The time domain linear sampling method (TD-LSM) solves inverse
scattering problems using time domain data by creating an indicator
function for the support of the unknown scatterer. It involves only
solving a linear integral equation called the near-field equation
using different data from sampling points that probe the domain where
the scatterer is located. To date, the method has been used for the
acoustic wave equation and has been tested for several different types
of scatterers, i.e. sound hard, impedance, and penetrable, and for
wave-guides. In this paper, we extend the TD-LSM to the time
dependent Maxwell’s system with impedance boundary conditions - a
similar analysis handles the case of a perfectly electrically
conducting (PEC) body. We provide an analysis that supports the use
of the TD-LSM for this problem, and preliminary numerical tests of the
algorithm. Our analysis relies on the Laplace transform approach
previously used for the acoustic wave equation. This is the first
application of the TD-LSM in electromagnetism.
keywords: Wave-based imaging, electromagnetism, impedance, linear sampling, time domain
The inverse scattering problem studied in this paper concerns the reconstruction of the shape of a bounded scatterer using time domain electromagnetic scattering data. In particular, we probe the scatterer using incident fields due to point sources located away from the scatterer, and the data for the inverse problem is the scattered field measured on a surface containing the unknown scatterer which could have multiple components. Reconstructing the shape of the scatterer from this data is a non-linear and ill-posed problem.
In the frequency domain, there are many possible techniques to solve this problem. For example, optimization based schemes determine the unknown shape by finding the best fit of the data using a suitable parametrization of the unknown surface. Obviously such a method requires a priori knowledge of the nature of the scatterer, including the number of scatterers and their topology. This is a very flexible technique able to handle many different measurement data (multistatic, bistatic etc), although generally computationally intensive. For theoretical progress and results for electromagnetic inverse problems using real measured data, see e.g.[9, 2, 3, 24, 13, 12].
In order to decrease the need for a priori data and mitigate the computational burden, an alternative approach is to use a qualitative method which seeks to determine the shape of the scatterer without determining its material properties. This type of approach started with the work of Colton and Kirsch  for the Helmholtz equation and has been expanded to include a variety of methods, as well as applications to electromagnetism and elasticity. Of particular relevance to our work is the Linear Sampling Method (LSM) [4, 17] for Maxwell’s equations (see also  for the related generalized Linear Sampling Method). This method solves a sequence of linear integral equations to construct an indicator function that can be used to determine the boundary of the scatterer.
While the frequency domain LSM is certainly easy to implement and can determine the shape of the scatterer for a variety of different types of scatterer, it requires a large amount of multi-static data. In an effort to decrease the number of source and receiver points, the Time Domain LSM (TD-LSM) was proposed, and analyzed for acoustic scattering by a sound soft object by Haddar et al. . Numerical results show that a coarser set of data can be used. The TD-LSM was then extended, with an improved numerical implementation, to scatterers with an impedance boundary condition by Marmorat et al. . Following encouraging numerical results , an analysis of the TD-LSM for penetrable acoustic scatterers was given by Cakoni at al. 33].
Our paper is devoted to extending the TD-LSM to the time domain electromagnetic inverse problem. We give the first analysis and preliminary numerical results for the method. The methods we use to prove the theoretical results are extensions of the techniques in [8, 16, 29] via Laplace transforms. We choose to analyze scattering by an impenetrable scatterer with an impedance boundary condition (suitable for an imperfect conductor). The same analysis can be adapted to the case of a perfect conductor and we shall show a numerical example of this case. Unfortunately, for Maxwell’s equations, it is not currently possible to analyze the case of a penetrable scatterer using Laplace transforms . As shown by Cakoni et al.  for the case of a spherically stratified medium, and by Vodev  for more general scatterers, there does not exist a suitable half plane of the complex plane that is devoid of transmission eigenvalues, and this rules out the simple use of Laplace transforms as was done for the Helmholtz equation . For a complete discussion of this issue, see .
Despite being unable to analyze the TD-LSM for penetrable scatterers, we can still test the algorithm in this case. In Section 4 we provide an example with a penetrable scatterer. The reconstruction is very similar to the case of perfect conducting or impedance scatterers, so suggesting that the TD-LSM may be applicable in that case.
To test the TD-LSM, we use synthetic data computed via a nodal Discontinuous Galerkin (DG) method coupled with a low-storage explicit Runge-Kutta time stepping scheme [19, 7]. The DG method provides an efficient numerical technique to numerically solve differential equations and has properties that make it well-suited for wave simulations, see e.g. [18, 35, 23, 22]. These features include, e.g., high-order accuracy, straightforward handling of large discontinuities in the material parameters, and support for complex problem geometries. In addition, the method has excellent parallelization properties in both CPU and GPU environments, see e.g. [21, 14, 27]. All of these are essential features for the numerical scheme to be used for solving complex wave problems.
The layout of this paper is as follows: in Section 2 we give details of the time dependent forward problem for electromagnetic scattering from a bounded scatterer with an impedance boundary condition, and derive relevant continuity estimates for the problem. In Section 3 we formulate the TD-LSM for Maxwell’s equations and prove analogues of the usual theorems regarding the performance of the TD-LSM. In Section 4 we provide some details of the implementation of the inversion technique and then give some numerical examples showing the performance of the method.
Throughout the paper, bold-face font will be used to represent vector quantities. For example
We define . For a Hilbert space , we consider the space of -valued distributions depending on ; we denote by the subset of causal distributions, that is, those distributions which vanish on .
2 Maxwell’s equations with an impedance boundary condition
We start by defining some notation and spaces. Let denote the scatterer. It is assumed to be a bounded domain whose complement is connected and whose boundary is Lipschitz continuous. We denote by the unit outward normal to on .
The appropriate solution space in the Laplace domain will be a subspace of
To define such subspace we need the space of tangential square integrable functions on :
Similarly, we consider
2.1 Time domain Maxwell’s equations
In the time domain, Maxwell’s equations for the causal electric field and magnetic field in are
where we assume that there is no imposed current and the material that fills is lossless (e.g. air or vacuum). We will also suppose that the relative electric permittivity and magnetic permeability are symmetric matrix functions with uniformly bounded entries that are piecewise in . They are also assumed to be uniformly positive definite almost everywhere in .
The electromagnetic field is assumed to be subject to the following impedance boundary condition, that models, for example, an imperfectly conducting body:
Here we denote by the tangential trace of the electric field , and this notation will be used in the sequel for such a trace of any smooth enough vector field. Concerning the datum , it is a causal tangential vector field that is usually obtained from the trace of a smooth incident field, as we will detail in the following sections. Moreover, the matrix valued function is assumed to be uniformly bounded and symmetric almost everywhere on . We also assume that maps tangential vectors to tangential vectors, for which it is uniformly positive definite. By this we mean that, for almost every on and for each vector that is tangential to at (i.e. ), also the vector is tangential to at (i.e. ) and it holds that .
The speed of light in vacuum is given by . Then, following , we rescale the electric and magnetic fields:
The rescaled electromagnetic field is still causal and satisfies
and is subject to the impedance boundary condition
Above we have set and , where is the impedance of free space. Notice that there is no need for a radiation condition in the time domain under the causality assumption. This is because of the finite speed of propagation of electromagnetic waves, so that at any time there is a large enough ball in for which the field vanishes outside of this ball.
The problem is typically rewritten in terms of either the rescaled electric or magnetic field. Here we opt for the former: More precisely, we rewrite the second equation as and use this in the time derivative of the remaining equations to obtain
2.2 Analysis of the forward problem based on the Fourier-Laplace transform
Let us first recall some basic facts about the Fourier-Laplace transform, cf. [25, 32, 16] that will be used here. For a Banach space , let and represent the space of -valued distributions and tempered distributions on the real line, respectively. For any , , we set . This allows us to consider the Laplace transform of any such that , defined by
where . In particular, when is a Sobolev space (e.g. ), for any we consider the Hilbert space
endowed with the norm . We will make use of the well-known Plancherel’s theorem, which relates the norm of a function in with the weighted norms of in
. Indeed, notice that the Fourier-Laplace transform can be rewritten in terms of the usual Fourier transform:
so that Plancherel’s theorem for the usual Fourier transform leads to
in some situations, this is useful to deduce bounds of time dependent fields, e.g.
We consider two Banach spaces and , and write to represent the space of linear and bounded operators from into . Let be an analytic function for which there exist and such that
Set , and the associated convolution operator. Then, for all , extends to a bounded operator from to .
In all the sequel, we will replace by in the above defined spaces in order to denote the corresponding subspaces of causal functions. In this sense, Paley-Wierner theory will be frequently applied to study casuality, cf. .
where we denote and . Notice that, provided , there is no need to impose a Silver-Müller radiation condition in the Fourier-Laplace domain fields but it suffices to require . Also notice that can be recovered from by taking the Fourier-Laplace transform of , which leads to .
Next we obtain a variational formulation of the scattering problem (5-6). To this end, we multiply both sides of equation (5) by the complex conjugate of a smooth test function of compact support and integrate by parts in to obtain:
where . We make use of the impedance boundary condition to rewrite the integral on as
Now we need to define the solution space
endowed with the norm . Using the density of in (cf. ), the variational form of the Fourier-Laplace domain forward problem is then to find that satisfies
for any .
In order to study this variational formulation, we suppose that for some fixed and consider the sesquilinear form associated to the left hand side:
Using the approach in , notice that
The imaginary part of this expression can be studied term by term under our assumptions on the coefficients to obtain the following inequality:
Here and in the sequel, , and are positive constants associated with the positive definiteness properties of the coefficient functions , and , respectively. Also notice that, for any fixed , the boundedness of the sesquilinear form follows from the assumed uniform boundedness of the coefficient functions and the definition of the space . Therefore, the Lax-Milgram lemma guarantees that there exists a unique solution such that
To allow us to go back to the time domain problem, we need bounds on which make explicit the dependence on . With this aim, we notice that, since
it follows that
By the Cauchy-Schwarz inequality, we get the bound
which allows us to apply Lemma 2.1 to go back to the time domain and guarantee the following result. Notice that causality preservation is straightforward by the Paley-Wiener theory.
We have thus proved the following result:
3 The inverse problem and the linear sampling method
We now formulate precisely the inverse problem we shall study. We assume that the unknown scattering object is illuminated by incident fields that are due to regularized point sources (see (7) below) which are a model of a source of electromagnetic waves. Each source point is placed on a fixed surface . We seek to reconstruct the scatterer from measurements of the scattered fields corresponding to those incident fields on a possibly different surface , which is a model for a measurement device. Both and are piecewise smooth surfaces, and are allowed to be open or closed. When either surface is closed we assume is enclosed by that surface. In the case when (or ) is open, we suppose that it is a subset of an analytic closed surface (or , respectively) that encloses .
In order to describe the regularized point sources that we consider, we fix a polarization , a source point , and a smooth function that models a modulation function in time. Then we take regularized incident magnetic dipoles defined in the classical sense for and by:
is the regularized counterpart of the fundamental solution of the time dependent wave equation
where denotes the Dirac delta distribution. In particular, these dipoles are divergence free away from the sources, that is, for . Also notice that they are the regularized time dependent counterparts of the magnetic dipoles proposed in  for the frequency domain. Moreover, they can be written as the convolution in time of the modulation function and the fundamental solution of the wave equation as follows:
Let us recall that the fundamental solution of the wave equation satisfies, in the distributional sense,
Thus, since the regularized dipole is divergence free away from the source point , we have in the distributional sense that
Let denote the scattered field corresponding to the incident field . The linearity of Maxwell’s equations (2-3) shows that the scattered field for a superposition of incident fields equals the superposition of the corresponding scattered fields. More generally, for a function with , we may consider the incident field
and the corresponding scattered field
In the following, we will make use of polarizations given by tangential fields on , and then measure the tangential component of the scattered field on . Accordingly, we define the near field operator by applied to a vector function by
where, as usual, the subscript refers to the tangential trace here taken on (i.e. ).
Concerning causality, we emphasize that even for a causal field , the corresponding incident field is not necessarily causal (and hence, nor is the scattered field ). However, the measured data that represents the kernel of the integral operator , are tangential components of causal electromagnetic waves.
For later use, we note that the incident field operator can be represented as the convolution in time of the modulation function with the vector potential defined by the non-regularized magnetic dipole operator. Indeed, for tangential densities , the non-regularized counterpart of is
which is the curl of the (non-regularized) retarded single layer potential for the wave equation defined over the surface ; see the paragraph 3.1.1 for more details about this integral operator. Then
The Time Domain Linear Sampling Method (TD-LSM) is an imaging technique that yields a picture of the scatterer by approximately solving, for each sampling point, a linear integral equation whose right hand side is the tangential trace of a point source placed at the point under study. Using the measured scattered field, we can compute the near field operator (10) applied to a vector function . Then, for each sampling point , polarization and delay , we seek an approximate solution of the near field equation
This is an ill-posed linear integral equation.
In our numerical tests, is kept constant for all sampling points in a test region that we choose a priori to search for the scatterer: Theoretically, one might also let vary depending on , but the assumption that this is not the case allows to neglect the dependence of on . Moreover, is typically fixed to be a unit vector (). Note that in their analysis of the TD-LSM for the wave equation , the authors argue that one may choose .
By solving the near field equation approximately for many sampling points we construct an indicator function for the scatterer. Details of this procedure are given in Remark 3.7.
3.1 Basic ingredients for the TD-LSM analysis
We next study some basic tools for the theoretical justification of the TD-LSM. More precisely, we start by recalling an integral operator related to the wave equation: the so called retarded single layer potential. Then we study the operator related to the superposition of incident fields, and the impedance trace operator that maps incident fields onto the associated boundary data in the impedance boundary condition (3) of the forward problem. These results lay the foundation to deduce some basic properties of the near field measurement operator .
3.1.1 Retarded single layer potential for the wave equation
The regularized retarded single layer potential for the wave equation defined over is
Notice that it is the regularization by means of the modulation function of the non-regularized retarded single layer potential for the wave equation defined over :
cf. . The latter defines a bounded operator from into that preserves causality. Moreover, the following result provides bounds on the single layer operator.
Let . Then is a bounded operator from into , and also into . Moreover, its trace on (that is, ) is continuous across .
A stronger result can be proved for the regularized single layer .
The Fourier-Laplace transform of satisfies a transmission problem in ; more precisely, in , and its trace is continuous across whereas its normal derivative has a jump equal to the density ; accordingly, a variational formulation of this transmission problem leads to
Taking the imaginary part of the product by , and using Cauchy’s generalized inequality and the trace theorem,
for any . In particular, if is small enough, we deduce that
where does not depend on . Now we can infer information back to the time domain using Lemma 2.1, and it follows that, for any , the single layer potential is bounded as a map from into for any ; in particular, it is bounded from into and into . ∎
3.1.2 Analysis of the operator associated to the superposition of incident fields
The aim of this paragraph is to study the operator defined in (9) between suitable Sobolev spaces. To accomplish this, we define the following closed subspace of associated to incident fields:
Then we have the following result:
Assume that . Then is a bounded and injective operator from into for any , and the range of its tangential traces is dense in .
We first study the operator based on explicit bounds in of its Fourier-Laplace transform
Remaining in the Fourier-Laplace domain, we note that the single layer potential for vector fields is defined to act componentwise. Then, the mapping
is bounded from into because from (14) we have that
where is independent of . Also notice that the compactness of the support of the modulation function guarantees that its Fourier-Laplace transform