Alexander Keimer

Alexander Keimer
Friedrich-Alexander-University of Erlangen-Nürnberg | FAU · Department of Mathematics

Dr.

About

50
Publications
11,493
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472
Citations
Citations since 2017
43 Research Items
445 Citations
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2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
Introduction
My research interest lies in the study of nonlocal conservation laws in theory and applications. I study existence and uniqueness of solutions, properties of the solutions and their applications in applied sciences. I also work on traffic flow modelling and routing on large scale networks based on real-time information provided by routing applications.
Additional affiliations
April 2016 - December 2021
University of California, Berkeley
Position
  • Senior Researcher
Description
  • Mathematical models of traffic flow Hamilton-Jacobi equations in traffic flow Routing operators on macroscopic traffic flow models
April 2008 - December 2015
Friedrich-Alexander-University of Erlangen-Nürnberg
Position
  • Research Assistant
Description
  • Assistance and tutorials to several lectures for students of engineering Assistance and tutorials to several lectures for students of mathematics (e.g. Optimization, (Optimal) Control Theory, Linear Algebra, Measure Theory)

Publications

Publications (50)
Preprint
Full-text available
Networks are essential models in many applications such as information technology, chemistry, power systems, transportation, neuroscience, and social sciences. In light of such broad applicability, a general theory of dynamical systems on networks may capture shared concepts, and provide a setting for deriving abstract properties. To this end, we d...
Article
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In this note, we extend the known results on the existence and uniqueness of weak solutions to conservation laws with nonlocal flux. In case the nonlocal term is given by a convolution γ∗q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usep...
Preprint
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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 fi...
Preprint
Full-text available
In this contribution, we study the "Bando-follow the leader" car-following model, a second-order ordinary differential equation, for its well-posedness. Under suitable conditions, we provide existence and uniqueness results, and also bounds on the higher derivatives, i.e. velocity and acceleration. We then extend the result to the "reaction" delay...
Chapter
The technological advancements in terms of vehicle on-board sensors and actuators, as well as for infrastructures, open an unprecedented scenario for the management of vehicular traffic. We focus on the problem of smoothing traffic by controlling a small number of autonomous vehicles immersed in the bulk traffic stream. Specifically, we aim at diss...
Preprint
Full-text available
Model predictive control (MPC) is a powerful control method that handles dynamical systems with constraints. However, solving MPC iteratively in real time, i.e., implicit MPC, has been a challenge for 1) systems with low-latency requirements, 2) systems with limited computational resources, and 3) systems with fast and complex dynamics. To address...
Preprint
Full-text available
We study nonlocal conservation laws with a discontinuous flux function of regularity L^{\infty} (R) in the spatial variable and show existence and uniqueness of weak solutions in C ([0, T ] ; L^{1}_{loc}(\R)) , as well as related maximum principles. We achieve this well-posedness by a proper reformulation in terms of a fixed-point problem. This fix...
Article
We consider a class of nonlocal conservation laws with a second-order viscous regularization term which finds an application in modelling macroscopic traffic flow. The velocity function depends on a weighted average of the density ahead, where the averaging kernel is of exponential type. We show that, as the nonlocal impact and the viscosity parame...
Article
We study the exact boundary controllability of a class of nonlocal conservation laws modeling traffic flow. The velocity of the macroscopic dynamics depends on a weighted average of the traffic density ahead and the averaging kernel is of exponential type. Under specific assumptions, we show that the boundary controls can be used to steer the syste...
Preprint
Full-text available
Traffic assignment is one of the key approaches used to model the congestion patterns that arise in transportation networks. Since static traffic assignment does not have a notion of time dynamics, it is not designed to represent the complex dynamics of transportation networks as usage changes throughout the day. Dynamic traffic assignment methods...
Preprint
Full-text available
This contribution analyzes the widely used and well-known "intelligent driver model" (briefly IDM), which is a second order car-following model governed by a system of ordinary differential equations. Although this model was intensively studied in recent years for properly capturing traffic phenomena and driver braking behavior, a rigorous study of...
Preprint
Full-text available
In this note, we extend the known results on the existence and uniqueness of weak solutions to conservation laws with nonlocal flux. In case the nonlocal term is given by a convolution γ * q, we weaken the standard assumption on the kernel γ ∈ L ∞ (0, T); W 1,∞ (R) to the substantially more general condition γ ∈ L ∞ ((0, T); BV (R)), which allows f...
Preprint
Full-text available
We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solu...
Article
We study the initial value problem and the initial boundary value problem for nonlocal conservation laws. The nonlocal term is realized via a spatial integration of the solution between specified boundaries and affects the flux function of a given “local” conservation law in a multiplicative way. For a strictly convex flux function and strictly pos...
Preprint
Full-text available
We consider a class of nonlocal conservation laws with a second-order viscous regularization term which finds an application in modelling macroscopic traffic flow. The velocity function depends on a weighted average of the density ahead, where the averaging kernel is of exponential type. We show that, as the nonlocal reach and the viscosity paramet...
Article
Full-text available
In this contribution, we study the existence and uniqueness of nonlocal transport equations. The term "nonlocal" refers to the fact that the flux function's derivative will be integrated over a neighborhood of the corresponding space-time coordinate. We will demonstrate existence and uniqueness of weak solutions for T V \cap L \infty initial datum...
Preprint
Full-text available
We consider a system of nonlocal balance laws where every single balance law is coupled with the remaining ones by a nonlocal velocity function which takes into account the averaged density of all other equations as well as by a right hand "semi-linear" term. We show existence and uniqueness of weak solutions for small time horizon and a maximum pr...
Preprint
Full-text available
We study the exact boundary controllability of a class of nonlocal conservation laws modeling traffic flow. The velocity of the macroscopic dynamics depends on a weighted average of the traffic density ahead and the averaging kernel is of exponential type. Under specific assumptions, we show that the boundary controls can be used to steer the syste...
Preprint
We prove existence of solutions to a conservation law with nonlocal discontinuous flux modeling material flow on a conveyor belt. The discontinuity is with respect to the unknown function and arises in a dynamic velocity field which is active only at high densities and takes into account the effect of colliding parts though the nonlocal operator. T...
Chapter
Processes in the field of chemical engineering do not consist of one single step, but typically a high number of strongly interconnected unit operations linked with recycling streams. This inherent complexity further exacerbates when distributed particle properties, i.e., dispersity, must be considered, noteworthy being the case whenever particulat...
Preprint
Full-text available
The spread of COVID-19 at the end of 2019 with its delay properties requires improved dynamical models to forecast and evaluate specific measures in politics. As these policy measures require real-time information on the evolution of the diseases, quantities like delay due to incubation time and infectious period can not be neglected and need to be...
Article
In this review, we discuss routing algorithms for the dynamic traffic assignment (DTA) problem that assigns traffic flow in a given road network as realistically as possible. We present a new class of so-called routing operators that route traffic flow at intersections based on either real-time information about the status of the network or histori...
Article
Full-text available
In order to obtain high‐quality particulate products with tailored properties, process conditions and their evolution in time must be chosen appropriately. Although the efficiency of these products depends on their dispersity in several dimensions, in established processes particle size is usually the decisive variable to adjust. As part of the syn...
Article
In this study we present a reformulation for a broad class of population balance equations that model nucleation and size dependent growth. This formulation enables the definition of new numerical methods, which have two advantages compared to existing schemes in the literature (e.g. finite volume type methods and methods based on the evolution of...
Article
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We introduce a model dealing with conservation laws on networks and coupled boundary conditions at the junctions. In particular, we introduce buffers of fixed arbitrary size and time-dependent split ratios at the junctions, which represent how traffic is routed through the network, Having defined the dynamics at the level of conservation laws, we l...
Article
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This study considers nonlocal conservation laws in which the velocity depends nonlocally on the solution not in real time but in a time-delayed manner. Nonlocal refers to the fact that the velocity of the conservation law depends on the solution integrated over a specific area in space. In every model modelling human’s behavior a time delay as reac...
Preprint
Full-text available
In this contribution we present a reformulation of population balance equations which model nucleation and size dependent growth. This formulation allows to define new numerical methods, significantly superior to already existent schemes in the literature in the following way: i) Higher precision due to non-smoothing ii) less run-time in comparison...
Article
The impact of the recent increase in routing apps usage on road traffic remains uncertain to this day. The article introduces, for the first time, a criterion to evaluate a distance between an observed state of traffic and the user equilibrium of the traffic assignment: the average marginal regret. The average marginal regret provides a quantitativ...
Article
We show that for monotone initial datum the solution of nonlocal conservation laws converges to the entropy solution of the corresponding local conservation laws when the nonlocal reach tends to zero. This particularly covers the principle cases of conservation laws: shocks and rarefactions. The considered problem is addressed by studying the Entro...
Article
This article presents an extensive theoretical framework to mathematically defined and information-based routing operators, applied to the continuous-time dynamic traffic assignment problem. Because of the difficulty of the mathematical framework required to provide existence and uniqueness proofs of the solution to the problem in the presence of a...
Article
We consider a nonlocal conservation law on a bounded spatial domain and show existence and uniqueness of weak solutions for nonnegative flux function and left-hand-side boundary datum. The nonlocal term is located in the flux function of the conservation law, averaging the solution by means of an integral at every spatial coordinate and every time,...
Article
Full-text available
We consider a system of nonlocal balance laws, modeling a multi-commodity supply edge. The coupling of the different goods is realized via a "weighted work in progress" (WWIP), such that the velocity of every balance law also depends on the "load" of the other commodities. This WWIP represents nonlocal impact and due to it, the proposed model offer...
Article
In this article, we generalize some existence and uniqueness results in [22] for scalar nonlocal balance laws to multi-dimensional nonlocal balance laws. Also in the multi-dimensional case it is possible to get rid of the Entropy condition which is usually postulated to guarantee a unique weak solution of (local) balance laws but which we prove to...
Article
Full-text available
Over the last decade, the rise of the mobile internet and the usage of mobile devices has enabled ubiquitous traffic information. With the increased adoption of specific smartphone applications, the number of users of routing applications has become large enough to disrupt traffic flow patterns in a significant manner. Similarly, but at a slightly...
Article
We consider a class of nonlocal balance laws as initial value problems on a finite time horizon and show existence and uniqueness of the corresponding weak solutions. The description “nonlocal” refers to the velocity of the balance law that depends on the weighted integral over an area in space at any given time. Existence of a weak solution for in...
Article
We consider a system of scalar nonlocal conservation laws on networks that model a highly re-entrant multi-commodity manufacturing system as encountered in semiconductor production. Every single commodity is mod-eled by a nonlocal conservation law, and the corresponding PDEs are coupled via a collective load, the work in progress. We illustrate the...
Article
Full-text available
In this contribution the optimal boundary control problem for a first order nonlinear, nonlocal hyperbolic PDE is studied. Motivated by various applications ranging from re-entrant manufacturing systems to particle synthesis processes, we establish the regularity of solutions for $W^{1,p}$-data. Based on a general $L^2$ tracking type cost functiona...
Thesis
Full-text available
In this work we consider a class of nonlinear, nonlocal hyperbolic conservation laws. These equations have already been used several times to model supply chains and thus their regularity and stability properties have also been a subject of recent research. We extend this class of models to so called ``multi-commodity'' models, i.e. models that si...
Article
Full-text available
Gefördert durch die Bayerische Forschungsstiftung http://www.forschungsstiftung.de Zusammenfassung Im Rahmen des Forschungsprojekts "Innovative Algorithmen zur Supply Chain Optimierung", gefördert durch die Bayerische Forschungsstiftung, wird auf Basis von realen Daten der Firma Henkel ein einfaches Referenz-modell, das zum Testen von geeigneten Al...
Article
The Lavrentiev regularization method is a tool to improve the regularity of the Lagrange multipliers in pde constrained optimal control problems with state constraints. It has already been used for problems with parabolic and elliptic systems. In this paper we consider Lavrentiev regularization for problems with a hyperbolic system, namely the scal...
Article
Full-text available
We review different approaches in Robust Optimization in order to find robust solutions to minimum cost flow problems. We will consider uncertainty in both costs and supply/demand. Our goal is to find a com-putationally tractable approach which delivers a robust solution to the minimum cost flow problem with uncertainty.
Article
Full-text available
Gefördert durch die Bayerische Forschungsstiftung http://www.forschungsstiftung.de Zusammenfassung Im Rahmen des Forschungsprojekts "Innovative Algorithmen zur Supply Chain Optimierung", gefördert durch die Bayerische Forschungsstiftung, werden auf Basis von realen Daten der Firma Henkel verschiedene Refe-renzmodelle entwickelt. Die Daten repräsent...

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