A regularization method based on level-sets for the problem of crack detection from electrical measurements

Abstract

We investigate regularization methods for solving the problem of crack detection in bounded planar domains from electrical measurements on the boundary. Based on the multiple level-set approach introduced in Álvarez et al (2009 J. Comput. Phys. 228 5710–21) and on the regularization strategy devised in De Cezaro et al (2009 Inverse Problems 25 035004), we propose a Tikhonov type method for stabilizing the inverse problem. Convergence and stability results for this Tikhonov method are proven. An iterative method of (multiple) level-set type is derived from the optimality conditions for the Tikhonov functional, and a relation between this method and the iterated Tikhonov method is established. The proposed level-set method is tested on the same benchmark problem considered in Álvarez et al (2009 J. Comput. Phys. 228 5710–21). The numerical experiments demonstrate its ability to identify cracks in different scenarios with high accuracy even in the presence of noise.