Bending energy regularization on shape spaces: a class of iterative methods on manifolds and applications to inverse obstacle problems

by Julian Eckhardt
Doctoral thesis
Date of Examination:2019-09-11
Date of issue:2019-12-13
Advisor:Prof. Dr. Thorsten Hohage
Referee:Prof. Dr. Thorsten Hohage
Referee:Prof. Dr. Max Wardetzky
Persistent Address: http://dx.doi.org/10.53846/goediss-7769

 

 

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Abstract

English

In applications such as nondestructive testing, geophysical exploration or medical imaging one often aims to reconstruct the boundary curve of a smooth bounded domain from indirect measurements. As a typical example we concentrate here on inverse obstacle scattering problems. We introduce a class of shape manifolds for describing admissible obstacles and we allow the reconstruction of general, not necessarily star-shaped, curves. By applying the bending energy as regularization term the Tikhonov regularization gain independence of the parameterization. Moreover, the structure of the shape manifold is investigated. It turns out to be a infinite-dimensional Riemannian manifold and therefore, geometry provides several tools, such as Levi-Civita connection, geodesics, Riemannian exponential map, Riemannian Hessian of a functional and parallel transport. One construction, we focus on, is the second fundamental form for which we give explicit formulas and prove local bounds. Furthermore, we introduce an iteratively regularized Gauss-Newton method on Riemannian manifolds. In each step we compute an update direction as an element in the tangent space using the derivative of the forward operator, the gradient and the Hessian of a regularizing functional. This update direction is mapped by the Riemannian exponential map onto the manifold. Under a general framework we prove convergence rates of this algorithm for exact and perturbed data. The assumptions appearing in the proof are discussed and mostly verified for inverse obstacle scattering problems. Numerical simulations demonstrate the benefits of the geometrical approach by using shape manifolds and bending-energy-based regularization to reconstruct non-star-shaped obstacles.
Keywords: inverse obstacle problems; obstacle scattering; nonlinear Tikhonov regularization; shape spaces; bending energy; shape manifold
 
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