Inverse Problems for a Class of Conditional Probability Measure-Dependent Evolution Equations

TL;DR

This paper addresses the inverse problem of identifying conditional probability measures in measure-dependent PDE models, establishing existence, stability, and discretization methods, with applications to bacterial flocculation dynamics.

Contribution

It introduces a novel framework for solving inverse problems involving measure-dependent evolution equations, including theoretical results and a practical discretization scheme.

Findings

Proved existence and well-posedness of the inverse problem.

Developed a stable discretization scheme for measure approximation.

Demonstrated the approach with numerical simulations in bacterial population models.

Abstract

We investigate the inverse problem of identifying a conditional probability measure in a measure-dependent dynamical system. We provide existence and well-posedness results and outline a discretization scheme for approximating a measure. For this scheme, we prove general method stability. The work is motivated by Partial Differential Equation (PDE) models of flocculation for which the shape of the post-fragmentation conditional probability measure greatly impacts the solution dynamics. To illustrate our methodology, we apply the theory to a particular PDE model that arises in the study of population dynamics for flocculating bacterial aggregates in suspension, and provide numerical evidence for the utility of the approach.