Elastic energy regularization for inverse obstacle scattering problems

Abstract

This paper is motivated by inverse problems in which the boundary curve of a smooth bounded domain has to be reconstructed from indirect measurements. As a classical example we study acoustic inverse obstacle scattering problems for cylindrical sound-soft scatterers using far-field measurements of scattered time-harmonic waves. By introducing a shape manifold as a solution set we allow the reconstruction of general, not necessarily star-shaped, curves. The bending energy is used as a stabilizing term in Tikhonov regularization to gain independence of the parametrization. Moreover, we discuss how self-intersections can be avoided by penalization with the Möbius energy and prove the regularizing property of our approach as well as convergence rates under variational source conditions.In a second part of the paper a discrete setting is introduced, and we describe a numerical method for finding the minimizer of the Tikhonov functional on the shape-manifold. Numerical examples demonstrate that our method can reconstruct non-star-shaped obstacles.