Identification of Starshape Constant Conductivity Inclusion from Boundary Measurement

Abstract

The problem of determining the interface separating a starshape support of constant conductivity inclusion from boundary measurement data of a solution of the corresponding PDEs is considered. An equivalent statement as a nonlinear integral equation is obtained. The problem is implemented using Fourier method. Numerical experiment based on simplified iteratively regularized Gauss-Newton method (sIRGNM) is presented. MSC classes: 31A25, 35R30, 65N21. Consider the following boundary value problemwhere Ω 1 is a unit circular domain with boundary ∂Ω 1 , and Ω S ⊂ Ω 1 is the support of S, and χ Ω S denoted the characteristic function of Ω S . The inverse problem consists in identifying the shape of Ω S given the Neumann data ∂u ∂n on ∂Ω 1 . Therefore, we define F as the operator mapping q to Assuming that the conductivity perturbation a is small, we linearize the equations w.r.t. the conductivity perturbation. The linearized equation for small a can be obtained by solvingwhere u = w − u 0 . Assume that supp(a) =Ω,Ω ⊂⊂ Ω, and suppose a ∈ L 2 (Ω). Then from (1.1) equivalently we haveHere g is applied voltage, u 0 is a harmonic reference potential, and u is fluctuation of the potential due to the inclusion a. From (1.2), we reduce our problem to finding the support of the source in the Poisson equation: ∆u = S(x) in Ω, u = 0 on ∂Ω, where the source is expressed formally as S(x) = − ∇ · a ∇ u 0 , and u 0 is the corresponding harmonic reference potential. The inverse linearized problem is to find a conductivity distribution a which fits the measured currents on the surface, ∂u ∂ν | ∂Ω 1 , of potential difference u satisfying (1.2)for the different applied voltages g. Starshape object inclusion problem as core identification.It is well known that the unique solution to the problem (0.1) is given bywhere G(x, ξ) is the Green's function of Ω 1 . Assume that a is a constant, Ω S to be star-shaped with respect to the origin. Then ∂Ω S = {q(t)(cos t, sin t) : t ∈ [0, 2π]} for q(t) > 0 and 2π−periodic. Consider the case a = χ Ω S , hence a| ∂Ω 1 = 0. Let Z 0 = Z\{0}, and let the Poisson Kernel: P (r, s) = 1 2π n∈Z (r) |n| e i ns , the Green's function of the unit disk Ω 1 in polar coordinates. We choose the harmonic reference potential u 0 of the form u 0 := U k (r, s) =k |k| e i ks , k ∈ Z 0 , which is the solution of Laplace equations with a scaled trigonometric function g = e i ks |k| , k ∈ Z 0 , as the Dirichlet