Abstract
The inverse scattering problem of an interior cavity with three different boundary conditions is considered. Bayesian method is used to reconstruct the shape of the cavity from scattered fields incited by point source(s) and measured on a closed curve inside the cavity. We prove the well-posedness in Bayesian perspective and present numerical examples to show the viability of the method.
1. Introduction
Recently, the inverse scattering problem of an interior cavity attracts many researchers’ attention due to its useful applications in industry. For example, in nondestructive testing the sources and receivers are sometimes placed inside the cavity to test the structural integrity [1]. Different from the typical inverse obstacle problem which is an exterior boundary value problem where the wave incidence and measurements are taken outside the obstacle, our problem is to recover the shape of the cavity from point source(s) and measurements on a closed curve inside the cavity. To be precise, we denote the cavity by a bounded simply connected domain with Lipschitz boundary . Our aim is to reconstruct the boundary from measurements (scattered waves) taken on a closed curve inside the cavity (see Figure 1). The point source(s) (incident waves) are also located on curve .
Some numerical methods have been proposed for this kind of inverse problem. The first paper related to this interior problem with Dirichlet boundary condition is Qin and Colton [2]. They proved the uniqueness and used a modification of linear sampling method to reconstruct the shape of the cavity. Then they developed this method for impedance boundary condition [3] and Hu et al. developed this method for mixed boundary condition [4]. In 2011, Qin and Cakoni [5] proposed a nonlinear integral equation method and Zeng et al. [6] developed the linear sampling method for inverse interior electromagnetic scattering problem. A decomposition method has been presented by Zeng et al. in [7]. Recently Liu [8] proposed a factorization method for this inverse interior cavity problem.
Bayesian theory is the central part of statistical inversion theory, so sometimes we just call the statistical inversion method Bayesian method. Different from the traditional approach which produces single estimate of the unknowns, Bayesian method produces a distribution which describes the behaviour of solution based on the prior information and the random measurement. All variables in the model are viewed as random variables and the randomness that is the degree of information about these variables is coded in probability distributions. Bayesian approach plays same role as regularization methods when dealing with the ill-posed inverse problems. Every prior distribution can be replaced by an appropriately chosen penalty and it tells us the information hidden in the penalty. Another advantage of Bayesian method is that it is a good interpretation of mathematical models, well understood, and generally accepted [9]. From numerical perspective, Bayesian approach is easy to encode when you already have the numerical method for direct problem. The process of solving direct problem can be viewed as a black box, because we are just concerned with the input and output. Although the computational amount is large for we should sample many points to describe the posterior distribution, Bayesian method is already used in many application areas with the development of computational capabilities. For some classical books and papers about Bayesian method we refer to [9–14].
Due to the advantages stated above, we choose Bayesian method to solve the inverse interior cavity problem. The study of using Bayesian approach for shape reconstruction in inverse scattering problem is a fairly new topic. It brings the randomness into the deterministic problem and provides us with a new perspective to view our problem. In the traditional method, the reconstructed geometries are different when the input of random noise varies and we do not know which one is more reliable. But in the Bayesian method we can directly obtain the distribution of our reconstructed result and it will not change when the distribution of noise is decided. From numerical simulation of the distribution we know which reconstructed geometries are more reliable (with high probability). Then we can use the statistic characteristics of this distribution, for example, the mean value, to estimate the result we wanted. Comparing to the sampling method, Bayesian method does not need to measure scattered fields corresponding to many point sources, respectively. Only measuring the scattered field once is enough. Of course, there are disadvantages of Bayesian method; for example, the computational efficiency is not high. But it gives us a new way to understand and cope with inverse interior cavity problem and it can be also extended to other inverse scattering problems, such as inverse open cavity problem and inverse rough surface problem. In our paper, we adopt the framework in [14] for an interior inverse scattering problem with three different boundary conditions. We prove the well-posedness of Bayesian method for this problem and present some numerical results.
The outline of this paper is as follows. In Section 2 we describe the problem and give the Bayesian formulation of our problem. In Section 3, we prove two important properties of the direct solution operator and get the well-posedness of Bayesian method. In Section 4, some numerical examples are presented to show the effectiveness of our method.
2. Bayesian Formulation of the Problem
In this paper we consider a TM polarized time harmonic electric dipole located inside an infinite cylinder. Let the cross section of the cylinder be a simply connected domain with boundary . We assume the point sources and the observational points all locate on curve inside the domain . Then the above scattering problem reduces to finding the scattered field which satisfieswhere is the wave number and is the incident wave given byHere, , is a fixed point on , and is the fundamental solution of two dimensional Helmholtz equations. is a linear operator defined byHere, and is the unit outward normal to .
The direct problem is solving the above equation to get scattered field for given . The inverse problem is recovering the shape for given measurements . To ensure the uniqueness of interior scattering problem, we assume that is not an eigenvalue for corresponding to boundary condition (2) of . For simplification, we assume is a starlike domain with respect to the origin. Thus, we can represent the boundary bywhere . We set just for the convenience of the proof in the next section. Then our model can be written aswhere ( is the number of measurements, i.e., ) is a finite dimensional observational operator corresponding to (1) and vector is the observational data with noise . We describe our prior information about , in terms of a probability measure . Here, is some function space to be chosen (see Section 3.1). And we use Bayes’ formula to calculate the posterior probability measure for given . Let and denote the probability density functions of measures and . By the knowledge in statistics the noise is often a mean zero random variable and we know little about as prior information in the interior inverse scattering problem. So we assume the noise with covariance matrix and the prior with mean function and bounded covariance operator . Of course, the prior measure may be various from the different prior information of the cavity geometry and we will present a numerical example in Section 4. According to Bayes’ formula, we can getwhere and means is proportional to . Then the Radon-Nikodym derivative related to the prior measure and posterior measure isIn this paper, we use MCMC (Markov Chain Monte Carlo) method to describe the posterior distribution according to and get the mean of . For the details of this method we refer to [12, 15].
3. Well-Posedness of Bayesian Method
In this section, our goal is to enable application of the framework in [14] for our inverse interior scattering problem. We divide this section into two subsections. In Section 3.1 we give the well-posedness of our Bayesian method and in Section 3.2 we prove that our observational operator satisfies the assumption in the well-posedness theorem.
3.1. Well-Posedness Theorem
According to Section  4 (common structure) of [14], we should choose a space and a prior measure such that and Assumption 2.7 in [14] holds on . Then we can get the well-posedness results of Bayesian method. So first we give the specific definition of prior measure on . Assume with the definition domain: Because in the representation of starlike domain is periodic, here and in the remaining part of the paper we use the square bracket in the definition of the domain to signify the periodicity. So is the usual Sobolev space with periodic condition ; that is, Now we assume a prior Gaussian measure on such that . Through simple calculation, we can get that the eigenvalues of are and the corresponding eigenfunctions are and . According to Karhunen-Loève expansion, we have where and are i.i.d. (independent and identically distributed) sequences with and . Clearly, this is in fact the Fourier expansion. Integrating we can get . is also a Gaussian measure because the integral operator is linear and continuous [16]. Using Lemma 6.25 in [14], we know that is almost surely in for any . In order to get , we integrate . The function is not unique when its second derivative is given. Here, we just use periodic form of in (13) the same as in [10]. More specifically, we define where is a Gaussian random variable. Then we can integrate the Fourier expansion of term by term to obtainHere, we define the Gaussian measure on based on its second derivative . There is also an alternative way to define the Gaussian measure on ; see [10].
Now we define with norm given in the following:where is defined as usual; that is, .
Then we obtain the following theorem.
Theorem 1. Assume defined by (13); then for .
Proof. From the analysis above, we know is almost surely in for any . So is almost surely in ; that is, .
From Theorem 4.1, Theorem 4.2, Theorem 6.31, and Lemma 2.8 in [14], in order to get the well-posedness theorem, we need to prove that the observational operator satisfies the following assumption.
Assumption 2 (Assumption 2.7 in [14]). (i) For every , there is an such that, for all ,(ii) For every , there is a such that, for all with ,
We will prove that satisfies Assumption 2 in the next subsection. Now we give the main theorem in our paper.
Theorem 3 (well-posedness). Let the observational operator satisfy Assumption 2 and with . Then one has the following.(i)The posterior measure is absolutely continuous with respect to and has Radon-Nikodym derivative given by(ii)The posterior measure is a well-defined probability measure on .(iii)The posterior measure is Lipschitz in the data , with respect to the Hellinger distance; that is, there exists a constant such that for all with . Here, the Hellinger distance is defined asprovided and are both absolutely continuous with .
The proof of Theorem 3 is just a result of application of Theorem 4.1, Theorem 4.2, Theorem 6.31, and Lemma 2.8 in [14]. In fact, from Theorem 4.2 in [14] we know the mean of is also continuous in . So if we use the mean of posterior distribution to approximate the solution of inverse interior scattering problem, it is continuous with the observational data .
3.2. Properties of Observational Operator
In this subsection, our task is to prove Assumption 2 when is defined by integral equation method in [17, 18]. We use the classical layer potential approach to formulate integral equations for direct interior scattering problem with three different boundary conditions. First, let us introduce the single- and double-layer operators and , given byand the normal derivative operator , given bywhere is the fundamental solution of Helmholtz equation and is the unit outward normal.
Then we look for a solution in the form offor the Dirichlet boundary condition, orfor the Neumann boundary condition or impedance boundary condition. From the jump relations, we see that is the solution of scattering problem (1)-(2), provided the density is a solution of the following integral equation:for the Dirichlet boundary condition, orfor the Neumann boundary condition, orfor the impedance boundary condition.
From [19] we known that the operators and are compact operators on , and is a compact operator on . If is not an eigenvalue of for all possible domain when changes, we can get the injectivity of , , and . For details we refer to [17]. In practice, the cavity is always bounded, so is also bounded. We can choose positive constants small enough and big enough such that . Then by Riesz-Fredholm theory [17], from the injectivity of the operators , , and , we know that they are bijective, their inverse operators are bounded on domain , and the bounds of the operators are only related to the constants and . From the analytic property of on , we obtain for Dirichlet boundary condition and for Neumann or impedance boundary condition. maps continuously into . As a result, and the trace for all the three boundary conditions [19].
Remark 4. Here, for the impedance boundary condition, when is a real number, is not an eigenvalue for because of the uniqueness of interior Helmholtz problem with impedance boundary condition. So the case of impedance boundary condition does not suffer from the existence of eigenfrequencies [5, 20].
For the Dirichlet boundary condition, we can find a small enough to avoid all Dirichlet eigenvalues. More specifically, for , the ball contains all possible domain . From the Rayleigh-Faber-Krahn inequality the smallest Dirichlet eigenvalue (>0) of for domain is no less than the smallest eigenvalue (>0) of . Then when , it is not an eigenvalue of for all possible domain . This method of avoiding eigenvalues is often used in inverse interior scattering problem [5].
For the Neumann boundary condition, we cannot give a similar method to avoid all the eigenvalues as Dirichlet boundary condition. The distribution of Neumann eigenvalues of interior scattering problem is still under investigation. So for Neumann boundary condition we just assume that is not an eigenvalue of for all possible domain . In fact, if the cavity is filled with some special material such that , then the interior Dirichlet and Neumann problems have at most one solution [17]. We will consider this situation in the future investigation.
Now we show that satisfies the property (i) in Assumption 2.
Theorem 5. For every , there is an such that, for all , where .
Proof. From the definition of , we just need to prove By (22) and (23), it is sufficient to showorNow we give the proof of (29); then the proof of (30) is similar. It is known thatHere taking Neumann boundary condition as an example, because the upper bounds of depend only on the bounds of and . From the analysis above Theorem 5, and are bounded when the domain is bounded. So we have .
Because is analytic on ,Now our goal is to bound the two factors in right hand of the above inequality.
When , .
When , according to Young’s inequality, On the other hand, Then we obtain the final result:
The next step is to prove That is, satisfies property (ii) in Assumption 2.
In order to get this Lipschitz continuity, we use the definition of domain derivative in [21]. There are also other ways to define and get the domain derivative, such as [22–24], but the results are similar. Assume is a bounded open domain in with smooth boundary . is a function defined on satisfying where and are some partial differential operators. Given a regular (in our application regular means smooth) vector field defined in , one denotes For small enough, is an open set close to with regular boundary. can be viewed as a variation of . Then we define and we call the local derivative the domain derivative with . Let be the local derivative of in a direction ; that is,where is a regular vector field.
From [21] we know that, with some suitable smoothness hypotheses (e.g., smooth), the derivative satisfies the following equation and boundary condition:where is the normal derivative of , , and .
In our problem, the smoothness of and and the regularity of satisfy the hypotheses to and in [21]. The operator , , , and . Clearly, and are linear bounded operators, so the derivatives are themselves; that is, Then we obtain the derivative in our problem satisfiesHere, because of the boundary condition (2), the tangential part of is 0; that is,
In our problem, is a starlike domain. The variation of is in fact the variation of . So we can denote , where is the direction of variation of . Then the boundary condition (44) changes to where .
Now we show that satisfies property (ii) in Assumption 2.
Theorem 6. For every , there is a such that, for all with ,
Proof. From the above analysis of derivative , it is easy to get that By mean value theorem [25] and the regularity of and , Then from the definition of , we obtain
4. Numerical Examples
We have already demonstrated the well-posedness of the posterior distribution. Now we give some numerical examples to show the effectiveness of our method. In order to describe the posterior distribution, we consider to adopt MCMC method to generate samples distributed according to (7). Then the average of these samples can be used to approximate the expectations with respect to the posterior distribution and hence to make predictions about the shape parameters [12]. Recently, a DRAM (Delayed Rejection Adaptive MCMC) algorithm is proposed to improve the efficiency of the standard MH (Metropolis-Hasting) algorithm. DRAM algorithm combines two powerful ideas in MCMC: adaptive Metropolis samplers and delayed rejection [15]. So in this paper, we use DRAM algorithm to generate samples.
We consider a peanut cavity which is the radial function parameterized by . The interior curve is a circle defined by . We take the source points and on . The wave number . In order to avoid the inverse crime we solve the direct problem by MFS (method of fundamental solution) to obtain the synthetic data (observational data ) on curve . Three different boundary conditions are considered, respectively, in Examples 1, 2, and 3. All the numerical experiments followed were performed using MATLAB software.
Example 1 (the sound-soft boundary condition). We truncate 6 terms of the series in (13) and assume the corresponding 13 coefficients obey Gaussian distribution . We take the measurement error , (the relative error is about 0.3%), and generate 100000 samples of the posterior distribution. The posterior distribution histograms of coefficients and are presented in Figure 2 and the Markov Chains of them are presented in Figure 3. From Figure 2 we can see the general shape of the posterior distributions of and . We omit the histograms for the other coefficients due to the space constraints. The trace plots of the Markov Chains in Figure 3 look like “fuzzy worms,” so as a rule of thumb the step size in our algorithm is appropriate [12].
Because the beginning draws of the chain represent poorly the distribution to be explored [12], we remove the first 20000 draws from the chain and take the mean value of the other draws. The reconstruction of the cavity is shown in Figure 4. The reconstruction of the radial function and the scattered field are displayed in Figure 5, where the -axis represents the angle value varying from 0 to . In Figure 4 and left one in Figure 5 the red line is the exact cavity shape and the blue line is the reconstruction. For the right figure of Figure 5 the solid line is the exact value and the “+” line is the reconstruction value. We use the same notations in the following experiments without extra illustrations. From these three reconstruction figures we can see our method is effective.
In the next experiment, we consider the effect of different levels of noise. In Figures 6 and 7, the noise (the relative error is about 0.6%). In Figures 8 and 9, the noise , (the relative error is about 3.3%). In the right one of Figure 9, the green line denotes the measurement data (the exact scattered field added with noise). From the reconstruction of scattered field, we know that, in the interior cavity scattering problems, the scattered field varies little (the difference between the maximum and minimum is about 0.25) from different measurement locations; that is, the scattered field is almost a constant. So the noise we add is small. If the additive noise is big corresponding to the variation, the reconstruction effect will be poor (see Figure 9). That is also why we choose two point sources illuminating together. The variation of scattered field is bigger than only one point source’s situation.
In the last experiment of sound-soft boundary condition part, we explore a different prior measure. Assume the noise , , and the prior density where can be chosen as different functions, such as norm function, norm function, or TV norm function corresponding to different prior measures, in fact different regularization strategies. Here, we choose the norm function and . This prior called impulse prior is corresponding to the regularization. We also generate 100000 samples for simulation. The reconstruction of the cavity (Figure 10), the radial function, and the scattered field (Figure 11) are as follows. This experiment shows that Gaussian prior measure is not the only choice. The other priors may be also effective in numerical simulations. We adopt Gaussian prior measure because we know little about radial function in advance, and the theoretical system of Gaussian prior measure is easy to build under the framework of [14]. The theoretical results of other prior measures are also under consideration.
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Example 2 (the sound-hard boundary condition). We truncate 6 terms of the series in (13) and assume the corresponding 13 coefficients obey Gaussian distribution . We take the measurement error , , and generate 100000 samples of the posterior distribution. For the constraints of space, we only present the figures of reconstructed cavity, radial function, and scattered field (Figures 12 and 13). From Example 2 we can see that the Bayesian method is also effective for inverse interior scattering problem with sound-hard boundary condition.
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Example 3 (the impedance boundary condition). We choose in this example and other parameters are the same as in Example 2. The reconstructions of cavity, radial function, and scattered field are in Figures 14 and 15. This example shows that our method is effective in the situation of impedance boundary condition.
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5. Conclusions
In this paper, we study the inverse interior scattering problem. Bayesian method is used to reconstruct the shape of the cavity from interior measurement. We prove the well-posedness and present some numerical examples to illustrate that our method is effective. In the future, we will consider Bayesian method for more complicated inverse scattering problem, for example, the open cavity embedded in the infinite ground plane.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The research was supported by the NSFC 11371172, 11271159.