arXiv subjects

Thomas Schillinger

Publications and source records attributed to Thomas Schillinger.

8 recordsLinked to original sources

arXiv2026

Optimal Inflow Control for Transport Equations with Uncertain Velocities and Demand

We study optimal inflow control for a linear transport equation subject to uncertainty in both time-dependent downstream demand and transport velocity. The random velocity induces a random travel time and thereby changes the relation between an inflow decision and the demand observed at its arrival time. For a fixed downstream observation window, we derive explicit optimal controls in the continuous-time setting and for piecewise constant controls. We decompose the irreducible stochastic error into contributions from demand and velocity uncertainty and show that the latter admits a bound that is linear in the velocity variance. As a computationally attractive alternative, we analyze a deterministic mean-velocity proxy whose additional performance loss admits a higher-order bound. We also establish Lipschitz stability of the optimal control with respect to perturbations of the velocity law in the Wasserstein distance. Numerical experiments illustrate the analytical results and investigate the proxy strategy for an uncertain nonlocal transport model beyond the linear theory.

math.OC↗

arXiv2026

A Hyperbolic Transport Model for Passenger Flow on Tram Networks

We introduce a modeling framework for an urban tram network based on a hyperbolic partial differential equation describing the transport of passengers along the network, coupled with a family of stochastic processes representing passenger boarding. Solutions are considered in a measure-valued sense. The system is further extended and subjected to uncertainties such as delays and service interruptions through a numerical study. Its robustness is assessed using appropriate risk measures.

math.NA↗

arXiv2024

Probabilistic modeling of car traffic accidents

We introduce a counting process to model the random occurrence in time of car traffic accidents, taking into account some aspects of the self-excitation typical of this phenomenon. By combining methods from probability and differential equations, we study this stochastic process in terms of its statistical moments and large-time trend. Moreover, we derive analytically the probability density functions of the times of occurrence of traffic accidents and of the time elapsing between two consecutive accidents. Finally, we demonstrate the suitability of our modelling approach by means of numerical simulations, which address also a comparison with real data of weekly trends of traffic accidents.

physics.soc-ph↗

arXiv2024

Inverse demand tracking in transportation networks

This paper deals with the reconstruction of the desired demand in an optimal control problem, stated over a tree-shaped transportation network which is governed by a linear hyperbolic conservation law. As desired demands typically undergo fluctuations due to seasonality or unexpected events making short-term adjustments necessary, such an approach can exemplary be used for forecasting from past data. We suggest to model this problem as a so-called inverse optimal control problem, i.e., a hierarchical optimization problem whose inner problem is the optimal control problem and whose outer problem is the reconstruction problem. In order to guarantee the existence of solutions in the function space framework, the hyperbolic conservation law is interpreted in weak sense allowing for control functions in Lebesgue spaces. For the computational treatment of the model, we transfer the hierarchical problem into a nonsmooth single-level one by plugging the uniquely determined solution of the inner optimal control problem into the outer reconstruction problem before applying techniques from nonsmooth optimization. Some numerical experiments are presented to visualize various features of the model including different types of noise in the demand and strategies of how to observe the network in order to obtain good reconstructions of the desired demand.

math.OC↗

arXiv2023

Data-inspired modeling of accidents in traffic flow networks using the Hawkes process

We consider hyperbolic partial differential equations (PDEs) for a dynamic description of the traffic behavior in road networks. These equations are coupled to a Hawkes process that models traffic accidents taking into account their self-excitation property which means that accidents are more likely in areas in which another accident just occurred. We discuss how both model components interact and influence each other. A data analysis reveals the self-excitation property of accidents and determines further parameters. Numerical simulations using risk measures underline and conclude the discussion of traffic accident effects in our model.

math.NA↗

arXiv2022

Control Strategies for Transport Networks under Demand Uncertainty

In this article, we consider transport networks with uncertain demands. Network dynamics are given by linear hyperbolic partial differential equations and suitable coupling conditions, while demands are incorporated as solutions to stochastic differential equations. For the demand satisfaction, we solve a constrained optimal control problem. Controls in terms of network inputs are then calculated explicitly for different assumptions. Numerical simulations are performed to underline the theoretical results.

math.OC↗

arXiv2022

Stochastic optimal control for nonlinear damped network dynamics

We present a stochastic optimal control problem for a tree network. The dynamics of the network are governed by transport equations with a special emphasis on the non-linear damping function. Demand profiles at the network sinks are modelled by a stochastic differential equations. An explicit optimal inflow into the network is determined and numerical simulations are presented to show the effects for different choices of the non-linear damping.

math.OC↗

arXiv2021

Microscopic and Macroscopic Traffic Flow Models including Random Accidents

We introduce microscopic and macroscopic stochastic traffic models including traffic accidents. The microscopic model is based on a Follow-the-Leader approach whereas the macroscopic model is described by a scalar conservation law with space dependent flux function. Accidents are introduced as interruptions of a deterministic evolution and are directly linked to the traffic situation. Based on a Lax-Friedrichs discretization convergence of the microscopic model to the macroscopic model is shown. Numerical simulations are presented to compare the above models and show their convergence behaviour.

math.PR↗