The inverse obstacle scattering problem to reconstruct the
boundary ¶D of a sound-soft obstacle D
from the knowledge of the far field pattern u∞
for the scattering of a time-harmonic plane wave ui can be
interpreted as a nonlinear ill-posed operator equation F(¶D)
= u∞. Here, the operator
F maps the boundary of the obstacle onto the far-field of the scattered
wave us. We will review recent results on regularized Newton
iteration methods as applied to the above equation. They provide a very popular
method for the numerical solution of the inverse obstacle scattering problem
with accurate reconstructions.
In the second part of the talk, as an alternative approach, we consider the
nonlinear operator equation G(¶D) +
ui = 0 by regularized Newton iterations. Here, the operator G
is defined in terms of the density j of the
single-layer potential over ¶D that has far
field pattern u∞. Then G
maps ¶D onto the trace of the single-layer
potential on ¶D. It intrigues that the Newton
iterations for this equation do not need the solution of the forward problem and
that the method resembles to some extent the least squares method for solving
the inverse obstacle problem due to Kirsch and Kress.
Copyright. All Rights Reserved.