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

Kuang Huang, Bangti Jin, Zhi Zhou

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.

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

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.
Paper Structure (13 sections, 8 theorems, 97 equations, 3 figures)

This paper contains 13 sections, 8 theorems, 97 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Well-posedness of the direct problem
  3. Inverse problem of recovering inflow rates
  4. Reformulation as an integral equation
  5. Well-posedness of the inverse problem
  6. Discussions on nonsmooth inflow distribution
  7. Reconstruction method for inflow rates
  8. Numerical method
  9. Error analysis
  10. Numerical results and discussions
  11. Numerical results for inflow rate recovery
  12. Inverse problem for inflow distribution
  13. Concluding remarks

Key Result

Theorem 2.2

\newlabelthm:wellposedness_strong0 Suppose that the initial data $\bar{k}\in\mathbf{L}^\infty([0,\infty);[0,\infty))$ is Lipschitz continuous, that $\phi\in\mathbf{L}^\infty([0,\infty)\times[0,\infty);[0,\infty))$ is Lipschitz continuous in space and its Lipschitz constant is locally bounded on $t for $t\in[0,\infty)$, where $C(t)$ depends on $t,\norm{f}_{\mathbf{L}^\infty}, \norm{V'}_{\mathbf{L}

Figures (3)

  • Figure 1: Numerical results for \ref{['exp1']} with exact data and noisy data. The left, mid and right panels are for the forward problem and inverse problem with exact and noisy data, respectively. The top and bottom rows are for cases (a) and (b), respectively.
  • Figure 2: Numerical results for the three cases of \ref{['exp2']}. The top and bottom rows are for the forward problem and inverse problem with noisy data, respectively.
  • Figure 3: Numerical results for the two cases of \ref{['exp4']} with noisy data.

Theorems & Definitions (22)

  • Theorem 2.2
  • Proof 1
  • Lemma 2.3
  • Proof 2
  • Definition 2.4
  • Theorem 2.5
  • Proposition 2.6
  • Proposition 3.2
  • Remark 3.3
  • Theorem 3.4
  • ...and 12 more