Detecting and approximating decision boundaries in low-dimensional spaces

Numerical Algorithms, vol. 95, pp. 1503–1537

Abstract

Abstract 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.

Authors 2

1. Matthias Grajewski corresponding Aachen Faculty of Medical Engineering and Technomathematics Institute for Data-Driven Technologies

   FH Aachen

   Affiliation as printed

   Faculty of Medical Engineering and Technomathematics, FH Aachen University of Applied Sciences, Heinrich-Mußmann-Str. 1, 52428, Jülich, Germany

   Institute for Data-Driven Technologies, FH Aachen University of Applied Sciences, Heinrich-Mußmann-Str., 52428, Jülich, Germany

2. Andreas Kleefeld Aachen Faculty of Medical Engineering and Technomathematics

   FH Aachen · Forschungszentrum Jülich · Jülich Supercomputing Centre

   Affiliation as printed

   Faculty of Medical Engineering and Technomathematics, FH Aachen University of Applied Sciences, Heinrich-Mußmann-Str. 1, 52428, Jülich, Germany

   Forschungszentrum Jülich GmbH, Jülich Supercomputing Centre, Wilhelm-Johnen-Str., 52425, Jülich, Germany