Numerical methods for solving ill-posed inverse problems with soft qualitative constraints
Many inverse problems that arise in the fields of physics experiments data analysis, economics data analysis etc. are ill-posed ones. In this work, we investigate such problems with multiple solutions or with solutions that non-continuously depend on input data. The regularization method is widely used to solve them, however, in this work, we propose a different approach that utilizes a priori information in the form of soft constraints. The notion of soft constraint is based on the concept of fuzzy set according to the qualitative possibility theory proposed by prof. Yu.P. Pyt’ev. Formulations of problems with soft qualitative constraints are given, and numerical methods for solving them in linear and non-linear cases are proposed. In the linear case, the problems can be transformed to the linear programming ones. In the non-linear case, we propose the gradient methods for solving them wherein TensorFlow library can be used to automatically calculate function derivatives.