arXiv subjects

Hendrik Kleikamp

Publications and source records attributed to Hendrik Kleikamp.

13 recordsLinked to original sources

arXiv2026

Robust lane-change intention anticipation under uncertainty based on a recursive Bayesian filtering approach

In this paper we discuss and analyze a method for lane-change intention anticipation of drivers on highways based on kinematic features and surrounding observations. The approach makes use of a recursive Bayesian filtering strategy, which can be interpreted as a hidden Markov model. We present two models for the likelihood term and show how they can handle missing observations. An additional focus of the work lies on calibrating the obtained probabilities in order to obtain reliable predictions. The introduced approach is evaluated in practice on the highD dataset and compared to several other baseline methods. Particular emphasize is put on evaluating the robustness and temporal consistency as well as calibration of the developed algorithm. Extensive numerical experiments allow for a careful performance assessment and detailed discussion of advantages and limitations of the different methodologies.

eess.SY↗

arXiv2026

Data-Driven Model Order Reduction with pyMOR

pyMOR is a free and open-source software library of model order reduction algorithms for the Python programming language. Designed with classical model-based reduction methods for large-scale parametric partial differential equation problems in mind, algorithms in pyMOR are implemented in terms of operations on abstract VectorArray, Operator and Model interfaces, allowing for a seamless integration with external solver codes implementing the full-order model. For cases where a tight integration with the full-order model code is not feasible, data-driven model order reduction algorithms, which only require simulation or measurement data of the full-order model, are an attractive alternative. In this work we discuss the data-driven methods that have been recently added to pyMOR, show practical examples of their application using pyMOR and compare their performance with classical model-based methods. We show that pyMOR serves as a unified framework for combining model-based and data-driven methods, enabling the construction of flexible and efficient hierarchical model reduction pipelines.

cs.MS↗

arXiv2026

Overcoming slow Kolmogorov width decay in parametric optimal control via neural network surrogates

In this paper we deal with parametric, linear-quadratic optimal control problems in which the solution can be uniquely characterized by the optimal final time adjoint state. As a motivating example, we establish theoretical results showing that for distributed control of the heat equation, the manifold of final time adjoints over the parameter space exhibits a slow decay of its Kolmogorov width if this was already the case for the parameter-dependent target states. Traditional linear reduced-order models would thus require a large reduced space in order to guarantee a sufficient accuracy, making them inefficient in this application. To overcome the limitation of linear models, we discuss a nonlinear surrogate based on U-Nets that maps parametric fields to approximate final time adjoints. We show that a suitable a posteriori error estimator remains applicable to the U-Net approximation and can be used to certify the surrogate results. Through two extensive numerical experiments, we show the potential of the U-Net surrogate and compare it with several linear and nonlinear methods from the literature. The results show that the U-Net consistently achieves the highest accuracy among the methods considered while requiring significantly fewer training samples.

math.OC↗

arXiv2026

Kernel Methods in the Deep Ritz framework: Theory and practice

In this contribution, kernel approximations are applied as ansatz functions within the Deep Ritz method. This allows to approximate weak solutions of elliptic partial differential equations with weak enforcement of boundary conditions using Nitsche's method. A priori error estimates are proven in different norms leveraging both standard results for weak solutions of elliptic equations and well-established convergence results for kernel methods. This availability of a priori error estimates renders the method useful for practical purposes. The procedure is described in detail, meanwhile providing practical hints and implementation details. By means of numerical examples, the performance of the proposed approach is evaluated numerically and the results agree with the theoretical findings.

math.NA↗

arXiv2025

Towards a Knowledge Graph for Models and Algorithms in Applied Mathematics

Mathematical models and algorithms are an essential part of mathematical research data, as they are epistemically grounding numerical data. In order to represent models and algorithms as well as their relationship semantically to make this research data FAIR, two previously distinct ontologies were merged and extended, becoming a living knowledge graph. The link between the two ontologies is established by introducing computational tasks, as they occur in modeling, corresponding to algorithmic tasks. Moreover, controlled vocabularies are incorporated and a new class, distinguishing base quantities from specific use case quantities, was introduced. Also, both models and algorithms can now be enriched with metadata. Subject-specific metadata is particularly relevant here, such as the symmetry of a matrix or the linearity of a mathematical model. This is the only way to express specific workflows with concrete models and algorithms, as the feasible solution algorithm can only be determined if the mathematical properties of a model are known. We demonstrate this using two examples from different application areas of applied mathematics. In addition, we have already integrated over 250 research assets from applied mathematics into our knowledge graph.

cs.AI↗

arXiv2024

Adaptive Model Hierarchies for Multi-Query Scenarios

In this contribution we present an abstract framework for adaptive model hierarchies together with several instances of hierarchies for specific applications. The hierarchy is particularly useful when integrated within an outer loop, for instance an optimization iteration or a Monte Carlo estimation where for a large set of requests answers fulfilling certain criteria are required.

math.NA↗

arXiv2024

Two-stage model reduction approaches for the efficient and certified solution of parametrized optimal control problems

In this contribution we develop an efficient reduced order model for solving parametrized linear-quadratic optimal control problems with linear time-varying state system. The fully reduced model combines reduced basis approximations of the system dynamics and of the manifold of optimal final time adjoint states to achieve a computational complexity independent of the original state space. Such a combination is particularly beneficial in the case where a deviation in a low-dimensional output is penalized. In addition, an offline-online decomposed a posteriori error estimator bounding the error between the approximate final time adjoint with respect to the optimal one is derived and its reliability proven. We propose different strategies for building the involved reduced order models, for instance by separate reduction of the dynamical systems and the final time adjoint states or via greedy procedures yielding a combined and fully reduced model. These algorithms are evaluated and compared for a two-dimensional heat equation problem. The numerical results show the desired accuracy of the reduced models and highlight the speedup obtained by the newly combined reduced order model in comparison to an exact computation of the optimal control or other reduction approaches.

math.NA↗

arXiv2024

Ontologies for Models and Algorithms in Applied Mathematics and Related Disciplines

In applied mathematics and related disciplines, the modeling-simulation-optimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data, the Mathematical Research Data Initiative has developed, merged and implemented ontologies and knowledge graphs. This contributes to making mathematical research data FAIR by introducing semantic technology and documenting the mathematical foundations accordingly. Using the concrete example of microfracture analysis of porous media, it is shown how the knowledge of the underlying mathematical model and the corresponding numerical algorithms for its solution can be represented by the ontologies.

cs.AI↗

arXiv2024

Application of an adaptive model hierarchy to parametrized optimal control problems

In this contribution we apply an adaptive model hierarchy, consisting of a full-order model, a reduced basis reduced order model, and a machine learning surrogate, to parametrized linear-quadratic optimal control problems. The involved reduced order models are constructed adaptively and are called in such a way that the model hierarchy returns an approximate solution of given accuracy for every parameter value. At the same time, the fastest model of the hierarchy is evaluated whenever possible and slower models are only queried if the faster ones are not sufficiently accurate. The performance of the model hierarchy is studied for a parametrized heat equation example with boundary value control.

math.OC↗

arXiv2023

Be greedy and learn: efficient and certified algorithms for parametrized optimal control problems

We consider parametrized linear-quadratic optimal control problems and provide their online-efficient solutions by combining greedy reduced basis methods and machine learning algorithms. To this end, we first extend the greedy control algorithm, which builds a reduced basis for the manifold of optimal final time adjoint states, to the setting where the objective functional consists of a penalty term measuring the deviation from a desired state and a term describing the control energy. Afterwards, we apply machine learning surrogates to accelerate the online evaluation of the reduced model. The error estimates proven for the greedy procedure are further transferred to the machine learning models and thus allow for efficient a posteriori error certification. We discuss the computational costs of all considered methods in detail and show by means of two numerical examples the tremendous potential of the proposed methodology.

math.OC↗

arXiv2023

Application of Deep Kernel Models for Certified and Adaptive RB-ML-ROM Surrogate Modeling

In the framework of reduced basis methods, we recently introduced a new certified hierarchical and adaptive surrogate model, which can be used for efficient approximation of input-output maps that are governed by parametrized partial differential equations. This adaptive approach combines a full order model, a reduced order model and a machine-learning model. In this contribution, we extend the approach by leveraging novel kernel models for the machine learning part, especially structured deep kernel networks as well as two layered kernel models. We demonstrate the usability of those enhanced kernel models for the RB-ML-ROM surrogate modeling chain and highlight their benefits in numerical experiments.

math.NA↗

arXiv2022

Nonlinear Model Order Reduction using Diffeomorphic Transformations of a Space-Time Domain

In many applications, for instance when describing dynamics of fluids or gases, hyperbolic conservation laws arise naturally in the modeling of conserved quantities of a system, like mass or energy. These types of equations exhibit highly nonlinear behaviors like shock formation or shock interaction. In the case of parametrized hyperbolic equations, where, for instance, varying transport velocities are considered, these nonlinearities and strong transport effects result in a highly nonlinear solution manifold. This solution manifold cannot be approximated properly by linear subspaces. To this end, nonlinear approaches for model order reduction of hyperbolic conservation laws are required. We propose a new method for nonlinear model order reduction that is especially well-suited for hyperbolic equations with discontinuous solutions. The approach is based on a space-time discretization and employs diffeomorphic transformations of the underlying space-time domain to align the discontinuities. To derive a reduced model for the diffeomorphisms, the Lie group structure of the diffeomorphism group is used to associate diffeomorphisms with corresponding velocity fields via the exponential map. In the linear space of velocity fields, standard model order reduction techniques, such as proper orthogonal decomposition, can be applied to extract a reduced subspace. For a parametrized Burgers' equation with two merging shocks, numerical experiments show the potential of the approach.

math.NA↗

arXiv2022

Adaptive machine learning based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery

In this contribution, we develop an efficient surrogate modeling framework for simulation-based optimization of enhanced oil recovery, where we particularly focus on polymer flooding. The computational approach is based on an adaptive training procedure of a neural network that directly approximates an input-output map of the underlying PDE-constrained optimization problem. The training process thereby focuses on the construction of an accurate surrogate model solely related to the optimization path of an outer iterative optimization loop. True evaluations of the objective function are used to finally obtain certified results. Numerical experiments are given to evaluate the accuracy and efficiency of the approach for a heterogeneous five-spot benchmark problem.

math.NA↗