@book{45191,
  editor       = {{Gräßler, Iris and Maier, Günter W. and Steffen, Eckhard and Roesmann, Daniel}},
  isbn         = {{9783031261039}},
  publisher    = {{Springer International Publishing}},
  title        = {{{The Digital Twin of Humans}}},
  doi          = {{10.1007/978-3-031-26104-6}},
  year         = {{2023}},
}

@article{45971,
  abstract     = {{<jats:title>Abstract</jats:title>
               <jats:p>An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.</jats:p>}},
  author       = {{Bartels, Sören and Kovács, Balázs and Wang, Zhangxian}},
  issn         = {{0272-4979}},
  journal      = {{IMA Journal of Numerical Analysis}},
  keywords     = {{Applied Mathematics, Computational Mathematics, General Mathematics}},
  publisher    = {{Oxford University Press (OUP)}},
  title        = {{{Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints}}},
  doi          = {{10.1093/imanum/drad037}},
  year         = {{2023}},
}

@article{53140,
  abstract     = {{We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. When combined with a method dependent Ritz projection for the initial data, unconditional stability can be shown while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.}},
  author       = {{Contri, Alessandro and Kovács, Balázs and Massing, André}},
  journal      = {{arXiv}},
  title        = {{{Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates}}},
  doi          = {{10.48550/ARXIV.2312.05511}},
  year         = {{2023}},
}

@article{53533,
  author       = {{Ernst, Alena and Schmidt, Kai-Uwe}},
  issn         = {{0305-0041}},
  journal      = {{Mathematical Proceedings of the Cambridge Philosophical Society}},
  keywords     = {{General Mathematics}},
  number       = {{1}},
  pages        = {{129--160}},
  publisher    = {{Cambridge University Press (CUP)}},
  title        = {{{Intersection theorems for finite general linear groups}}},
  doi          = {{10.1017/s0305004123000075}},
  volume       = {{175}},
  year         = {{2023}},
}

@article{55277,
  author       = {{Klahn, B. and Technau, Marc}},
  journal      = {{Int. J. Number Theory}},
  number       = {{10}},
  pages        = {{2443–2450}},
  title        = {{{Galois groups of (ⁿ₀)+(ⁿ₁)X+…+(ⁿ₆)X⁶}}},
  doi          = {{10.1142/S1793042123501208}},
  volume       = {{19}},
  year         = {{2023}},
}

@article{55279,
  author       = {{Minelli, P. and Sourmelidis, A. and Technau, Marc}},
  journal      = {{Math. Ann.}},
  pages        = {{291–320}},
  title        = {{{Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums}}},
  doi          = {{10.1007/s00208-022-02452-2}},
  volume       = {{387}},
  year         = {{2023}},
}

@inproceedings{42163,
  abstract     = {{The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler--Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.}},
  author       = {{Offen, Christian and Ober-Blöbaum, Sina}},
  booktitle    = {{Geometric Science of Information}},
  editor       = {{Nielsen, F and Barbaresco, F}},
  keywords     = {{System identification, discrete Lagrangians, travelling waves}},
  location     = {{Saint-Malo, Palais du Grand Large, France}},
  pages        = {{569--579}},
  publisher    = {{Springer, Cham.}},
  title        = {{{Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves}}},
  doi          = {{10.1007/978-3-031-38271-0_57}},
  volume       = {{14071}},
  year         = {{2023}},
}

@article{27426,
  abstract     = {{Regularization is used in many different areas of optimization when solutions
are sought which not only minimize a given function, but also possess a certain
degree of regularity. Popular applications are image denoising, sparse
regression and machine learning. Since the choice of the regularization
parameter is crucial but often difficult, path-following methods are used to
approximate the entire regularization path, i.e., the set of all possible
solutions for all regularization parameters. Due to their nature, the
development of these methods requires structural results about the
regularization path. The goal of this article is to derive these results for
the case of a smooth objective function which is penalized by a piecewise
differentiable regularization term. We do this by treating regularization as a
multiobjective optimization problem. Our results suggest that even in this
general case, the regularization path is piecewise smooth. Moreover, our theory
allows for a classification of the nonsmooth features that occur in between
smooth parts. This is demonstrated in two applications, namely support-vector
machines and exact penalty methods.}},
  author       = {{Gebken, Bennet and Bieker, Katharina and Peitz, Sebastian}},
  journal      = {{Journal of Global Optimization}},
  number       = {{3}},
  pages        = {{709--741}},
  title        = {{{On the structure of regularization paths for piecewise differentiable regularization terms}}},
  doi          = {{10.1007/s10898-022-01223-2}},
  volume       = {{85}},
  year         = {{2023}},
}

@article{44857,
  abstract     = {{Ancestral reconstruction is a classic task in comparative genomics. Here, we study the genome median problem, a related computational problem which, given a set of three or more genomes, asks to find a new genome that minimizes the sum of pairwise distances between it and the given genomes. The distance stands for the amount of evolution observed at the genome level, for which we determine the minimum number of rearrangement operations necessary to transform one genome into the other. For almost all rearrangement operations the median problem is NP-hard, with the exception of the breakpoint median that can be constructed efficiently for multichromosomal circular and mixed genomes. In this work, we study the median problem under a restricted rearrangement measure called c4-distance, which is closely related to the breakpoint and the DCJ distance. We identify tight bounds and decomposers of the c4-median and develop algorithms for its construction, one exact ILP-based and three combinatorial heuristics. Subsequently, we perform experiments on simulated data sets. Our results suggest that the c4-distance is useful for the study the genome median problem, from theoretical and practical perspectives.}},
  author       = {{Silva, Helmuth O.M. and Rubert, Diego P. and Araujo, Eloi and Steffen, Eckhard and Doerr, Daniel and Martinez, Fábio V.}},
  issn         = {{0399-0559}},
  journal      = {{RAIRO - Operations Research}},
  keywords     = {{Management Science and Operations Research, Computer Science Applications, Theoretical Computer Science}},
  number       = {{3}},
  pages        = {{1045--1058}},
  publisher    = {{EDP Sciences}},
  title        = {{{Algorithms for the genome median under a restricted measure of rearrangement}}},
  doi          = {{10.1051/ro/2023052}},
  volume       = {{57}},
  year         = {{2023}},
}

@unpublished{44859,
  author       = {{Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak Hieronymus}},
  booktitle    = {{arXiv:2305.08619}},
  title        = {{{Sets of r-graphs that color all r-graphs}}},
  year         = {{2023}},
}

@unpublished{45498,
  abstract     = {{We present a novel method for high-order phase reduction in networks of
weakly coupled oscillators and, more generally, perturbations of reducible
normally hyperbolic (quasi-)periodic tori. Our method works by computing an
asymptotic expansion for an embedding of the perturbed invariant torus, as well
as for the reduced phase dynamics in local coordinates. Both can be determined
to arbitrary degrees of accuracy, and we show that the phase dynamics may
directly be obtained in normal form. We apply the method to predict remote
synchronisation in a chain of coupled Stuart-Landau oscillators.}},
  author       = {{von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}},
  booktitle    = {{arXiv:2306.03320}},
  pages        = {{29}},
  title        = {{{A parametrisation method for high-order phase reduction in coupled  oscillator networks}}},
  year         = {{2023}},
}

@article{46256,
  author       = {{Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak Hieronymus}},
  issn         = {{0895-4801}},
  journal      = {{SIAM Journal on Discrete Mathematics}},
  keywords     = {{General Mathematics}},
  number       = {{3}},
  pages        = {{1548--1565}},
  publisher    = {{Society for Industrial & Applied Mathematics (SIAM)}},
  title        = {{{Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs}}},
  doi          = {{10.1137/22m1500654}},
  volume       = {{37}},
  year         = {{2023}},
}

@article{29240,
  abstract     = {{The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,
we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.}},
  author       = {{Ober-Blöbaum, Sina and Offen, Christian}},
  issn         = {{0377-0427}},
  journal      = {{Journal of Computational and Applied Mathematics}},
  keywords     = {{Lagrangian learning, variational backward error analysis, modified Lagrangian, variational integrators, physics informed learning}},
  pages        = {{114780}},
  publisher    = {{Elsevier}},
  title        = {{{Variational Learning of Euler–Lagrange Dynamics from Data}}},
  doi          = {{10.1016/j.cam.2022.114780}},
  volume       = {{421}},
  year         = {{2023}},
}

@article{29236,
  abstract     = {{The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.}},
  author       = {{McLachlan, Robert and Offen, Christian}},
  journal      = {{Journal of Geometric Mechanics}},
  keywords     = {{variational integrators, backward error analysis, Euler--Lagrange equations, multistep methods, conjugate symplectic methods}},
  number       = {{1}},
  pages        = {{98--115}},
  publisher    = {{AIMS Press}},
  title        = {{{Backward error analysis for conjugate symplectic methods}}},
  doi          = {{10.3934/jgm.2023005}},
  volume       = {{15}},
  year         = {{2023}},
}

@article{37654,
  abstract     = {{Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when
learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite
the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we
enhance the HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach
allows to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples,
a pendulum on a cart and a two-body problem from astrodynamics are considered.}},
  author       = {{Dierkes, Eva and Offen, Christian and Ober-Blöbaum, Sina and Flaßkamp, Kathrin}},
  issn         = {{1054-1500}},
  journal      = {{Chaos}},
  number       = {{6}},
  publisher    = {{AIP Publishing}},
  title        = {{{Hamiltonian Neural Networks with Automatic Symmetry Detection}}},
  doi          = {{10.1063/5.0142969}},
  volume       = {{33}},
  year         = {{2023}},
}

@article{23428,
  abstract     = {{The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems in recent years, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still quite scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points; for both ordinary and stochastic differential equations. Moreover, we extend our analysis to nonlinear control-affine systems using either ergodic trajectories or i.i.d.
samples. Here, we exploit the linearity of the Koopman generator to obtain a bilinear system and, thus, circumvent the curse of dimensionality since we do not autonomize the system by augmenting the state by the control inputs. To the
best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the proposed approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.}},
  author       = {{Nüske, Feliks and Peitz, Sebastian and Philipp, Friedrich and Schaller, Manuel and Worthmann, Karl}},
  journal      = {{Journal of Nonlinear Science}},
  title        = {{{Finite-data error bounds for Koopman-based prediction and control}}},
  doi          = {{10.1007/s00332-022-09862-1}},
  volume       = {{33}},
  year         = {{2023}},
}

@article{21600,
  abstract     = {{Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML.}},
  author       = {{Dellnitz, Michael and Hüllermeier, Eyke and Lücke, Marvin and Ober-Blöbaum, Sina and Offen, Christian and Peitz, Sebastian and Pfannschmidt, Karlson}},
  journal      = {{SIAM Journal on Scientific Computing}},
  number       = {{2}},
  pages        = {{A579--A595}},
  title        = {{{Efficient time stepping for numerical integration using reinforcement  learning}}},
  doi          = {{10.1137/21M1412682}},
  volume       = {{45}},
  year         = {{2023}},
}

@inbook{16296,
  abstract     = {{Multiobjective optimization plays an increasingly important role in modern
applications, where several objectives are often of equal importance. The task
in multiobjective optimization and multiobjective optimal control is therefore
to compute the set of optimal compromises (the Pareto set) between the
conflicting objectives. Since the Pareto set generally consists of an infinite
number of solutions, the computational effort can quickly become challenging
which is particularly problematic when the objectives are costly to evaluate as
is the case for models governed by partial differential equations (PDEs). To
decrease the numerical effort to an affordable amount, surrogate models can be
used to replace the expensive PDE evaluations. Existing multiobjective
optimization methods using model reduction are limited either to low parameter
dimensions or to few (ideally two) objectives. In this article, we present a
combination of the reduced basis model reduction method with a continuation
approach using inexact gradients. The resulting approach can handle an
arbitrary number of objectives while yielding a significant reduction in
computing time.}},
  author       = {{Banholzer, Stefan and Gebken, Bennet and Dellnitz, Michael and Peitz, Sebastian and Volkwein, Stefan}},
  booktitle    = {{Non-Smooth and Complementarity-Based Distributed Parameter Systems}},
  editor       = {{Michael, Hintermüller and Roland, Herzog and Christian, Kanzow and Michael, Ulbrich and Stefan, Ulbrich}},
  isbn         = {{978-3-030-79392-0}},
  pages        = {{43--76}},
  publisher    = {{Springer}},
  title        = {{{ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation}}},
  doi          = {{10.1007/978-3-030-79393-7_3}},
  year         = {{2022}},
}

@inbook{30294,
  abstract     = {{With the ever increasing capabilities of sensors and controllers, autonomous driving is quickly becoming a reality. This disruptive change in the automotive industry poses major challenges for manufacturers as well as suppliers as entirely new design and testing strategies have to be developed to remain competitive. Most importantly, the complexity of autonomously driving vehicles in a complex, uncertain, and safety-critical environment requires new testing procedures to cover the almost infinite range of potential scenarios.}},
  author       = {{Peitz, Sebastian and Dellnitz, Michael and Bannenberg, Sebastian}},
  booktitle    = {{German Success Stories in Industrial Mathematics}},
  editor       = {{Bock, H. G. and Küfer, K.-H. and Maas, P. and Milde, A. and Schulz, V.}},
  isbn         = {{9783030814540}},
  issn         = {{1612-3956}},
  publisher    = {{Springer International Publishing}},
  title        = {{{Efficient Virtual Design and Testing of Autonomous Vehicles}}},
  doi          = {{10.1007/978-3-030-81455-7_23}},
  volume       = {{35}},
  year         = {{2022}},
}

@article{30490,
  author       = {{Cresson, Jacky and Jiménez, Fernando and Ober-Blöbaum, Sina}},
  journal      = {{AIMS}},
  pages        = {{57--89}},
  title        = {{{Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations}}},
  volume       = {{14(1)}},
  year         = {{2022}},
}