arXiv subjects

Simone Göttlich

Publications and source records attributed to Simone Göttlich.

At least 37 records · Page 2Linked to original sources

arXiv2023

Lyapunov stabilization for nonlocal traffic flow models

Using a nonlocal second-order traffic flow model we present an approach to control the dynamics towards a steady state. The system is controlled by the leading vehicle driving at a prescribed velocity and also determines the steady state. Thereby, we consider both, the microscopic and macroscopic scales. We show that the fixed point of the microscopic traffic flow model is asymptotically stable for any kernel function. Then, we present Lyapunov functions for both, the microscopic and macroscopic scale, and compute the explicit rates at which the vehicles influenced by the nonlocal term tend towards the stationary solution. We obtain the stabilization effect for a constant kernel function and arbitrary initial data or concave kernels and monotone initial data. Numerical examples demonstrate the theoretical results.

math.OC↗

arXiv2022

Conservation laws with nonlocal velocity -- the singular limit problem

We consider conservation laws with nonlocal velocity and show for nonlocal weights of exponential type that the unique solutions converge in a weak or strong sense (dependent on the regularity of the velocity) to the entropy solution of the local conservation law when the nonlocal weight approaches a Dirac distribution. To this end, we establish first a uniform total variation estimate on the nonlocal velocity which enables it to prove that the nonlocal solution is entropy admissible in the limit. For the entropy solution, we use a tailored entropy flux pair which allows the usage of only one entropy to obtain uniqueness (given some additional constraints). For general weights, we show that monotonicity of the initial datum is preserved over time which enables it to prove the convergence to the local entropy solution for rather general kernels and monotone initial datum as well. This covers the archetypes of local conservation laws: Shock waves and rarefactions. It also underlines that a ``nonlocal in the velocity'' approximation might be better suited to approximate local conservation laws than a nonlocal in the solution approximation where such monotonicity does only hold for specific velocities.

math.AP↗

arXiv2022

Conservation laws with discontinuous flux function on networks: a splitting algorithm

In this article, we present an extension of the splitting algorithm proposed in [22] to networks of conservation laws with piecewise linear discontinuous flux functions in the unknown. We start with the discussion of a suitable Riemann solver at the junction and then describe a strategy how to use the splitting algorithm on the network. In particular, we focus on two types of junctions, i.e., junctions where the number of outgoing roads does not exceed the number of incoming roads (dispersing type) and junctions with two incoming and one outgoing road (merging type). Finally, numerical examples demonstrate the accuracy of the splitting algorithm by comparisons to the exact solution and other approaches used in the literature.

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

Parameter calibration with Consensus-based Optimization for interaction dynamics driven by neural networks

We calibrate parameters of neural networks that model forces in interaction dynamics with the help of the Consensus-based global optimization method (CBO). We state the general framework of interaction particle systems driven by neural networks and test the proposed method with a real dataset from the ESIMAS traffic experiment. The resulting forces are compared to well-known physical interaction forces. Moreover, we compare the performance of the proposed calibration process to the one in [4] which uses a stochastic gradient descent algorithm.

math.OC↗

arXiv2021

A combined first and second order model for a junction with ramp buffer

Second order macroscopic traffic flow models are able to reproduce the so-called capacity drop effect, i.e., the phenomenon that the outflow of a congested region is substantially lower than the maximum achievable flow. Within this work, we propose a first order model for a junction with ramp buffer that is solely modified at the intersection so that the capacity drop is captured. Theoretical investigations motivate the new choice of coupling conditions and illustrate the difference to purely first and second order models. The numerical example considering the optimal control of the onramp merging into a main road highlights that the combined model generates similar results as the second order model.

physics.soc-ph↗

arXiv2021

Network models for nonlocal traffic flow

We present a network formulation for a traffic flow model with nonlocal velocity in the flux function. The modeling framework includes suitable coupling conditions at intersections to either ensure maximum flux or distribution parameters. Based on an upwind type numerical scheme, we prove the maximum principle and the existence of weak solutions on networks. We also investigate the limiting behavior of the proposed models when the nonlocal influence tends to infinity. Numerical examples show the difference between the proposed coupling conditions and a comparison to the Lighthill-Whitham-Richards network model.

math.NA↗

arXiv2021

Penalty alternating direction methods for mixed-integer optimal control with combinatorial constraints

We consider mixed-integer optimal control problems with combinatorial constraints that couple over time such as minimum dwell times. We analyze a lifting and decomposition approach into a mixed-integer optimal control problem without combinatorial constraints and a mixed-integer problem for the combinatorial constraints in the control space. Both problems can be solved very efficiently with existing methods such as outer convexification with sum-up-rounding strategies and mixed-integer linear programming techniques. The coupling is handled using a penalty-approach. We provide an exactness result for the penalty which yields a solution approach that convergences to partial minima. We compare the quality of these dedicated points with those of other heuristics amongst an academic example and also for the optimization of electric transmission lines with switching of the network topology for flow reallocation in order to satisfy demands.

math.OC↗

arXiv2021

Space mapping-based optimization with the macroscopic limit of interacting particle systems

We propose a space mapping-based optimization algorithm for microscopic interacting particle dynamics which are inappropriate for direct optimization. This is of relevance for example in applications with bounded domains such that the microscopic optimization is difficult. The space mapping algorithm exploits the relationship of the microscopic description of the interacting particle system and the corresponding macroscopic description as partial differential equation in the "many particle limit". We validate the approach with the help of a toy problem that allows for direct optimization. Then we study the performance of the algorithm in two applications. An evacuation dynamic is considered and the transportation of goods on a conveyor belt is optimized. The numerical results underline the feasibility of the proposed approach.

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↗

arXiv2021

Optimal control for interacting particle systems driven by neural networks

We propose a neural network approach to model general interaction dynamics and an adjoint based stochastic gradient descent algorithm to calibrate its parameters. The parameter calibration problem is considered as optimal control problem that is investigated from a theoretical and numerical point of view. We prove the existence of optimal controls, derive the corresponding first order optimality system and formulate a stochastic gradient descent algorithm to identify parameters for given data sets. To validate the approach we use real data sets from traffic and crowd dynamics to fit the parameters. The results are compared to forces corresponding to well-known interaction models such as the Lighthill-Whitham-Richards model for traffic and the social force model for crowd motion.

math.OC↗

arXiv2021

On the relation of powerflow and Telegrapher's equations: continuous and numerical Lyapunov stability

In this contribution we analyze the exponential stability of power networks modeled with the Telegrapher's equations as a system of balance laws on the edges. We show the equivalence of periodic solutions of these Telegrapher's equations and solutions to the well-established powerflow equations. In addition we provide a second-order accurate numerical scheme to integrate the powerflow equations and show (up to the boundary conditions) Lyapunov stability of the scheme.

math.NA↗

arXiv2021

Density dependent diffusion models for the interaction of particle ensembles with boundaries

The transition from a microscopic model for the movement of many particles to a macroscopic continuum model for a density flow is studied. The microscopic model for the free flow is completely deterministic, described by an interaction potential that leads to a coherent motion where all particles move in the same direction with the same speed known as a flock. Interaction of the flock with boundaries, obstacles and other flocks leads to a temporary destruction of the coherent motion that macroscopically can be modeled through density dependent diffusion. The resulting macroscopic model is an advection-diffusion equation for the particle density whose diffusion coefficient is density dependent. Examples describing i) the interaction of material flow on a conveyor belt with an obstacle that redirects or restricts the material flow and ii) the interaction of flocks (of fish or birds) with boundaries and iii) the scattering of two flocks as they bounce off each other are discussed. In each case, the advection-diffusion equation is strictly hyperbolic before and after the interaction while the interaction phase is described by a parabolic equation. A numerical algorithm to solve the advection-diffusion equation through the transition is presented.

cond-mat.stat-mech↗

arXiv2020

Nonlocal approaches for multilane traffic models

We present a multilane traffic model based on balance laws, where the nonlocal source term is used to describe the lane changing rate. The modelling framework includes the consideration of local and nonlocal flux functions. Based on a Godunov type numerical scheme, we provide BV estimates and a discrete entropy inequality. Together with the $L^1$-contractivity property, we prove existence and uniqueness of weak solutions. Numerical examples show the nonlocal impact compared to local flux functions and local sources.

math.NA↗

arXiv2020

Input-to-state stability of a scalar conservation law with nonlocal velocity

In this paper, we study input-to-state stability (ISS) of an equilibrium for a scalar conservation law with nonlocal velocity and measurement error arising in a highly re-entrant manufacturing system. By using a suitable Lyapunov function, we prove sufficient and necessary conditions on ISS. We also analyze the numerical discretization of ISS for a discrete scalar conservation law with nonlocal velocity and measurement error. A suitable discretized Lyapunov function is also analyzed to provide ISS of an equilibrium for the numerical approximation. Finally, we show numerical simulations to validate the theoretical findings.

math.OC↗

arXiv2020

A two-dimensional multi-class traffic flow model

The aim of this work is to introduce a two-dimensional macroscopic traffic model for multiple populations of vehicles. Starting from the paper [20], where a two-dimensional model for a single class of vehicles is proposed, we extend the dynamics to a multi-class model leading to a coupled system of conservation laws in two space dimensions. Besides the study of the Riemann problems we also present a Lax-Friedrichs type discretization scheme recovering the theoretical results by means of numerical tests. We calibrate the multi-class model with real data and compare the fitted model to the real trajectories. Finally, we test the ability of the model to simulate the overtaking of vehicles.

math.NA↗

arXiv2020

The food seeking behavior of slime mold: a macroscopic approach

Starting from a particle model we derive a macroscopic aggregation-diffusion equation for the evolution of slime mold under the assumption of propagation of chaos in the large particle limit. We analyze properties of the macroscopic model in the stationary case and study the behavior of the slime mold between food sources. The efficient numerical simulation of the aggregation-diffusion equation allows for a detailed analysis of the interplay between the different regimes drift, interaction and diffusion.

nlin.AO↗