Level set methods for geometric inverse problems in linear elasticity

Abstract

In this paper, we investigate the regularization and numerical solution of geometric inverse problems related to linear elasticity with minimal assumptions on the geometry of the solution. In particular, we consider the probably severely ill-posed reconstruction problem of a two-dimensional inclusion from a single boundary measurement. In order to avoid parametrizations, which would introduce a priori assumptions on the geometric structure of the solution, we employ the level set method for the numerical solution of the reconstruction problem. With this approach, we construct an evolution of shapes with a normal velocity chosen depending on the shape derivative of the corresponding least-squares functional in order to guarantee its descent. Moreover, we analyse penalization by perimeter as a regularization method, based on recent results on the convergence of Neumann problems and a generalization of Golab's theorem. The behaviour of the level set method and of the regularization procedure in the presence of noise are tested in several numerical examples. It turns out that reconstructions of good quality can be obtained only for simple shapes or for unreasonably low noise levels. However, it seems reasonable that the quality of reconstructions improves by using more than a single boundary measurement, which is an interesting topic for future research.