The probabilistic inverse problem and its solving method based on probability density evolution theory and convex optimization algorithms

Abstract

A probabilistic inverse problem-solving method based on the framework of Probability Density Evolution Theory and convex optimization algorithms is proposed. This method reformulates the identification of the random source as a quadratic programming problem with linear constraints, identifying the probability density function of the random source in a physical stochastic system even when the distribution type of the random source is entirely unknown. Through singular value decomposition of the quadratic matrix, an error analysis is performed, revealing that the solvability of the probabilistic inverse problem fundamentally depends on the injectivity of the mapping from the random source space to the response space. Case studies confirm that the proposed method is not sensitive to prior information and does not require any predefined assumptions about the distribution type. Meanwhile, it can preliminarily determine whether the inverse problem is solvable before the computational process begins.