A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme

作者：Burkard C^*; Potthast R

来源：Inverse Problems, 2010, 26(4): 045007.

DOI：10.1088/0266-5611/26/4/045007

We study the reconstruction of three-dimensional rough surfaces via a potential approach. Using single-layer potentials for the solution of shape reconstruction problems was first suggested by Kirsch and Kress in the case of bounded obstacles. The basic idea of this method is the reformulation of the inverse problem in an optimization problem, where the field is approximated by a single-layer potential and the unknown surface is simultaneously fitted as a zero curve of the sum of scattered and incident fields. The key difficulty in the inverse rough surface scattering problem is the unboundedness and non-compactness of the scatterer and the non-compactness of the associated single-layer potential. This significantly changes the setup and rigorous analysis of the optimization problem. We suggest a multi-section approach to the field and surface reconstruction problem. First, we carry out the full convergence analysis of the Kirsch-Kress method for rough surfaces in three dimensions using a semi-finite approach. In particular, we show the existence of an optimal surface and the convergence of the method towards a solution of the inverse problem. Then, we provide the convergence analysis for the multi-section method. This extends recent results of Heinemeyer et al (2008 SIAM J. Numer. Anal. 46 (4) 1780-98) from the direct to the inverse problem. Finally, we study reconstructions of a finite section of the unknown scatterer. The numerical results prove the feasibility of the approach.

• 出版日期2010-4