Convex Optimization Framework for Probabilistic Inverse Problem Identification Based on the Principle of Probability Preservation

This paper investigates the probabilistic inverse problem in physical stochastic systems, which seeks to infer the underlying probability structure of an unobservable random source based on observed system responses. Grounded in the Principle of Preservation of Probability, a unified convex optimization framework is proposed that requires minimal prior assumptions. The core approach discretizes the random source space into subdomains, transforming the identification task into a quadratic programming problem with linear constraints to determine the probability measure assigned to each subdomain. Two complementary implementations are introduced. The first directly utilizes the transient probability density function (PDF) of the system response. The second, termed the Diffusion Manifold Method, operates directly on raw response data by projecting it onto an intrinsic Riemannian manifold, enabling effective handling of high-dimensional outputs and measurement noise. A singular value decomposition (SVD) of the resulting system transition probability matrix provides a criterion for well-posedness, revealing that solvability depends on the injectivity of the system mapping. In cases where the problem is ill-posed, the PDF of the random source is reconstructed as a linear combination of eigenmodes. A two-step optimization procedure is then applied: first, the identifiable eigenmodes of the source PDF are recovered from the response; second, the unidentifiable modes are regularized using quality metrics, such as maximum entropy. Numerical examples demonstrate that the proposed methods reliably recover complex and irregular probability structures, including joint distributions, even in the presence of noise and with limited data-without relying on strong prior assumptions.

Keywords: Probabilistic Inverse Problem, Principle of Preservation of Probability, Convex Optimization, Ill-posed Problem, Riemannian Manifold.