Abstract
In the paper, we put forward a new topological taxonomy which allows to distinguish and separate multiple solutions to the ill-conditioned parametric inverse problems appearing in engineering, geophysics, medicine, etc. This taxonomy distinguishes the areas of insensitivity to parameters, called landforms of the misfit landscape, be it around minima (lowlands), maxima (uplands), or stationary points (shelves). We proved their important separability and completeness conditions. In particular, lowlands, uplands and shelves are pairwise disjoint and there are no other subsets of the positive measure in the admissible domain on which misfit function takes a constant value. The topological taxonomy is related to the second, 'local' one, which characterizes the types of ill-conditioning of the particular solutions. We hope that the proposed results will be helpful for a better and more precise formulation of the ill-conditioned inverse problems and for selecting and profiling complex optimization strategies used to solve these problems.
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