Over the past few years, the Factorization Method of Kirsch has proven to be
a successful new method for solving inverse scattering problems. Not only is
there a well-founded theoretical justification, but the method has also been
applied to large classes of problems, many of them not at all related to the
bounded obstacle scattering problem the method was originally designed for.
What makes the method so interesting is that it reconstructs an obstacle
without solving any direct scattering problems simply from the data using
spectral properties of the so-called far field operator.
Inverse problems for unbounded scatterers are much more difficult to solve
than for bounded obstacles. Of particular interest are periodic surfaces.
Contrary to the case of the bounded obstacle problem, here, the far field
patterns do no provide sufficient data to determine a surface uniquely. To be
able to uniquely determine the scatterer, nearfield data has to be collected.
Nevertheless, the Factorization method can be applied to such problems. In
the talk, the approach will be presented in detail, putting an emphasis on
what modifications are necessary to treat the situation of a diffraction
grating. These theoretical results will be illustrated by several numerical
examples of successful reconstructions. An outlook will discuss the
possibility of extending these results to more general rough surfaces.
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