  A method for detecting and approximating fault lines or surfaces,
respectively, or decision curves in two and three dimensions with guaranteed
accuracy is presented. Reformulated as a classification problem, our method
starts from a set of scattered points along with the corresponding
classification algorithm to construct a representation of a decision curve by
points with prescribed maximal distance to the true decision curve. Hereby, our
algorithm ensures that the representing point set covers the decision curve in
its entire extent and features local refinement based on the geometric
properties of the decision curve. We demonstrate applications of our method to
problems related to the detection of faults, to Multi-Criteria Decision Aid
and, in combination with Kirsch's factorization method, to solving an inverse
acoustic scattering problem. In all applications we considered in this work,
our method requires significantly less pointwise classifications than
previously employed algorithms.
