Huygens' principle and iterative methods in inverse obstacle scattering (Q609549)

The inverse problem the authors consider is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, Huygens' principle will be interpreted as a system of two integral equations. The scattering of a time harmonic incident wave \(u^{i}\) by a sound-soft obstacle which is a bounded domain \(D\subset \mathbb{R}^{m}, m=2,3\) with a connected complement is treated. This is modeled by an exterior boundary value problem for the Helmholtz equation \[ \triangle u^{s}+k^{2}u^{s}=0 \quad\text{in} \quad \mathbb{R}^{m}\setminus \overline{D} \qquad k>0, \] with the Dirichlet condition \[ u^{i}+u^{s}=0 \quad \text{on}\quad \partial D \] and the Sommerfeld radiation condition, uniformly in all directions. \[ \frac{\partial u^{s}}{\partial r}-iku^{s}=o\biggr(\frac{1}{r^{\frac{m-1}{2}}}\biggl), \quad r=|x|\rightarrow\infty. \] It is supposed that \(u^{i}=e^{ik x}\cdot d\) where the unit vector \(d\) is the direction of propagation. So the Huygens' principle can be viewed as a system of two integral equations for the unknown boundary \(\partial D\) and the induced surface flux: \[ -\frac{\partial u}{\partial \nu} \quad \text{on} \quad \partial D, \] where \(\nu\) is the interior normal vector. Next sections deal with the iteration scheme, which is of Newton type.