Inverse Obstacle Scattering
It is well known that the propagation of an acoustic wave in a homogeneous, isotropic, and inviscid fluid is approximately described by a velocity potential \(U({\bf x},t)\) satisfying the wave equation \(U_{tt} = c^2\Delta U\). Here, \(c\) denotes the speed of sound, \(v = U_{\bf x}\) is the velocity field, and \(p = U_t\)is the pressure. If \(U\) is time harmonic, that is \(U({\bf x},t) = \operatorname{Re}\big(u({\bf x})e^{i\omega t}\big)\), \(\omega>0\), in complex notation, then the complexvalued spacedependent \(u\) satisfies the Helmholtz equation \[ \Delta u + \kappa^2 u = 0\quad\text{in}\ \mathbb{R}^3\setminus\overline{\Omega}. \] The domain \(\Omega\subset\mathbb{R}^3\) describes an obstacle and \(\kappa = \omega/c\) is the wave number. We assume that \(\Omega\) is simply connected and bounded with smooth boundary \(\Gamma = \partial\Omega\). For soundsoft obstacles, the pressure \(p\) vanishes on \(\Gamma\), which leads to the Dirichlet boundary condition \[ u=0\quad\text{on}\ \Gamma. \]We shall consider the situation that \(u = u_i+u_s\) is composed of a known incident plane wave \(u_i({\bf x}) = e^{i\kappa {\bf d}\cdot{\bf x}}\) with direction \({\bf d}\), and a scattered wave \(u_s\). The scattered field satisfies the Sommerfeld radiation condition \[ \lim_{r\to\infty} r \left\{ \frac{\partial u_s}{\partial r}  i\kappa u_s\right\} = 0,\quad r = \{\bf x}\, \] which implies the asymptotic behaviour \[ u_s({\bf x}) = \frac{e^{i\kappa\{\bf x}\}}{\{\bf x}\} \left\{u_\infty\bigg(\frac{\bf x}{\{\bf x}\}\bigg) + \mathcal{O}\bigg(\frac{1}{\{\bf x}\}\bigg)\right\}, \quad \{\bf x}\\to\infty. \] The function \(u_\infty:\mathbb{S}^2 :=\{{\bf x}: \{\bf x}\=1\}\to\mathbb{C}\) is called the farfield pattern.
The inverse obstacle scattering problem reads as follows: Given the direction \({\bf d}\), the far field pattern \(u_\infty\) of the scattered wave, find the domain \(\Omega\). In cooperation with Thorsten Hohage (GeorgAugustUniversity of Göttingen, Germany), we are developing efficient methods to solve this severely illposed inverse problem. Generating for example synthetic data from the following dolphin
we gain the reconstruction
Selected Publications

H. Harbrecht and T. Hohage.
Fast methods for threedimensional inverse obstacle scattering.
J. Integral Equations Appl., 19(3):237260, 2007.

H. Harbrecht and T. Hohage.
A Newton method for reconstructing non starshaped domains in
electrical impedance tomography.
Inverse Probl. Imaging, 3(2):353371, 2009.