Michael Herty

RWTH Aachen University · Faculty of Mathematics, Computer Science and Natural Sciences

We are interested in the feedback stabilization of general linear multi-dimensional first order hyperbolic systems $\mathbb{R}^d$. Using a novel Lyapunov function taking into account the multi-dimensional geometry we show stabilization in $L^2$ for the arising system using a suitable feedback control. We show the applicability discussing the barotr...

We study the turnpike phenomenon for optimal control problems with mean field dynamics that are obtained as the limit $N\rightarrow \infty$ of systems governed by a large number $N$ of ordinary differential equations. We show that the optimal control problems with with large time horizons give rise to a turnpike structure of the optimal state and t...

Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of kernels and function spaces generated by kernels, so called reproducing kernel Hilbert spaces. Motivated by re...

The application of machine learning methods has great potential for accelerating technical processes and reducing costs. Based on existing data, neural networks can be trained to predict the results of expensive experimental studies or time-consuming simulations reliably and thus replace them. This article introduces a new neural network that is ba...

The mathematical modeling and the stability analysis of multi-lane traffic in the macroscopic scale is considered. We propose a new first order model derived from microscopic dynamics with lane changing, leading to a coupled system of hyperbolic balance laws. The macroscopic limit is derived without assuming ad hoc space and time scalings. The anal...

Residual deep neural networks (ResNets) are mathematically described as interacting particle systems. In the case of infinitely many layers the ResNet leads to a system of coupled system of ordinary differential equations known as neural differential equations. For large scale input data we derive a mean–field limit and show well–posedness of the r...

Traffic flow on networks requires knowledge on the behavior across traffic intersections. For macroscopic models based on hyperbolic conservation laws there exist nowadays many ad-hoc models describing this behavior. Based on car trajectory data we propose a novel framework combining data-fitted models with the requirements of consistent coupling c...

We present several recent results and generalizations that have been obtained in the theory of collision-freeness studied in Zerz and Herty (2019). A nonlinear ODE system ẋ(t) = f(x(t)) is called collision-free if the solution to the initial value problem with x(0) = x⁰ has distinct components for all times t whenever the initial state x⁰ has disti...

This chapter aims to present a review of existing results on theoretical and numerical aspects of the control of hyperbolic balance laws. Several aspects will be covered including the differential calculus in the presence of weak entropic discontinuous solutions in the scalar and system's case as well as results on the nonconservative adjoint equat...

This review aims to present recent developments in modeling and control of multiagent systems. A particular focus is set on crowd dynamics characterized by complex interactions among agents, also called social interactions, and large-scale systems. Specifically, in a crowd each individual agent interacts with a field generated by the other agents a...

We consider control strategies for large--scale interacting agent systems under uncertainty. The particular focus is on the design of robust controls that allow to bound the variance of the controlled system over time. To this end we consider $\mathcal{H}_\infty$ control strategies on the agent and mean field description of the system. We show a bo...

We propose novel less diffusive schemes for conservative one- and two-dimensional hyperbolic systems of nonlinear partial differential equations (PDEs). The main challenges in the development of accurate and robust numerical methods for the studied systems come from the complicated wave structures, such as shocks, rarefactions and contact discontin...

We propose a novel scheme to numerically solve scalar conservation laws on networks without the necessity to solve Riemann problems at the junction. The scheme is derived using the relaxation system introduced in [Jin and Xin, Comm. Pure Appl. Math. 48(3), 235-276 (1995)] and taking the relaxation limit also at the nodes of the network. The scheme...

CALL FOR PAPERS: We are pleased to announce that the 9th International Conference on Modeling, Simulation and Applied Optimization will be held in Marrakesh, Morocco on April 26th-28th 2023 (www.icmsao.org/). ICMSAO provides a venue for engineers, mathematicians, and scientists from all over the world to share their latest research results in the f...

In this work we are interested in stochastic particle methods for multi-objective optimization. The problem is formulated using parametrized, single-objective sub-problems which are solved simultaneously. To this end a consensus based multi-objective optimization method on the search space combined with an additional heuristic strategy to adapt par...

Machine learning methods have shown potential for the optimization of production processes. Due to the complex relationships often inherent in those processes, the success of such methods is often uncertain and unreliable. Therefore, understanding the (algorithmic) behavior and results of machine learning methods is crucial to improve the predictio...

To enable safe operations in applications such as rocket combustion chambers, the materials require cooling to avoid material damage. Here, transpiration cooling is a promising cooling technique. Numerous studies investigate possibilities to simulate and evaluate the complex cooling mechanism. One naturally arising question is the amount of coolant...

We are interested in the feedback stabilization of systems described by Hamilton-Jacobi type equations in $\R^n$. A reformulation leads to a a stabilization problem for a multi-dimensional system of $n$ hyperbolic partial differential equations. Using a novel Lyapunov function taking into account the multi-dimensional geometry we show stabilization...

We propose novel less diffusive schemes for conservative one- and two-dimensional hyperbolic systems of nonlinear partial differential equations (PDEs). The main challenges in the development of accurate and robust numerical methods for the studied systems come from the complicated wave structures, such as shocks, rarefactions and contact discontin...

Neuroscience models commonly have a high number of degrees of freedom and only specific regions within the parameter space are able to produce dynamics of interest. This makes the development of tools and strategies to efficiently find these regions of high importance to advance brain research. Exploring the high dimensional parameter space using n...

Among the class of nonlinear particle filtering methods, the Ensemble Kalman Filter (EnKF) has gained recent attention for its use in solving inverse problems. We review the original method and discuss recent developments in particular in view of the limit for infinitely particles and extensions towards stability analysis and multi–objective optimi...

Among the class of nonlinear particle filtering methods, the Ensemble Kalman Filter (EnKF) has gained recent attention for its use in solving inverse problems. We review the original method and discuss recent developments in particular in view of the limit for infinitely particles and extensions towards stability analysis and multi--objective optim...

We introduce an approach and a software tool for solving coupled energy networks composed of gas and electric power networks. Those networks are coupled to stochastic fluctuations to address possibly fluctuating demand due to fluctuating demands and supplies. Through computational results, the presented approach is tested on networks of realistic s...

We present a multi-agent algorithm for multi-objective optimization problems, which extends the class of consensus-based optimization methods and relies on a scalarization strategy. The optimization is achieved by a set of interacting agents exploring the search space and attempting to solve all scalar sub-problems simultaneously. We show that thos...

A multiresolution analysis for solving stochastic conservation laws is proposed. Using a novel adaptation strategy and a higher dimensional deterministic problem, a discontinuous Galerkin (DG) solver is derived. A multiresolution analysis of the DG spaces for the proposed adaptation strategy is presented. Numerical results show that in the case of...

We introduce a bilevel problem of the optimal control of an interacting agent system that can be interpreted as as Stackelberg game with a large number of followers. It is shown that the model is well posed by providing conditions that allow to formally reduce the problem to a single level unconstrained problem. The mean-field limit is derived form...

We are interested in ensemble methods to solve multi-objective optimization problems. An ensemble Kalman method is proposed to solve a formulation of the nonlinear problem using a weighted function approach. An analysis of the mean field limit of the ensemble method yields an explicit update formula for the weights. Numerical examples show the impr...

To enable safe operations in applications such as rocket combustion chambers, the materials require cooling to avoid material damage. Here, transpiration cooling is a promising cooling technique. Numerous studies investigate possibilities to simulate and evaluate the complex cooling mechanism. One naturally arising question is the amount of coolant...

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 microsco...

We present a positive and negative stabilization results for a semilinear model of gas flow in pipelines. For feedback boundary conditions, we obtain unconditional stabilization in the absence and conditional instability in the presence of the source term. We also obtain unconditional instability for the corresponding quasilinear model given by the...

Neuroscience models commonly have a high number of degrees of freedom and only specific regions within the parameter space are able to produce dynamics of interest. This makes the development of tools and strategies to efficiently find these regions of high importance to advance brain research. Exploring the high dimensional parameter space using n...

We are interested in numerical schemes for the simulation of large scale gas networks. Gas transport is described by a simplified Euler equation with a general equation of state for the pressure, including in particular the isentropic as well as the isothermal case. The numerical scheme is based on transformation of conservative variables in Rieman...

In this chapter we survey recent progress on mathematical results on gas flow in pipe networks with a special focus on questions of control and stabilization. We briefly present the modeling of gas flow and coupling conditions for flow through vertices of a network. Our main focus is on gas models for spatially one-dimensional flow governed by hype...

We consider kinetic vehicular traffic flow models of BGK type [24]. Considering different spatial and temporal scales, those models allow to derive a hierarchy of traffic models including a hydrodynamic description. In this paper, the kinetic BGK–model is extended by introducing a parametric stochastic variable to describe possible uncertainty in t...

Nowadays, neural networks are widely used in many applications as artificial intelligence models for learning tasks. Since typically neural networks process a very large amount of data, it is convenient to formulate them within the mean-field and kinetic theory. In this work we focus on a particular class of neural networks, i.e. the residual neura...

We study a hierarchy of models based on kinetic equations for the descriptions of traffic flow in presence of autonomous and human–driven vehicles. The autonomous cars considered in this paper are thought of as vehicles endowed with some degree of autonomous driving which decreases the stochasticity of the drivers' behavior. Compared to the existin...

We propose a novel scheme to numerically solve scalar conservation laws on networks without the necessity to solve Riemann problems at the junction. The scheme is derived using the relaxation system introduced in [Jin and Xin, Comm. Pure. Appl. Math. 48 (1995), 235-276] and taking the relaxation limit also at the nodes of the network. The scheme is...

Residual deep neural networks (ResNets) are mathematically described as interacting particle systems. In the case of infinitely many layers the ResNet leads to a system of coupled system of ordinary differential equations known as neural differential equations. For large scale input data we derive a mean--field limit and show well--posedness of the...

In this work we are interested in the construction of numerical methods for high dimensional constrained nonlinear optimization problems by particle-based gradient-free techniques. A consensus-based optimization (CBO) approach combined with suitable penalization techniques is introduced for this purpose. The method relies on a reformulation of the...

We present a novel approach to determine the evolution of level sets under uncertainties in the velocity fields. This leads to a stochastic description of the level sets. To compute the quantiles of random level sets, we use the stochastic Galerkin method for a hyperbolic reformulation of the level-set equations. A novel intrusive Galerkin formulat...

We present a Stackelberg game with a large number of followers where every player-leader and followers-has its own state and control. We derive the mean field limit of infinitely many followers and address aspects of consistent control. Finally, we propose a numerical method based on the derived model and present numerical results.

We consider kinetic vehicular traffic flow models of BGK type. Considering different spatial and temporal scales, those models allow to derive a hierarchy of traffic models including a hydrodynamic description. In this paper, the kinetic BGK-model is extended by introducing a parametric stochastic variable to describe possible uncertainty in traffi...

Die Anwendung von Methoden des Maschinellen Lernens birgt großes Potenzial für die Beschleunigung technischer Prozesse und die damit verbundene Reduktion von Kosten. Auf Basis vorhandener Daten können neuronale Netze trainiert werden, die Ergebnisse teurer experimenteller Untersuchungen oder zeitaufwändiger Simulationen zuverlässig vorhersagen und...

We introduce an approach and a software tool for solving coupled energy networks composed of gas and electric power networks. Those networks are coupled to stochastic fluctuations to address possibly fluctuating demand due to fluctuating demands and supplies. Through computational results the presented approach is tested on networks of realistic si...

We study a hierarchy of models based on kinetic equations for the descriptions of traffic flow in presence of automated and human--driven vehicles. For a sufficiently large penetration rate of autonomous vehicles stabilization of traffic phenomena, like stop--and--go waves, is observed. The influence of the penetration rate is investigated analytic...

Models for the evolution of hidden microstructural states are needed for fast prediction and closed-loop control of workpiece properties. Machine learning allows to obtain models by learning from experimental data, avoiding the limitations of explicitly defined physics-based models. However, the identification of the parameters of deep network stru...

We investigate the propagation of uncertainties in the Aw-Rascle-Zhang model, which belongs to a class of second order traffic flow models described by a system of nonlinear hyperbolic equations. The stochastic quantities are expanded in terms of wavelet-based series expansions. Then, they are projected to obtain a deterministic system for the coef...

We propose a novel framework for model-order reduction of hyperbolic differential equations. The approach combines a relaxation formulation of the hyperbolic equations with a discretization using shifted base functions. Model-order reduction techniques are then applied to the resulting system of coupled ordinary differential equations. On computati...

We are interested in the control of forming processes for nonlinear material models. To develop an online control we derive a novel feedback law and prove a stabilization result. The derivation of the feedback control law is based on a Laypunov analysis of the time-dependent viscoplastic material models. The derivation uses the structure of the und...

Close-die forging usually has the goal of shaping a workpiece at reduced forming forces and to set the properties for the application at hand utilizing the microstructural changes occurring at high homologous temperatures. The evolution of the property-determining microstructure can be treated as a dynamic system during hot forming. It is controlle...

We investigate the propagation of uncertainties in the Aw-Rascle-Zhang model, which belongs to a class of second order traffic flow models described by a system of nonlinear hyperbolic equations. The stochastic quantities are expanded in terms of wavelet-based series expansions. Then, they are projected to obtain a deterministic system for the coef...

The modelling of gas networks requires the development of coupling techniques at junctions. Recent work on the coupling of hyperbolic systems based on solving two half Riemann problems can be useful also for the coupling issue in gas networks. This strategy is exemplified here for the coupling of a fluid with a solid modelled by the Euler equations...

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 propose a numerical discretization of the scalar conse...

We are interested in the development of an algorithmic differentiation framework for computing approximations to tangent vectors to scalar and systems of hyperbolic partial differential equations. The main difficulty of such a numerical method is the presence of shock waves that are resolved by proposing a numerical discretization of the calculus i...

The synthesis of control laws for interacting agent-based dynamics and their mean-field limit is studied. A linearization-based approach is used for the computation of sub-optimal feedback laws obtained from the solution of differential matrix Riccati equations. Quantification of dynamic performance of such control laws leads to theoretical estimat...

This edited monograph offers a summary of future mathematical methods supporting the recent energy sector transformation. It collects current contributions on innovative methods and algorithms. Advances in mathematical techniques and scientific computing methods are presented centering around economic aspects, technical realization and large-scale...

In this work we investigate the ability of a kinetic approach for traffic dynamics to predict speed distributions obtained through rough data. The present approach adopts the formalism of uncertainty quantification, since reaction strengths are uncertain and linked to different types of driver behaviour or different classes of vehicles present in t...

In this paper, properties of a recently proposed mathematical model for data flow in large-scale asynchronous computer systems are analyzed. In particular, the existence of special weak solutions based on propagating fronts is established. Qualitative properties of these solutions are investigated, both theoretically and numerically.

We develop online feedback control for forming processes governed by nonlinear material models. A novel approach based on a description using partial differential equation governing time-dependent viscoplastic deformations is proposed. The derivation of the feedback control law is based on a Lyapunov analysis of the linearised partial differential...

The heterogeneous microstructure in metallic components results in locally varying fatigue strength. Metal fatigue strongly depends on size and shape of non-metallic inclusions and pores, commonly referred to as defects. Nodular cast iron (NCI) contains graphite inclusions (nodules) whose shape and frequency influence the fatigue strength. Fatigue...

We present a linear-quadratic Stackelberg game with a large number of followers and we also derive the mean field limit of infinitely many followers. The relation between optimization and mean-field limit is studied and conditions for consistency are established. Finally, we propose a numerical method based on the derived models and present numeric...

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...

We survey recent results on controlled particle systems. The control aspect introduces new challenges in the discussion of properties and suitable mean field limits. Some of the aspects are highlighted in a detailed discussion of a particular controlled particle dynamics. The applied techniques are shown on this simple problem to illustrate the bas...

We analyse the existence of Nash equilibria for a class of quadratic multi-leader-follower games using the nonsmooth best response function. To overcome the challenge of nonsmoothness, we pursue a smoothing approach resulting in a reformulation as a smooth Nash equilibrium problem. The existence and uniqueness of solutions are proven for all smooth...

In this article we survey recent progress on mathematical results on gas flow in pipe networks with a special focus on questions of control and stabilization. We briefly present the modeling of gas flow and coupling conditions for flow through vertices of a network. Our main focus is on gas models for spatially one-dimensional flow governed by hype...

The efficient numerical treatment of convex quadratic mixed-integer optimization poses a challenging problem. Therefore, we introduce a method based on the duality principle for convex problems to derive suitable lower bounds that can used to select the next node to be solved within the branch-and-bound tree. Numerical results indicate that the new...

We are interested in the development of an algorithmic differentiation framework for computing approximations to tangent vectors to scalar and systems of hyperbolic partial differential equations. The main difficulty of such a numerical method is the presence of shock waves that are resolved by proposing a numerical discretization of the calculus i...

The ensemble Kalman filter belongs to the class of iterative particle filtering methods and can be used for solving control--to--observable inverse problems. In recent years several continuous limits in the number of iteration and particles have been performed in order to study properties of the method. In particular, a one--dimensional linear stab...

We develop a class of numerical methods for solving optimal control problems governed by nonlinear conservation laws in two space dimensions. The relaxation approximation is used to transform the nonlinear problem to a semi-linear diagonalizable system with source terms. The relaxing system is hyperbolic and it can be numerically solved without nee...

In this paper, a supply chain with a manufacturer and an Internet sharing platform who leases products to customers is considered. We solve the optimization problem in two structures, namely, without and with recycling. In basic structure, manufacturer produces new products and sells them to the platform. Then platform leases products to customers...

This paper deals with the Aw-Rascle-Zhang model for traffic flow on uni-directional road networks. For the conservation of the mass and the generalized momentum, we construct weak solutions for Riemann problems at the junctions. We particularly focus on a novel approximation to the homogenized pressure by introducing an additional equation for the...

We are interested in numerical schemes for the simulation of large scale gas networks. Typical models are based on the isentropic Euler equations with realistic gas constant. The numerical scheme is based on transformation of conservative variables in Riemann invariants and its corresponding numerical dsicretization. A particular, novelty of the pr...

The heterogeneous microstructure in metallic components results in locally varying fatigue strength. Metal fatigue strongly depends on size and shape of non-metallic inclusions and pores, commonly referred to as "defects". Nodular cast iron (NCI) contains graphite inclusions (nodules) whose shape and frequency influence the fatigue strength. Fatigu...

We introduce new coupling conditions for isentropic flow on networks based on an artificial density at the junction. The new coupling condition can be formally derived from a kinetic model by imposing a condition on energy dissipation. Existence and uniqueness of solutions to the generalized Riemann and Cauchy problem are proven. The result for the...

We develop a class of numerical methods for solving optimal control problems governed by nonlinear conservation laws in two space dimensions. The relaxation approximation is used to transform the non-linear problem to a semi-linear diagonalizable system with source terms. The relaxing system is hyperbolic and it can be numerically solved without ne...

We introduce new coupling conditions for isentropic flow on networks based on an artificial density at the junction. The new coupling conditions can be derived from a kinetic model by imposing a condition on energy dissipation. Existence and uniqueness of solutions to the generalized Riemann and Cauchy problem are proven. The result for the general...

The central nervous system (CNS) is an immune-privileged compartment that is separated from the circulating blood and the peripheral organs by the blood–brain and the blood–cerebrospinal fluid (CSF) barriers. Transmigration of lymphocyte subsets across these barriers and their activation/differentiation within the periphery and intrathecal compartm...

We introduce a novel Lyapunov function for stabilization of linear Vlasov--Fokker--Planck type equations with stiff source term. Contrary to existing results relying on transport properties to obtain stabilization, we present results based on hypocoercivity analysis for the Fokker--Planck operator. The existing estimates are extended to derive suit...

We consider the flow of gas through networks of pipelines. A hierarchy of models for the gas flow is available. The most accurate model is the pde system given by the 1-d Euler equations. For large-scale optimization problems, simplifications of this model are necessary. Here we propose a new model that is derived for high-pressure flows that are c...