. The wave field is represented in terms of Airy functions and an eigenvalue equation is derived by matching the boundary conditions and the radiation condition in the outer cladding layer. The complex propagation constant is obtained by solving the eigenvalue equation with the Newton-Raphson method, and the imaginary part of the propagation constant gives directly the bending loss of the bent waveguide. The results are compared with the previous experimental and numerical results and are shown to be highly accurate and effective. The phase enhancement due to the bending is also studied.

A simple and efficient analytical method is presented for studying leaky modes in a buried rectangular waveguide on a high-refractive-index substrate. The cross-sectional profile of the refractive index for the buried rectangular waveguide is decomposed into three parts. The two main parts correspond to two independent multilayered slab waveguide structures and the remaining part is treated as a small perturbation term. The contributions from all three parts to the leakage loss are given analytically. Numerical results are presented and compared with those calculated with the semi-vectorial finite difference method. Fast speed, satisfactory accuracy and simplicity are the main advantages of the present method.

A simple and efficient method for studying the leakage loss in a conventional silica-on-silicon planar waveguide is presented. The magnitude of the leakage loss is obtained by calculating the complex propagation constant of the leaky waveguide structure. The dependence of the leakage loss on the thickness of the silica buffer layer (as well as the thickness of the silica upper-cladding layer) is investigated. Numerical results are presented and compared to those calculated with the beam propagation method.

An optical mammographic imaging problem for multiple small breast cancers is considered in the frequency domain. The diffusion model approximation with an extrapolated boundary condition for the compressed breast is used. An explicit identification scheme based on a weighted Fourier transform is introduced to determine the multiple cancers in a layer-by-layer fashion. Numerical results are presented.

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.

Some mammography-related inverse problems for the time-dependent diffusion equation are considered. The extrapolated boundary condition is used and the problem is linearized. Uniqueness theorems for two different types of inverse problems are given, namely the determination of the diffusion or absorption coefficients from the reflection data and the simultaneous determination of the diffusion and absorption coefficients from the transmission data. The inverse problem for the latter is reduced to two tomography problems which can be solved successively.