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.

翻译：本文提出一种在二维与三维空间中，以可保证精度检测并逼近断层线/曲面或决策曲线的方法。该方法将问题重构为分类问题，从一组散点及其对应的分类算法出发，通过预设与真实决策曲线最大距离的点来构建决策曲线的表征。在此过程中，算法确保表征点集完整覆盖决策曲线，并基于决策曲线的几何特性实现局部细化。我们展示了该方法在多个问题中的应用，包括断层检测、多准则决策辅助，以及与Kirsch因子分解方法结合求解声波逆散射问题。在所涉及的所有应用中，该方法所需的逐点分类次数均显著少于此前采用的算法。