Residual correction techniques for the efficient solution of inverse scattering problems

Abstract

We consider the scattering of time-harmonic electromagnetic waves by penetrable inhomogeneous obstacles. In particular, we study the numerical solution of an inverse scattering problem, where the refractive index of the obstacle is computed from some knowledge of the scattered waves, generated by the obstacle itself, and known incident waves. This problem can be formulated by a pair of non-linear integral equations, and its numerical solution is usually a time-consuming computation. We propose an efficient solution of this problem by taking into account a linearization of the integral equation under consideration. The proposed method is tested by a numerical experiment, where the inverse scattering problem is numerically solved for different obstacles.

Introduction

We introduce the notation. Let $R$, $C$ be the set of real numbers, and complex numbers, respectively. Let $R^N$, $C^N$ be the N-dimensional real Euclidean space, and the N-dimensional complex Euclidean space, respectively. Let $\underset{\_}{x}$, $\underset{\_}{y}\in R^N$, we denote with ${\underset{\_}{x}}^{⊺}\cdot \underset{\_}{y}$ the Euclidean scalar product of $\underset{\_}{x}$ and $\underset{\_}{y}$, the superscript ⊺ means transposed, and $\left|\right|\underset{\_}{x}\left|\right|$ denotes the Euclidean norm of $\underset{\_}{x}$. Let $S=\{\underset{\_}{x}\in R^2:\left|\right|\underset{\_}{x}\left|\right|=1\}$. Let $z\in C$, we denote with $\overline{z}$ the complex conjugate of z, with Re(z), Im(z) the real and the imaginary part of z respectively. We denote with ı the imaginary unit. We denote with $C^{M\times N}$ the space of complex matrices having M rows and N columns.

We consider a two-dimensional inhomogeneous isotropic medium. The inhomogeneity of the medium is contained in a compact set $D\subset R^2$. Let $n:R^2\to C$ be the refractive index of the medium, so that $n(\underset{\_}{x})=1$ for $\underset{\_}{x}\in R^2\setminus D$, and $Re(n(\underset{\_}{x}))\ge 1$, $Im(n(\underset{\_}{x}))\ge 0$ for $\underset{\_}{x}\in D$.

We consider an electromagnetic wave propagating on this medium. We suppose that this is a time-harmonic plane wave, so that the space-variables dependent part is given by

$$u^i(\underset{\_}{x}\text{,}\underset{\_}{d}\text{,}k)=e^{ık{\underset{\_}{d}}^{⊺}\cdot \underset{\_}{x}}\text{,}\underset{\_}{x}\in R^2\text{,}$$

where k > 0 is the wave number, $\underset{\_}{d}\in S$ is the propagation direction. The interaction of the inhomogeneity in the medium and the incident plane wave uⁱ generates a scattered wave. This scattered wave is supposed harmonic in time and with the same time-frequency of the incident wave; we denote with $u^{s\text{,}n}(\underset{\_}{x}\text{,}\underset{\_}{d}\text{,}k)$, $\underset{\_}{x}\in R^2$ its space-variables dependent part.

Function $u^{s\text{,}n}(\cdot \text{,}\underset{\_}{d}\text{,}k)$ is the solution of the following boundary-value problem:

$${\Delta }_{\underset{\_}{x}}{u}^{s\text{,}n}(\underset{\_}{x}\text{,}\underset{\_}{d}\text{,}k)+k^{2}n(\underset{\_}{x})u^{s\text{,}n}(\underset{\_}{x}\text{,}\underset{\_}{d}\text{,}k)=k^{2}m(\underset{\_}{x})u^i(\underset{\_}{x}\text{,}\underset{\_}{d}\text{,}k)\text{,}\underset{\_}{x}\in R^2\text{,}$$

$$\underset{\left|\right|\underset{\_}{x}\left|\right|\to +\infty }{\lim }\sqrt{\left|\right|\underset{\_}{x}\left|\right|}[{\underset{\_}{\hat{x}}}^{⊺}\cdot {\nabla }_{\underset{\_}{x}}{u}^{s\text{,}n}(\underset{\_}{x}\text{,}\underset{\_}{d}\text{,}k)-ık{u}^{s\text{,}n}(\underset{\_}{x}\text{,}\underset{\_}{d}\text{,}k)]=0\text{,}\underset{\_}{\hat{x}}\in S\text{,}$$

where ${\Delta }_{\underset{\_}{x}}$ is the Laplace operator with respect to the $\underset{\_}{x}$ variables, ${\nabla }_{\underset{\_}{x}}$ is the gradient operator with respect to the $\underset{\_}{x}$ variables, $\underset{\_}{\hat{x}}=\underset{\_}{x}/\left|\right|\underset{\_}{x}\left|\right|$ for $\underset{\_}{x}\ne \underset{\_}{0}$, and m = 1 − n is the contrast function; note that, boundary condition (3) is the so called Sommerfeld radiation condition, see [6, p. 206] for a general presentation of this problem. Problems (2), (3) provides a reduced formulation of the electromagnetic scattering problem; this formulation holds for Transverse Magnetic (TM) waves, see [15, Chapter 6] for details. We can assume a TM symmetry when the refractive index of the obstacle and the incident field are independent of a coordinate variable; note that, for a plane wave this holds when it is linearly polarized, and has the electric field along to a coordinate axis, see [7], [12] for details on a practical situation where these assumptions hold. Here, to fix the ideas, we assume that this symmetry holds with respect to the x₃ coordinate, so problem (2), (3) are given in terms of coordinates $\underset{\_}{x}={(x_1\text{,}{x}_2)}^{⊺}\in R^2$.

The solution u^s,n, of problem (2), (3), has the following asymptotic behaviour:

$$u^{s\text{,}n}(\underset{\_}{x}\text{,}\underset{\_}{d}\text{,}k)=\frac{e^{ık\left|\right|\underset{\_}{x}\left|\right|}}{\left|\right|\underset{\_}{x}|{|}^{1/2}}{u}^{0\text{,}n}(\underset{\_}{\hat{x}}\text{,}\underset{\_}{d}\text{,}k)+O\left(\frac{1}{\left|\right|\underset{\_}{x}|{|}^{3/2}}\right)\text{,}\left|\right|\underset{\_}{x}\left|\right|\to +\infty \text{,}$$

where u^0,n is the so called far field pattern, see [6, p.66] for details.

As in the classical scattering theory we can consider two different kinds of problems: the direct scattering problem and the inverse scattering problem. The direct scattering problem requires the computation of $u^{s\text{,}n}(\underset{\_}{x}\text{,}\underset{\_}{d}\text{,}k)$, $\underset{\_}{x}\in R^2$, solution of boundary-value problem (2), (3), from the knowledge of k, $\underset{\_}{d}$, and of the refractive index n. The inverse scattering problem requires the computation of the refractive index $n(\underset{\_}{x})$, $\underset{\_}{x}\in R^2$ from some knowledge of the scattered wave u^s,n. In particular, several applications consider the scattered wave in the far field zone; this is a region, far from the obstacle, where the far field pattern provides an accurate approximation of the scattered wave. So, the following inverse problem is usually considered.

Problem 1

(Inverse scattering problem) Let $I=\{{\underset{\_}{d}}_j\in S\text{,}j=1\text{,}2\text{,}\dots \text{,}J\}$ be the set of the incidence directions; for j = 1, 2, …, J, let $M({\underset{\_}{d}}_j)=\{{\underset{\_}{\hat{x}}}_{j\text{,}l}\in S\text{,}l=1\text{,}2\text{,}\dots \text{,}L\}$, be the set of the measurement directions associated to ${\underset{\_}{d}}_j\in I$. From the knowledge of $u^{0\text{,}n}({\underset{\_}{\hat{x}}}_{j\text{,}l}\text{,}{\underset{\_}{d}}_j\text{,}k)\text{,}l=1\text{,}2\text{,}\dots \text{,}L\text{,}j=1\text{,}2\text{,}\dots \text{,}J$ compute the refractive index $n(\underset{\_}{x})\text{,}\underset{\_}{x}\in R^2$.

Methods for the solution of Problem 1 are usually based on the integral formulation of boundary-value problem (2), (3). This formulation provides a pair of integral equations involving n, u^0,n, and uⁿ = uⁱ + u^s,n, that is the total field. These integral equations, due to the presence of uⁿ, are nonlinear integral equations for the unknown n. A simple approach to solve these equations is given by the Born approximation, where it is assumed that the total field uⁿ, in D, is well approximated by uⁱ. This approximation is not well-accurate when the contrast function m and/or the diameter of D are large with respect to 1/k; in such situations the corresponding solution of Problem 1 is usually affected by large errors. Several different methods have been proposed to avoid this drawback of the Born approximation, such as for example the iterative Born approximation [16], the distorted Born approximation [5], [13], the modified gradient method [2], the Newton–Kantorovich method [14], and the dual space method [6, p. 266].

We study the numerical solution of Problem 1. The integral formulation of problem (2), (3) are used to obtain, through a suitable discretization scheme, a non-linear system. The solution of this system is given by the discretization variables of refractive index n. So, an approximate solution of Problem 1 can be computed from the least squares solution of such a non-linear system. Note that this can be seen as a general scheme to solve Problem 1. Different choices can be considered in the discretization of integral equations or in the numerical solution of the resulting non-linear system, but a time-consuming procedure is always obtained.

We propose the use of a residual correction technique for the efficient solution of Problem 1. In particular, this technique allows an efficient solution of the integral equation arising from problem (2), (3), and it can be profitably used in the solution of Problem 1. This technique is based on a linearization, with respect to function n, of the integral formulation of problem (2), (3). Note that, a similar linearization formula has been already used in a derivation of the Newton–Kantorovich method [13]. In the present paper, this technique is used together with the inversion method presented in [9], [8] in order to obtain an efficient numerical solution of Problem 1. In particular, the resulting method can be seen as an improvement of the method proposed in [9], [8].

In Section 2 we describe the integral formulation of problem (2), (3), and the numerical solution of Problem 1. In Section 3 we describe the residual correction technique and its use in the numerical solution of Problem 1. In Section 4 we show some numerical results obtained in the solution of Problem 1. Finally, in Section 5 we give conclusions and future possible developments of this paper.