Recovery of an interface from boundary measurement in an elliptic differential equation (Q421367)

Let \(\Omega\) be a smooth bounded domain in \(\mathbb R^{2}\) and \(S\subset \Omega\) a subdomain. This paper considers the elliptic boundary value problem \(\Delta u(x)=p(x)u(x)\) in \(\Omega\), \(\partial u(x )/\partial \nu=g(x)\) on \(\partial \Omega\), where \(\nu\) denotes the unit outward normal to \(\partial \Omega\) and \(p(x)=p_{0}\chi_{S}(x)\), which arises from a semiconductor transistor model. This paper studies the inverse problem of recovering an interior interface from a boundary measurement in this boundary value problem. The authors set up a nonlinear least-squares formulation for solving the inverse problem, and establish the necessary derivatives with respect to the interface. Then they propose both the Gauss-Newton iterative method and the conjugate gradient method for the least-squares problem and present implementation of these methods using integral equations.