Analysis of a Linear Sampling Method for Identification of Obstacles

Abstract

AnalysisDuring the last decade there are many papers published, in which linear sampling methods for identification of an obstacle D are proposed. Such methods (see, for example, [2,3,6]) are based on a numerical verification of the inclusion of some function f := f (α, z), z ∈ R 3 , α ∈ S 2 , in the range R(B) of a certain operator B. It is proved in this paper that the methods, proposed in the above papers, have essential difficulties. This will also be demonstrated by numerical experiments. Although it is true that f ∈ R(B) when z ∈ D, it turns out that in any neighborhood of f , however small, there are elements from R(B). Also, although f ∈ R(B) when z ∈ D, there are elements in every neighborhood of f , however small, which do not belong to R(B) even if z ∈ D. Therefore it is not possible to construct a stable numerical method for the identification of D based on checking the inclusions f ∈ R(B) and f ∈ R(B).We prove below that the range R(B) is dense in the space L 2 (S 2 ). Assumption (A). We assume throughout that k 2 is not a Dirichlet eigenvalue of the Laplacian in D.Let us introduce some notations: N (B) and R(B) are, respectively, the null-space and the range of a linear operator B, D ∈ R 3 is a bounded domain (obstacle) with a smooth boundary S, D = R 3 \ D, u 0 = e ikα·x , k =const> 0, α ∈ S 2 is a unit vector, N is the unit normal to S pointing into D , g = g(x, y, k) := g(|x − y|) := e ik|x−y| 4π|x−y| , f := e −ikα ·z , where z ∈ R 3 and α ∈ S 2 , α := xr −1 , r = |x|, u = u (x, α, k) is the scattering solution: Manuscript

Optimization Methods in Direct and Inverse Scattering

“…The following claim follows easily from the results in [50], [55] (cf [42] Consider B : L 2 (S) → L 2 (S 2 ), and A : L 2 (S 2 ) → L 2 (S 2 ), where B is defined in (8.6) and Aq := S 2 A(α ′ , α)q(α)dα. Then one proves (see [76]): Theorem 1. The ranges R(B) and R(A) are dense in L 2 (S 2 ) Remark 1.…”

“…Examples of such methods include [14], [16], [42]. However, one can show that the methods proposed in the above papers have essential difficulties, see [76]. Although it is true that f ∈ R(B) when z ∈ D, it turns out that in any neighborhood of f there are elements from R(B).…”