arXiv subjects

Claudia Totzeck

Publications and source records attributed to Claudia Totzeck.

At least 19 recordsLinked to original sources

arXiv2026

Structure-Preserving Data-Driven Identification of Port-Hamiltonian Differential-Algebraic Systems

We present a data-driven approach to identifying linear index-1 differential-algebraic pH systems (pH-DAEs) based on input-output measurements. In comparison to the identification of port-Hamiltonian (pH) systems, the algebraic constraint and the index condition pose additional challenges. First, we establish a structure-preserving formulation of the considered pH-DAE class and derive an implicit midpoint discretization that preserves the algebraic constraints and discrete dissipation inequality. We formulate the identification problem as a regularized least-squares minimization problem subject to the pH-DAE dynamics. Exploiting the index-1 structure, we reduce the constrained problem to an unconstrained optimization problem over the system parameters while preserving the port-Hamiltonian structure. Next, we derive an adjoint-based formulation to efficiently evaluate the gradient of the resulting reduced cost functional. This enables us to use gradient-based optimization methods for parameter estimation. Under suitable assumptions on the admissible parameter set, the existence of a minimizer is established. Numerical experiments demonstrate that the proposed approach can identify surrogate pH-DAE systems that accurately reproduce the input-output behavior of reference systems. Further investigations show the approach's potential for identifying reduced-order surrogate models. Cross-validation with independent input signals confirms the predictive capability of the identified models.

math.NA↗

arXiv2026

Feedback approaches for set-point stabilization of interacting particle systems

We explore the problem of asymptotically stabilizing a class of interacting particle systems to a prescribed particle configuration with zero velocity. The class of interacting particle systems is motivated by the Cucker-Smale model on the mean-field level. Rather than designing a feedback by manually solving an IDA-PBC matching equation, we derive a feedback using an instantaneous rolling horizon control algorithm. We show that this feedback preserves the port-Hamiltonian structure of the interacting particle system and therefore allows for an interpretation in the IDA-PBC framework. Additionally, the energy balance of the closed-loop Hamiltonian provides an important tool to prove convergence of the particles' velocities and interaction forces in discrete and continuous time. If the control operator is surjective, we additionally prove convergence of the particle positions to the prescribed configuration.

math.OC↗

arXiv2026

A Consensus-based optimization algorithm using Gaussian processes for global optimization problems in Sobolev spaces

We propose an algorithm to approximate solutions of global optimization problems in Sobolev spaces that follows the spirit of Consensus-based algorithms in finite dimensions. The main ingredient are Gaussian processes. In fact, we exploit their rich toolbox in order to draw sample functions from Sobolev spaces that satisfy initial values, boundary conditions or state constraints. Well-known marginalization properties of Gaussian processes help us to discretize the algorithm, that is stated in infinite dimensions, appropriately. We illustrate the performance of the algorithm and show its feasibility for nonlinear boundary value problems with state constraints as well as nonlinear optimal control problems constrained by a system of ordinary differential equations with several numerical results.

math.OC↗

arXiv2025

Emergent Cooperative Driving Strategies for Stop-and-Go Wave Mitigation via Multi-Agent Reinforcement Learning

Stop-and-go waves in traffic flow pose a persistent challenge, compromising safety, efficiency, and environmental sustainability. This paper introduces a novel mitigation strategy discovered through training multi-agent deep reinforcement learning (DRL) agents in a simulated ring-road environment. The agents autonomously develop a cooperative driving policy, where most vehicles maintain minimal headways to maximize throughput, while a single "buffer" vehicle adopts a larger headway to absorb perturbations and prevent wave propagation. This strategy enhances stability without sacrificing overall flow. We further demonstrate that adapting this cooperative strategy to classical car-following models, such as the Intelligent Driver Model (IDM), yields improved stability and traffic efficiency. Furthermore, we show within a parametrised linear framework, that the cooperative strategy can optimise system performance under stability constraints. Our findings offer promising insights for future autonomous vehicle systems and highway management.

physics.soc-ph↗

arXiv2025

A Space Mapping approach for the calibration of financial models with the application to the Heston model

We present a novel approach for parameter calibration of the Heston model for pricing an Asian put option, namely space mapping. Since few parameters of the Heston model can be directly extracted from real market data, calibration to real market data is implicit and therefore a challenging task. In addition, some of the parameters in the model are non-linear, which makes it difficult to find the global minimum of the optimization problem within the calibration. Our approach is based on the idea of space mapping, exploiting the residuum of a coarse surrogate model that allows optimization and a fine model that needs to be calibrated. In our case, the pricing of an Asian option using the Heston model SDE is the fine model, and the surrogate is chosen to be the Heston model PDE pricing a European option. We formally derive a gradient descent algorithm for the PDE constrained calibration model using well-known techniques from optimization with PDEs. Our main goal is to provide evidence that the space mapping approach can be useful in financial calibration tasks. Numerical results underline the feasibility of our approach.

math.NA↗

arXiv2024

Localized KBO with genetic dynamics for multi-modal optimization

In this paper, we introduce a novel approach to multi-modal optimization by enhancing the recently developed kinetic-based optimization (KBO) method with genetic dynamics (GKBO). The proposed method targets objective functions with multiple global minima, addressing a critical need in fields like engineering design, machine learning, and bioinformatics. By incorpo rating leader-follower dynamics and localized interactions, the algorithm efficiently navigates high-dimensional search spaces to detect multiple optimal solutions. After providing a binary description, a mean-field approximation is derived, and different numerical experiments are conducted to validate the results.

math.NA↗

arXiv2024

CBX: Python and Julia packages for consensus-based interacting particle methods

We introduce CBXPy and ConsensusBasedX.jl, Python and Julia implementations of consensus-based interacting particle systems (CBX), which generalise consensus-based optimization methods (CBO) for global, derivative-free optimisation. The raison d'\^etre of our libraries is twofold: on the one hand, to offer high-performance implementations of CBX methods that the community can use directly, while on the other, providing a general interface that can accommodate and be extended to further variations of the CBX family. Python and Julia were selected as the leading high-level languages in terms of usage and performance, as well as their popularity among the scientific computing community. Both libraries have been developed with a common ethos, ensuring a similar API and core functionality, while leveraging the strengths of each language and writing idiomatic code.

math.OC↗

arXiv2023

Time-continuous microscopic pedestrian models: an overview

We give an overview of time-continuous pedestrian models with a focus on data-driven modelling. Starting from pioneer, reactive force-based models we move forward to modern, active pedestrian models with sophisticated collision-avoidance and anticipation techniques through optimisation problems. The overview focuses on the mathematical aspects of the models and their different components. We include methods used for data-based calibration of model parameters, hybrid approaches incorporating neural networks, and purely data-based models fitted by deep learning. The conclusion outlines some development perspectives that we expect to grow in the coming years.

physics.soc-ph↗

arXiv2023

Kinetic based optimization enhanced by genetic dynamics

We propose and analyse a variant of the recently introduced kinetic based optimization method that incorporates ideas like survival-of-the-fittest and mutation strategies well-known from genetic algorithms. Thus, we provide a first attempt to reach out from the class of consensus/kinetic-based algorithms towards genetic metaheuristics. Different generations of genetic algorithms are represented via two species identified with different labels, binary interactions are prescribed on the particle level and then we derive a mean-field approximation in order to analyse the method in terms of convergence. Numerical results underline the feasibility of the approach and show in particular that the genetic dynamics allows to improve the efficiency, of this class of global optimization methods in terms of computational cost.

math.OC↗

arXiv2023

A new perspective on dynamic network flow problems via port-Hamiltonian systems

We suggest a global perspective on dynamic network flow problems that takes advantage of the similarities to port-Hamiltonian dynamics. Dynamic minimum cost flow problems are formulated as open-loop optimal control problems for general port-Hamiltonian systems with possibly state-dependent system matrices. We prove well-posedness of these systems and characterize optimal controls by the first-order optimality system, which is the starting point for the derivation of an adjoint-based gradient descent algorithm. Our theoretical analysis is complemented by a proof of concept, where we apply the proposed algorithm to static minimum cost flow problems and dynamic minimum cost flow problems on a simple directed acyclic graph. We present numerical results to validate the approach.

math.OC↗

arXiv2023

Modeling Minimum Cost Network Flows With Port-Hamiltonian Systems

We give a short overview of advantages and drawbacks of the classical formulation of minimum cost network flow problems and solution techniques, to motivate a reformulation of classical static minimum cost network flow problems as optimal control problems constrained by port-Hamiltonian systems (pHS). The first-order optimality system for the port-Hamiltonian system-constrained optimal control problem is formally derived. Then we propose a gradient-based algorithm to find optimal controls. The port-Hamiltonian system formulation naturally conserves flow and supports a wide array of further modeling options as, for example, node reservoirs, flow dependent costs, leaking pipes (dissipation) and coupled sub-networks (ports). They thus provide a versatile alternative to state-of-the art approaches towards dynamic network flow problems, which are often based on computationally costly time-expanded networks. We argue that this opens the door for a plethora of modeling options and solution approaches for network flow problems.

math.OC↗

arXiv2023

Asymptotic Analysis for Optimal Control of the Cattaneo Model

We consider an optimal control problem with tracking-type cost functional constrained by the Cattaneo equation, which is a well-known model for delayed heat transfer. In particular, we are interested the asymptotic behaviour of the optimal control problems for a vanishing delay time $\tau \rightarrow 0$. First, we show the convergence of solutions of the Cattaneo equation to the ones of the heat equation. Assuming the same right-hand side and compatible initial conditions for the equations, we prove a linear convergence rate. Moreover, we show linear convergence of the optimal states and optimal controls for the Cattaneo equation towards the ones for the heat equation. We present numerical results for both, the forward and the optimal control problem confirming these linear convergence rates.

math.OC↗

arXiv2023

Erratum and original of Port-Hamiltonian structure of interacting particle systems and its mean-field limit

We derive a minimal port-Hamiltonian formulation of a general class of interacting particle systems driven by alignment and potential-based force dynamics which include the Cucker-Smale model with potential interaction and the second order Kuramoto model. The port-Hamiltonian structure allows to characterize conserved quantities such as Casimir functions as well as the long-time behaviour using a LaSalle-type argument on the particle level. It is then shown that the port-Hamiltonian structure is preserved in the mean-field limit and an analogue of the LaSalle invariance principle is studied in the space of probability measures equipped with the 2-Wasserstein-metric. The results on the particle and mean-field limit yield a new perspective on uniform stability of general interacting particle systems. Moreover, as the minimal port-Hamiltonian formulation is closed we identify the ports of the subsystems which admit generalized mass-spring-damper structure modelling the binary interaction of two particles. Using the information of ports we discuss the coupling of difference species in a port-Hamiltonian preserving manner. The erratum corrects an error in our paper on the port-Hamiltonian structure of interacting particle systems. While convergence of the gradient of the Hamiltonian remains valid under the original assumptions, relative compactness of the system trajectories in the 2-Wasserstein space does not hold without an additional attractivity assumption on the binary interaction force. We provide a proof for the convergence of the gradient of the Hamiltonian based on Barbalat Lemma. A counterexample is given for the relative compactness of the system trajectories for repulsive binary interactions. In the case of short-range repulsion and long-range attraction we show several numerical studies that underpin our conjecture of relatively compact trajectories.

math.AP↗

arXiv2023

Data-driven adjoint-based calibration of port-Hamiltonian systems in time domain

We present a gradient-based identification algorithm to identify the system matrices of a linear port-Hamiltonian system from given input-output time data. Aiming for a direct structure-preserving approach, we employ techniques from optimal control with ordinary differential equations and define a constrained optimization problem. The input-to-state stability is discussed which is the key step towards the existence of optimal controls. Further, we derive the first-order optimality system taking into account the port-Hamiltonian structure. Indeed, the proposed method preserves the skew-symmetry and positive (semi)-definiteness of the system matrices throughout the optimization iterations. Numerical results with perturbed and unperturbed synthetic data, as well as an example from the PHS benchmark collection demonstrate the feasibility of the approach.

math.OC↗

arXiv2023

Structure-preserving identification of port-Hamiltonian systems -- a sensitivity-based approach

We present a gradient-based calibration algorithm to identify a port-Hamiltonian system from given time-domain input-output data. The gradient is computed with the help of sensitivities and the algorithm is tailored such that the structure of the system matrices of the port-Hamiltonian system (skew-symmetry and positive semi-definiteness) is preserved in each iteration of the algorithm. As we only require input-output data, we need to calibrate the initial condition of the internal state of the port-Hamiltonian system as well. Numerical results with synthetic data show the feasibility of the approach.

math.NA↗

arXiv2022

Consensus-Based Optimization for Multi-Objective Problems: A Multi-Swarm Approach

We propose a multi-swarm approach to approximate the Pareto front of general multi-objective optimization problems that is based on the Consensus-based Optimization method (CBO). The algorithm is motivated step by step beginning with a simple extension of CBO based on fixed scalarization weights. To overcome the issue of choosing the weights we propose an adaptive weight strategy in the second modelling step. The modelling process is concluded with the incorporation of a penalty strategy that avoids clusters along the Pareto front and a diffusion term that prevents collapsing swarms. Altogether the proposed $K$-swarm CBO algorithm is tailored for a diverse approximation of the Pareto front and, simultaneously, the efficient set of general non-convex multi-objective problems. The feasibility of the approach is justified by analytic results, including convergence proofs, and a performance comparison to the well-known non-dominated sorting genetic algorithm (NSGA2) and the recently proposed one-swarm approach for multi-objective problems involving Consensus-based Optimization.

math.OC↗

arXiv2022

Multi-scale description of pedestrian collective dynamics with port-Hamiltonian systems

Port-Hamiltonian systems (PHS) theory is a recent but already well-established modelling approach for non-linear physical systems. Some studies have shown lately that PHS frameworks are relevant for modelling and control of swarm and multi-agent systems. We identify in this contribution a general class of microscopic force-based pedestrian models that can be formulated as a port-Hamiltonian system. The pedestrian PHS has linear structure and dissipation components. Non-linear effects come from isotropic pedestrian interactions. Simulation results on a torus with disordered initial states show that the port-Hamiltonian pedestrian model can exhibit different types of dynamics. They range from relaxed speed models with no interaction, dynamical billiards, or crystallization dynamics to realistic pedestrian collective behaviors, including lane and strip formation for counter and crossing flow. The port-Hamiltonian framework is a natural multiscale description of pedestrian dynamics as the Hamiltonian turns out to be a generic order parameter that allows us to identify specific behaviours of the dynamics from a macroscopic viewpoint. Particular cases even enable through energy balance to determine the Hamiltonian behavior without requiring the tedious computation of the microscopic dynamics. Using PHS theory, we systematically identify a critical threshold value for the Hamiltonian, which relies only on exogenous input and can be physically interpreted.

physics.soc-ph↗

arXiv2022

Ensemble-based gradient inference for particle methods in optimization and sampling

We propose an approach based on function evaluations and Bayesian inference to extract higher-order differential information of objective functions {from a given ensemble of particles}. Pointwise evaluation $\{V(x^i)\}_i$ of some potential $V$ in an ensemble $\{x^i\}_i$ contains implicit information about first or higher order derivatives, which can be made explicit with little computational effort (ensemble-based gradient inference -- EGI). We suggest to use this information for the improvement of established ensemble-based numerical methods for optimization and sampling such as Consensus-based optimization and Langevin-based samplers. Numerical studies indicate that the augmented algorithms are often superior to their gradient-free variants, in particular the augmented methods help the ensembles to escape their initial domain, to explore multimodal, non-Gaussian settings and to speed up the collapse at the end of optimization dynamics.} The code for the numerical examples in this manuscript can be found in the paper's Github repository (https://github.com/MercuryBench/ensemble-based-gradient.git).

stat.ML↗