Regularized Newton iteration method for a penetrable cavity with internal measurements in inverse scattering problem

Abstract

In this paper, we consider the inverse scattering problem of determining the shape of a cavity with a penetrable inhomogeneous medium of compact support from one source and a knowledge of measurements placed on a curve inside the cavity. First, the boundary value problem of the partial differential equations can be transformed into an equivalent system of nonlinear and ill‐posed integral equations for the unknown boundary. Then, we apply the regularized Newton iterative method to reconstruct the boundary and prove the injectivity for the linearized system. Finally, we present some numerical examples to show the feasibility of our method.

The interior inverse electromagnetic scattering for an inhomogeneous cavity

“…Let v = 0, then 3 , we have that w satisfies equation ( 16) in the distributional sense. Then by duality pairing between H − 1 2 (div, S R ) and H − 1 2 (curl, S R ), we can derive equation (20). In the exact same way, we can have v satisfies equations ( 17) and (21).…”

“…For a survey of these methods we refer to [7,11,16]. There have been recent interests in the inverse scattering problem for cavities using measurements inside [15,20,21,[24][25][26]28]. Such inverse scattering problems have potential applications in monitoring the structural integrity of the fusion reactor.…”

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

“…In the case of constant or time-dependent diffusion coefficient, the classical backward heat conduction equation has been studied in many works, see for example, other studies. [8][9][10] However, to the best of our knowledge, there are not any result on (1)(2)(3). Our paper is the first study on this direction for parabolic systems with random model.…”

“…Some work related to the problem which is studied such as that of Hapuarachchi and Xu 1 constructed an approximation problem and use a quasi‐boundary value method to regularize nonlinear heat equation by using a small parameter. In addition, in 2019, the Liu et al 2 considered that the boundary value problem of the partial differential equations can be transformed into an equivalent system of nonlinear and ill‐posed integral equations for the unknown boundary. Then, they applied the regularized Newton iterative method to reconstruct the boundary for the linearized system.…”

“…{ t u 1 =  1 (u)Δu 1 + (x, t), (x, t) ∈  × (0, T), t u 2 =  2 (u)Δu 2 + g(x, t), (x, t) ∈  × (0, T), (1) subject to the final conditions u 1 (x, T) = (x), u 2 (x, T) = (x), x ∈ , (2) and homogeneous Dirichlet boundary condition u 1 (x, t) = u 2 (x, t) = 0, (x, t) ∈  × (0, T),…”