Elastic energy regularization for inverse obstacle scattering problems. (13th September 2019)

• Record Type: Journal Article
• Authors: Eckhardt, J; Hiptmair, R; Hohage, T; Schumacher, H; Wardetzky, M
• Is Part Of: Inverse problems. Volume 35:Number 10(2019)
• Publication Date: 2019-09-13
• Subjects: shape spaces -- inverse obstacle scattering -- nonlinear Tikhonov regularization; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ab3034
• Languages: English
• ISSNs: 0266-5611

Abstract

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.