## Abstract

The purpose of this chapter is to discuss some of the highlights of the mathematical theory of direct and inverse scattering and inverse source scattering problem for acoustic, elastic and electromagnetic waves. We also briefly explain the uniqueness of the external source for acoustic, elastic and electromagnetic waves equation. However, we must first issue a caveat to the reader. We will also present the recent results for inverse source problems. The resents results including a logarithmic estimate consists of two parts: the Lipschitz part data discrepancy and the high frequency tail of the source function. In general, it is known that due to the existence of non-radiation source, there is no uniqueness for the inverse source problems at a fixed frequency.

### Keywords

- scattering theory
- inverse scattering theory
- Helmholtz equation
- Bessel functions

## 1. Introduction

This chapter tries to provide some results and materials on inverse scattering, direct scattering theory and inverse source scattering problems. There have been many scientists who have contributed to the different components of this field, such as linearity or non-linearity of the inverse source problem, computational and numerical solution to the inverse source problem and analytical aspects of the problem, which have their own interests. We obviously cannot give a complete account of inverse scattering here from all angles. Hence, instead of attempting the impossible, we have chosen to present inverse scattering theory from the of our own interests and research program. Particularly, we will focus on inverse source problems for acoustic, elastics and electromagnetic waves. In other words, certain areas of inverse scattering theory are either ignored.

Scattering theory has played a central role in twentieth century mathematical physics and applied mathematics. Indeed, from Rayleigh’s explanation of why the sky is blue, to Rutherford’s discovery of the atomic nucleus, through the modern medical and clinical applications of computerized tomography, scattering phenomena have attracted scientists and mathematicians for over a hundred years. Broadly speaking, scattering theory is concerned with the effect an inhomogeneous medium has on an incident particle or wave. In particular, if the total field is viewed as the sum of an incident field

## 2. The direct and inverse scattering problem

The stationary incoming wave

(

with the Dirichlet boundary data

or the Neumann boundary data

The function

where

The electromagnetic scattering problem corresponding to the electric field

with the Silver- Muller radiation condition;

where (7) are the time-harmonic Maxwell equations and

where

where

The representation (14) follows from the fact that any solution

where

As showed above, the direct scattering problem has been thoroughly investigated and a considerable amount of information is available concerning its solution. In contrast, the inverse scattering problem has only recently progressed. It is worth to mention that the inverse problem is inherently nonlinear. In areas such as radar, sonar, geophysical exploration, medical imaging and nondestructive testing. As in with direct problem, the first question in inverse scattering problem is, how about uniqueness?. The first result in uniqueness brought up to the attention by Schiffer [5] who showed for the problem (2)–(5) the far field pattern

**Theorem 1.1.** If

**Proof.** There is a weak solution to Eq. (1) in

using the condition

since

### 2.1 Helmholtz equation

Studying an inverse problem always requires a solid knowledge of the theory for the corresponding direct problem. Therefore in this section is devoted to presenting the foundations of obstacle scattering problems for time harmonic acoustic waves. The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. Colton and Kress showed that [11] how one can derived the Helmholtz equation from the Euler’s equation. Then the domain of the solution is outside a bounded open set

where the wave number

and it is a well-posed problem if

Let

where

where

and

Then, we can solve the single layer potential integral equation

Alternatively, we can solve the double layer potential integral equation

In this case the scattered solution outside

### 2.2 Inverse source scattering problem

Motivated by the significant applications, the inverse source problems, as an important research subject in inverse scattering theory, have continuously attracted much attention by many researchers. Consequently, a great deal of mathematical and numerical results are available. In general, it is known that there is no uniqueness for the inverse source problem at a fixed frequency due to the existence of non-radiation sources. Hence, additional information is required for the source in order to obtain a unique solution, such as to seek the minimum energy solution. From the numerical and computational point of view, a more challenging issue is lack of stability. A small variation of the data might lead to a huge error in the reconstruction. Recently, it has been realized that the use of multi-frequency data is an effective approach to overcome the difficulties of non- uniqueness and instability which are encountered at a single frequency. An attempt was made in [13] to extend the stability results to the inverse random source of the one-dimensional stochastic Helmholtz equation. The inverse source problem seeks for the right hand side of a partial differential equation from boundary data. The inverse source problems are also considered as a basic mathematical tool for solving many imaging problems including reflection tomography, diffusion-based optical tomography, lidar imaging for chemical and biological threat detection, and fluorescence microscopy. In general, a feature of inverse problems for elliptic equations is a logarithmic type stability estimate which results in a robust recovery of only few parameters describing the source and hence yields very low resolution numerically.

For the Helmholtz equations, the results have shown increasing (getting nearly Lipschitz) stability when the Dirichlet data or Cauchy data are given on the whole boundary and K is getting large. Similar results are obtained for the time periodic solutions of the more complicated dynamical elasticity system. For elastic waves, the inverse source problem is to determine the external force that produces the measured displacement. The inverse source scattering problem for Maxwell equation arises in many scientific areas such as medical imaging. More specifically, Magnetoencephalography (MEG), the imaging modality is a non-invasive neurophysiological technique that measures the electric or magnetic fields generated by neuronal activity of the brain. For electromagnetic waves, the inverse source problem is to reconstruct the electric current density from tangential trace of electric field. As we know in [14], the inverse source problem does not have a unique solution at a single or at finitely many wave numbers. On the other hand, if we use all wave numbers in

First increasing stability results were obtained in [15] by using the spatial Fourier transform. In [16, 17] more general and sharp results were obtained in sub-domain of

Let the radiated wave field

Both

The stability of functions

where

**Theorem 1.2.** Let

Then there exist a constant

for all

While Bao, Li and Lu used Dirichlet to Neumann map to simplify the boundary conditions for two dimensional and three dimensional domains (disks and balls), Isakov, Lu, Chang and Entekhabi used the Fourier transform and observability bound for corresponding hyperbolic initial value boundary problem (wave equation) for two and three dimensional domain with

where

**Theorem 1.3.** There exists a generic constant

for all

In papers [21, 22], authors considered inverse source scattering problems for double layers medium. The results in the papers [23, 24] showed an stability estimate for elastic and electromagnetic waves. Also authors in [23] proved a stability estimate using just Dirichlet data. Increasing stability for the Schrodinger potential from the complete set of the boundary data (the Dirichlet-to Neumann map) was demonstrated in [25, 26]. They showed that the boundary condition for elastic waves they considered the following equation

where

where

To achieve the result, authors used Helmholtz decomposition. The decomposition was allowed them to break the Navier-Lame equation to two elliptic equations.

The results for discrete data for inverse source problem which was obtained in [27] are as follows:

**Theorem 1.4.** Let

with radiation condition (37),

Then

where

and

The stability increases as

#### 2.2.1 Uniqueness of source function

To achieve the uniqueness, we introduced two different approaches. The first approach is using the estimate for the source function. Letting the norm of the boundary data goes to zero, then the proof is complete. For instance, consider Theorem 1.2 and let

**Theorem 1.5.** Let

**Proof.** Denote by

Under their assumptions, there is a unique solution to the problem (40) with

Now let

Due to the properties of

Due to the assumption, the function

Due to the uniqueness in the lateral Cauchy problem for the wave equation (40) with the Cauchy data on

## 3. Conclusions

In this section, the scattering and inverse scattering theory, inverse source scattering problem were considered briefly. The recent results such stability estimates for external source and electric current density from boundary measurements of radiated wave field and uniqueness for source function for Helmholtz equation, Elasticity and Maxwell system have showed. We also show some result for discrete data. In addition, we also showed some results of using just Dirichlet data for improving stability which was a big improvement. There are still many challenges remain in this field. For instance, studying the stability in the inverse source problems for inhomogeneous media where the analytical Green tensors are not available and the present method may not be directly applicable. Another interesting topic in stability of the external source is to consider the governing equation in the time domain. The non-linear case is also is a very challenging problem. The direct and inverse scattering problems when both the source and the linear load are random is also an open problem. Another challenging problem is to study the random source scattering problem for three dimensional elastic wave equation. As I mentioned before, there are many scientist and researcher have been working on inverse scattering and more specifically on inverse source problems. To expand your knowledge and further mathematical development in this field of research, please see the result authors in [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], which were discussed different aspects of the problems.

## Acknowledgments

Without doubt, our world is a beautiful place full of questions and challenges thanks to people who want to develop and overcome these challenges who share the gift of their time and passion to mentor future generation. Thank you to everyone who strives to grow and help others grow. To all the individuals I have had the opportunity to lead, be led by, or watch their leadership and mentoring from afar, I want to say thank you for being the inspiration.

I am grateful to all of those with whom I have had the pleasure to work and those who help me to grow and learn. I would especially like to thank Dr. Victor Isakov, my PhD adviser who has provided me extensive personal and professional guidance and taught me a great deal about both scientific research and life in general. He has taught me more than I could ever give him credit for here. He has shown me, by his example, what a good scientist (and person) should be. I also would like to thank Professor Alexander Bukhgeym and Professor Thomas K. DeLillo for their help and advice.

I would like to thank my mother and belated father, whose love and guidance are with me in whatever I pursue. Many thanks to my sisters Marjan, Mona and Mina and my brother Jamshid for their constant support and unending inspiration.

This chapter is supported in part by NSF Award HRD-1824267.