Linear sampling methods for inverse boundary value problems in potential theory
Let \(D\) be a bounded domain in \(\mathbb{R}^2\) with connected \(C^2\) boundary \(\partial D\) and outward unit normal \(\nu\) and \(y\) a point in \(\mathbb{R}^2\setminus \overline{D}\). Let \(\Phi(x,y) = 1/(2\pi)\ln(1/|x-y|)\) be the fundamental solution of Laplace's equation in \(\mathbb{R}^2\). Further, assume that \(B\) is a bounded domain with connected \(C^2\) boundary \(\partial B\) that contains \(\overline{D}\). The authors consider the inverse problem of determining the shape of \(D\), i.e., the boundary \(\partial D\) from the knowledge of \(w(x,y)\) for all \(x,y\) in \(\partial B\) with \(w(x,y)\) being a harmonic function in \(\mathbb{R}^2\setminus \overline{D}\) satisfying \(\partial w/\partial \nu = - \partial \Phi(\cdot,y)/\partial \nu\) on \(\partial D\). They develop linear sampling methods suggested by \textit{D. Colton} and \textit{A. Kirsch} [Inverse Probl. 12, 383-393 (1996; Zbl 0859.35133)] for the problem and describe both the mathematical foundation and the numerical implementation of the method.
scientific article; zbMATH DE number 1879057