International Journal of Partial Differential Equations

Volume 2015 (2015), Article ID 968529, 9 pages

http://dx.doi.org/10.1155/2015/968529

## Direct and Inverse Scattering Problems for Domains with Multiple Corners

^{1}Department of Mathematics and Statistics and Center of Applied Physics, Louisiana Tech University, Ruston, LA 71272, USA^{2}Department of Mathematics and Statistics, Louisiana Tech University, Ruston, LA 71272, USA

Received 31 August 2014; Accepted 1 January 2015

Academic Editor: Chi K. Lin

Copyright © 2015 Songming Hou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We proposed numerical methods for solving the direct and inverse scattering problems for domains with multiple corners. Both the near field and far field cases are considered. For the forward problem, the challenges of logarithmic singularity from Green’s functions and corner singularity are both taken care of. For the inverse problem, an efficient and robust direct imaging method is proposed. Multiple frequency data are combined to capture details while not losing robustness.

#### 1. Introduction

Direct and inverse scattering problems have important applications in radar, sonar, and geophysical exploration, in medical imaging, and in nondestructive testing [1]. However, many theoretical and numerical challenges are associated with these problems, especially when the boundary has multiple corners. For direct scattering problem, the singularity of the Green’s function for the Helmholtz equation and the corner singularity of the boundary both require special treatment. For inverse scattering problem, the nonlinearity and ill-posedness challenge the design of an accurate, efficient, and robust numerical algorithm. There are two types of methods for solving the problem. The direct methods [2–6] are efficient but less accurate; the iterative methods [7–15] are accurate but more expensive. Typically the forward and adjoint problems have to be solved in each iteration. For the direct and inverse scattering problems, the near field and far field cases should be treated differently, although the main procedure remains to be the same.

For the direct scattering problem, we extended the work in [1] from the case with one corner to dealing with domain with multiple corners. We managed to make the system well conditioned. For the inverse scattering problem, we used a method similar to the MUSIC algorithm in [3, 4]. However, unlike the MUSIC algorithm which is a projection, we keep the phase information so that multiple frequency data can be combined to obtain both accurate and robust result. Efficiency is also obtained since it is a direct imaging algorithm with no iteration needed. We study both the near field and far field cases.

The organization of the paper is as follows. In Section 2, we will discuss the forward scattering problem for domains with multiple corners. In Section 3, a direct imaging algorithm is presented to solve the inverse problem using response matrix data. In Section 4, numerical examples are presented to demonstrate the efficiency, robustness, and accuracy of our methods. We will conclude in Section 5.

#### 2. The Forward Problem

Consider a time-harmonic plane wave, , incident on a scatterer with multiple corners, where is the wavenumber and is the incident direction. Let be the boundary of the scatterer. set . We consider the obstacle scattering problem. The total field satisfies the Helmholtz equation: The total field consists of the incident field and the scattered field : The incident field satisfies the homogeneous equation: It follows from (1), (2), and (3) that the scattered field satisfies In addition, the scattered field is required to satisfy the following Sommerfeld radiation condition: uniformly in . The uniqueness of the solution to the obstacle scattering problem is discussed in [1].

We use the combined single and double layer potential approach [16] so that the integral equation is uniquely solvable. For simplicity of notation, we first assume there is one corner at ; then By using the jump relation [1], we have the integral equation

We use change of variable and trapezoidal rule as follows [1]. Roughly speaking, about half of the points are equally distributed while the other half are accumulated near the corner. Consider

These lead to a linear system: where , , , , are given in [1].

In [1], the algorithm using graded mesh for solving direct scattering problem for sound-soft obstacle with one corner is presented. However, if we follow the same steps to treat the problem with multiple corners, large condition number for the linear system is observed; see Section 4. We propose a method to reduce the condition number. In [1], a notationally advantageous modification is made to the integral equation by inserting terms involving the fundamental solution to the Laplace equation. We observed that if this particular modification is not made, then the condition number is reduced significantly. In other words, although the notationally advantageous modification in [1] is helpful in theoretical discussion, in practice for domain with one corner it works well; however, for domain with multiple corners, it would lead to a linear system with large condition number. In this case we get around by removing this modification. See Section 4 for numerical comparison.

By solving the integral equation, we could compute both the near field and far field data. There are four cases:(1)plane incident wave, far field data,(2)plane incident wave, near field data,(3)point source, far field data,(4)point source, near field data.

The formula for far field data is

The formula for near field data is

The following formulas are implementation details for near field: Another issue is that for point source incident instead of plan wave we replace with where We have the following response matrix relations: For . For .

The corresponding singular values have the relations

We will verify these relations numerically in Section 4.

#### 3. The Inverse Problem

Shape reconstruction has important applications in radar, sonar, and geophysical exploration, in medical imaging, and in nondestructive testing [1]. The nonlinearity and ill-posedness make it a challenging problem. There are two types of methods for solving the problem. The direct methods [2–6] are efficient but less accurate; the iterative methods [7–15] are accurate but more expensive. Typically the forward and adjoint problems have to be solved in each iteration.

Figure 1 shows a typical configuration for such a problem. The background medium is assumed to be homogeneous.