The domain derivative for semilinear elliptic inverse obstacle problems

<p style='text-indent:20px;'>We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.</p>

Shape Reconstruction for Unsteady Advection-Diffusion Problems by Domain Derivative Method

This paper is concerned with the numerical simulation for shape reconstruction of the unsteady advection-diffusion problems. The continuous dependence of the solution on variations of the boundary is established, and the explicit representation of domain derivative of corresponding equations is derived. This allows the investigation of iterative method for the ill-posed problem. By the parametric method, a regularized Gauss-Newton scheme is employed to the shape inverse problem. Numerical examples indicate that the proposed algorithm is feasible and effective for the practical purpose.