On an Inverse Problem of the Generalized Bathtub Model of Network Trip Flows

TL;DR

This paper studies a nonlocal transport equation model for network trip flows, proving well-posedness, analyzing an inverse source problem with stability, and developing a numerical reconstruction method validated by experiments.

Contribution

It establishes the well-posedness of the generalized bathtub model, analyzes the inverse problem's stability, and introduces a practical numerical reconstruction approach.

Findings

Well-posedness of the model for classical and weak solutions

Conditional Lipschitz stability of the inverse problem

Effective numerical reconstruction with error analysis

Abstract

In this work, we investigate the generalized bathtub model, a nonlocal transport equation for describing network trip flows served by privately operated vehicles inside a road network. First, we establish the well-posedness of the mathematical model for both classical and weak solutions. Then we consider an inverse source problem of the model with model parameters embodying particular traffic situations. We establish a conditional Lipschitz stability of the inverse problem under suitable a priori regularity assumption on the problem data, using a Volterra integral formulation of the problem. Inspired by the analysis, we develop an easy-to-implement numerical method for reconstructing the flow rates, and provide the error analysis of the method. Further we present several numerical experiments to complement the theoretical analysis.