Indicator Functions for Shape Reconstruction Related to the Linear Sampling Method

We provide exact shape reconstruction formulas in the spirit of the Linear Sampling method for a class of inverse problems in shape determination in the context of timeindependent partial differential equations. To this end, we prove a general theorem how, and under which assumptions, domain characterizations based on the range of the square root of an operator transform into domain characterizations based on the operator itself. To show the flexibility of this general theory we apply this general principle to a variety of shape determination problems in inverse acoustic and electromagnetic scattering theory and inverse elliptic boundary value problems. Further, we also establish a regularization strategy for noisy measurement operators.