Analysis of the Hessian for Inverse Scattering Problems. Part I: Inverse Shape Scattering of Acoustic Waves

We derive expressions for the shape Hessian operator of the data misfit functional corresponding to the inverse problem of inferring the shape of a scatterer from reflected acoustic waves, using a Banach space setting and the Lagrangian approach. The shape Hessian is then analyzed in both Hölder and Sobolev spaces. Using an integral equation approach and compact embeddings in Hölder and Sobolev spaces, we show that the shape Hessian can be decomposed into four components, of which the Gauss-Newton part is a compact operator, while the others are not. Based on the Hessian analysis, we are able to express the eigenvalues of the Gauss-Newton Hessian as a function of the smoothness of the shape space, which shows that the smoother the shape is, the faster the decay rate. Analytical and numerical examples are presented to validate our theoretical results. The implication of the compactness of the Gauss-Newton Hessian is that for small data noise and model error, the discrete Hessian can be approximated by a low-rank matrix. This in turn enables fast solution of an appropriately regularized inverse problem, as well as Gaussian-based quantification of uncertainty in the estimated shape.