Search

A new second order Newton method for reconstructing the shape of a sound soft scatterer from the measured far-field pattern for scattering of time harmonic plane waves is presented. This method extends a hybrid between regularized Newton iterations and decomposition methods that has been suggested and analyzed in a number of papers by Kress and Serranho [Inverse Problems 19: 91–104, 2003, Inverse Problems 21: 773–784, 2005, J. Comput. Appl. Math. 204: 418–427, 2007, Inverse Problems 22: 663–680, 2006, Inverse Problems and Imaging 1: 691–712, 2007] and has some features in common with the second degree method for ill-posed nonlinear problems as considered by Hettlich and Rundell [SIAM J.Numer. Anal. 37: 587–620, 2000]. The main idea of our iterative method is to use Huygen's principle, i.e., represent the scattered field as a single-layer potential. Given an approximation for the boundary of the scatterer, this leads to an ill-posed integral equation of the first kind that is solved via Tikhonov regularization. Then, in a second order Taylor expansion, the sound soft boundary condition is employed to update the boundary approximation. In an iterative procedure, these two steps are alternated until some stopping criterium is satisfied. We describe the method in detail and illustrate its feasibility through examples with exact and noisy data. [ABSTRACT FROM AUTHOR]

We consider the problem of determining the shape and location of cracks from Cauchy data on the boundary of semi-infinite domains modeling the reconstruction of cracks within a heat conducting medium from temperature and heat flux measurements. Our reconstructions are based on a pair of nonlinear integral equations for the unknown crack and the unknown flux jump on the crack that are linear with respect to the flux and nonlinear with respect to the crack. We propose two different iteration methods employing the following idea: Given an approximate reconstruction for the crack we first solve one of the equations for the flux and subsequently linearize the other equation for updating the crack. The foundations for this approach for solving the inverse problem in semi-infinite domains are provided and numerical experiments exhibit the feasibility of both methods and their stability with respect to noisy data. [ABSTRACT FROM AUTHOR]