Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
Title Numerical methods for inverse scattering problems
Participant Klaus Giebermann
Key words scattering theory, integral equations, ill-posed problems, sampling method

Inverse Scattering Problems

The aim of inverse obstacle scattering problem is to determine the shape of one ore more obstacles by the knowledge of the scattered wave far away. This problem turns out to be an ill-posed , non-linear problem. This means that we want to invert a non-linear operator which has, unfortunately, no bounded inverse. Therefore, we have to regularize the problem.
Typically, the problem is formulated as a non-linear optimization problem. In this project, we focus on a new class of methods to deal with the inverse-scattering problem.

Regularized Sampling Method (RSM)

The method can be charaterized as follows: for every point x in try solve a linear integral equation where the right hand side depends on x. The kernel of the integral operator is given by the measured farfield. If the integral equation can be solved easily, i.e., if the norm of the solution is small, then x is assumed to lie inside the obstacle, otherwise outside.

The farfield for 42 incident waves is measured in 42 directions, where the directions are given as points on the unit sphere.

The experiment leads to the following farfield-data:

Using the regularized sampling method we get the following reconstructions:
Step 1
Step 2
Step 3
Final step
Original geometry

We can use the RSM for the reconstruction of multiple obstacles, too:

  • D. Colton, R. Kress: Inverse Acoustic and Electromagnetic Scattering Theory , Springer 1998
  • D. Colton, K. Giebermann, P. Monk: The regularized sampling method for solving three dimensional inverse scattering problems (1998)
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