On the reconstruction of Dirichlet-to-Neumann map in inverse scattering problems with stability estimates

Abstract

Consider the determination of Dirichlet-to-Neumann (D-to-N) map from the far-field pattern in inverse scattering problems, which is the key step in some recently developed inversion schemes such as probe method. Essentially, this problem is related to the reconstruction of the scattered wave from its far-field data. We firstly prove the well-known uniqueness result of the D-to-N map from the far-field pattern using a new scheme based on the mixed reciprocity principle. The advantage of this new proof scheme is that it provides an efficient algorithm for computing the D-to-N map, avoiding the numerical differentiation for the scattered wave. Then combining with the classical potential theory, a simple and feasible regularizing reconstruction scheme for the D-to-N map is proposed. Finally the stability estimate for the reconstruction with noisy input data is rigorously analyzed. Keywordsinverse scattering, Dirichlet-to-Neumann map, regularization, mixed reciprocity principle, stability MSC(2000): 35J25, 45A05, 45Q05, 65F22 Citation: Wang H B, Liu J J. On the reconstruction of Dirichlet-to-Neumann map in inverse scattering problems with stability estimates.

Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

“…However, we should remark that, in this example, the absolute error of the input data corrupted via (3.4) is still small even if we take the maximum random relative error as 10%, since the values of exact far-field patterns are not so large. According to the stability estimate of ∂ D in [18], the numerical results with good accuracy is reasonable. Of course, as the error level δ becomes large, the reconstructions are also deteriorated.…”

“…However, we can estimate the blowing-up rates due to the special structure of the fundamental solution on the right-hand side of (2.37) and (2.38). We state this result as follows (see [18]). which hold uniformly for all x, y ∈ ∂ .…”

“…With the help of the Green's representation theorem and properties of layer potentials, we can easily justify the following result [11,18]. In the proof of this lemma, we have shown that the Neumann data ∂u ∂ν ∂ = ∂ D f is the unique solution to the integral equation We are now prepared to derive two simple and efficient algorithms for computing (2.2) for x, y ∈ ∂ .…”

“…It is worthwhile to point out that the error estimate for this algorithm can be established by the same arguments as those in [18].…”

“…The ill-posedness of the original inverse problem consists in the determination of integral kernels from the far-field measurements. Given the domain , a stability estimate for reconstructing the D-to-N map for unknown D from the noisy far-field data {w ∞ δ (θ, d) : θ, d ∈ S} is recently established by the authors in [18]. However, the numerical realization schemes for the D-to-N map using the far-field data directly are still open.…”