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Stochastic nonlocal traffic flow models with Markovian noise

Timo Böhme, Simone Göttlich, Andreas Neuenkirch

Abstract

We extend our recently introduced stochastic nonlocal traffic flow model to more general random perturbations, including Markovian noise derived from a discretized Jacobi-type stochastic differential equation. Invoking a deterministic stability estimate, we show that the arising random weak entropy solutions are measurable, ensuring that quantities such as the expectation are well-defined. We show that the proposed Jacobi-type noise is of particular interest as it ensures interpretability, preserves boundedness, and significantly alters the stochastic realizations compared to the previous white noise approach. Moreover, we introduce a local solution operator which provides information on the local effect of the noise and utilize it to derive a mean-value hyperbolic nonlocal PDE, which serves as a proxy for the mean value of the exact solution. The quality of this proxy and the impact of the noise process are analyzed in several simulation studies.

Stochastic nonlocal traffic flow models with Markovian noise

Abstract

We extend our recently introduced stochastic nonlocal traffic flow model to more general random perturbations, including Markovian noise derived from a discretized Jacobi-type stochastic differential equation. Invoking a deterministic stability estimate, we show that the arising random weak entropy solutions are measurable, ensuring that quantities such as the expectation are well-defined. We show that the proposed Jacobi-type noise is of particular interest as it ensures interpretability, preserves boundedness, and significantly alters the stochastic realizations compared to the previous white noise approach. Moreover, we introduce a local solution operator which provides information on the local effect of the noise and utilize it to derive a mean-value hyperbolic nonlocal PDE, which serves as a proxy for the mean value of the exact solution. The quality of this proxy and the impact of the noise process are analyzed in several simulation studies.
Paper Structure (17 sections, 3 theorems, 103 equations, 6 figures)

This paper contains 17 sections, 3 theorems, 103 equations, 6 figures.

Table of Contents

  1. The class of sNV models and their properties
  2. A deterministic nonlocal velocity model (NV)
  3. A stochastic nonlocal velocity model (sNV)
  4. Existence, uniqueness and stability of the sNV model
  5. Measurability of solutions
  6. Local solution operator
  7. Mean velocity function and mean value proxy
  8. Markov-type error processes
  9. Numerical discretization
  10. Noise sampling and evaluation
  11. Numerical scheme for the sNV model
  12. Numerical scheme for the EsNV model
  13. Numerical results
  14. Impact of autocorrelated Markovian noise
  15. Validation of (EsNV) and benefit of nonlocality
  16. ...and 2 more sections

Key Result

Theorem 1.4

Let $\rho_0$ as in (eq:CP_BV) and assume that assumptions (A), (B) and (C) hold. Then, for any $T>0$ and any fixed $\omega \in \Omega$ a weak entropy solution $\rho(t,x)(\omega)$, in the sense of Def. def:nonlocal_weak_entropy_sol, to the Cauchy Problem of (eq:sNV), i.e., with $f_{\epsilon}(t,x,\rho)=\rho \bigl(W_\eta * v_\epsilon(\rho,\cdot)\bigr)(t,x)$, exists and is unique. Further, it holds f

Figures (6)

  • Figure 1: Characteristics based on $\rho_0^{\text{high}}$ to \ref{['eq:sNV']} in grey and \ref{['eq:EsNV']} in green.
  • Figure 2: Evolution of $\hat{f}_n$ ($\alpha=4, \sigma=1, \tau=0.5$) on a 601-node Chebyshev grid
  • Figure 3: Realizations (gray), Monte Carlo average and pointwise $\{5, 50, 95\}\%$-quantiles (blue, $M=2\cdot 10^3$) of \ref{['eq:sNV']}, including mean-proxy \ref{['eq:EsNV']} (green) at $T=1$. Left: white noise. Right: Jacobi-type noise. $\Delta x=10^{-3}$, $\Delta t$ acc. to \ref{['eq:sCFL_det']}
  • Figure 4: Left: Realizations (gray), Monte Carlo average and pointwise $\{5, 50, 95\}\%$-quantiles (blue, $M=2\cdot 10^3$) of \ref{['eq:sNV']}, including mean-proxy \ref{['eq:EsNV']} (green) at $T=2$ for Jacobi-type noise. Right: Zoom in on multiple averages for increasing numbers of samples. $\Delta x=3\cdot10^{-3}$, $\Delta t$ acc. to \ref{['eq:sCFL_det']}
  • Figure 5: Characteristics of Figure \ref{['fig:noise_comparison_high']}. The Characteristic Monte Carlo average ($M=2\cdot10^3$) in blue, with \ref{['eq:EsNV']} mostly overlaying the average in dashed green lines. $\Delta x=3\cdot10^{-3}$, $\Delta t$ acc. to \ref{['eq:sCFL_det']}.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Definition 1.1: Nonlocal weak entropy solution
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Existence, uniqueness and properties of (sNV)
  • Lemma 1.5: Stability estimate
  • Lemma 1.6: Measurability of solutions
  • proof
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • ...and 8 more