A novel quantitative inverse scattering scheme using interior resonant modes

Abstract

This paper is devoted to a novel quantitative imaging scheme of identifying impenetrable obstacles in time-harmonic acoustic scattering from the associated far-field data. The proposed method consists of two phases. In the first phase, we determine the interior eigenvalues of the underlying unknown obstacle from the far-field data via the indicating behaviour of the linear sampling method. Then we further determine the associated interior eigenfunctions by solving a constrained optimization problem, again only involving the far-field data. In the second phase, we propose a novel iteration scheme of Newton’s type to identify the boundary surface of the obstacle. By using the interior eigenfunctions determined in the first phase, we can avoid computing any direct scattering problem at each Newton’s iteration. The proposed method is particularly valuable for recovering a sound-hard obstacle, where the Newton’s formula involves the geometric quantities of the unknown boundary surface in a natural way. We provide rigorous theoretical justifications of the proposed method. Numerical experiments in both 2D and 3D are conducted, which confirm the promising features of the proposed imaging scheme. In particular, it can produce quantitative reconstructions of high accuracy in a very efficient manner.

A novel Newton method for inverse elastic scattering problems

“…where φδ p,n and φδ s,n can be derived similarly to (11) by replacing the right hand side in ( 11) with…”

“…To consider the convergence, we assume that the solution u can be analytically extended to Γ m . Here, this analytical extension may be possibly through the interior of D. Within this extension, we can represent the exact scattered field v in the form of (5), with ϕ ξ , ξ ∈ {p, s} in (5) given in form of (7) and the Fourier coefficients φξ,n , ξ ∈ {p, s} being the solution to (11).…”

“…For instance, [16] presents a hybrid method for the interior inverse scattering problem. In addition, the interior eigenfunctions are integrated into the Newton method to formulate an imaging scheme for identifying obstacles from the far-field data [11].…”