@article{45854, abstract = {{In a previous paper the authors developed an algorithm to classify certain quaternary quadratic lattices over totally real fields. The present article applies this algorithm to the classification of binary Hermitian lattices over totally imaginary fields. We use it in particular to classify the 48-dimensional extremal even unimodular lattices over the integers that admit a semilarge automorphism.}}, author = {{Kirschmer, Markus and Nebe, Gabriele}}, issn = {{1058-6458}}, journal = {{Experimental Mathematics}}, keywords = {{General Mathematics}}, number = {{1}}, pages = {{280--301}}, publisher = {{Informa UK Limited}}, title = {{{Binary Hermitian Lattices over Number Fields}}}, doi = {{10.1080/10586458.2019.1618756}}, volume = {{31}}, year = {{2022}}, } @article{19941, abstract = {{In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby ‘modified’ equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.}}, author = {{McLachlan, Robert I and Offen, Christian}}, journal = {{Journal of Geometric Mechanics}}, number = {{3}}, pages = {{447 -- 471}}, publisher = {{AIMS}}, title = {{{Backward error analysis for variational discretisations of partial differential equations}}}, doi = {{10.3934/jgm.2022014}}, volume = {{14}}, year = {{2022}}, } @article{23382, abstract = {{Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data needs to be approximated based on the trajectory observations. Moreover, the numerical integrator introduces an additional discretisation error. In this paper, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data avoided. The inverse modified data compensates for the discretisation error such that the discretisation error is eliminated. The technique is developed for Gaussian Processes.}}, author = {{Offen, Christian and Ober-Blöbaum, Sina}}, journal = {{Chaos: An Interdisciplinary Journal of Nonlinear Science}}, publisher = {{AIP}}, title = {{{Symplectic integration of learned Hamiltonian systems}}}, doi = {{10.1063/5.0065913}}, volume = {{32(1)}}, year = {{2022}}, } @article{24169, author = {{Nüske, Feliks and Gelß, Patrick and Klus, Stefan and Clementi, Cecilia}}, issn = {{0167-2789}}, journal = {{Physica D: Nonlinear Phenomena}}, title = {{{Tensor-based computation of metastable and coherent sets}}}, doi = {{10.1016/j.physd.2021.133018}}, year = {{2021}}, } @article{24170, author = {{Klus, Stefan and Gelß, Patrick and Nüske, Feliks and Noé, Frank}}, issn = {{2632-2153}}, journal = {{Machine Learning: Science and Technology}}, title = {{{Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry}}}, doi = {{10.1088/2632-2153/ac14ad}}, year = {{2021}}, } @article{21195, author = {{Goelz, Christian and Mora, Karin and Stroehlein, Julia Kristin and Haase, Franziska Katharina and Dellnitz, Michael and Reinsberger, Claus and Vieluf, Solveig}}, journal = {{Cognitive Neurodynamics}}, title = {{{Electrophysiological signatures of dedifferentiation differ between fit and less fit older adults}}}, doi = {{10.1007/s11571-020-09656-9}}, year = {{2021}}, } @article{21337, abstract = {{We present a flexible trust region descend algorithm for unconstrained and convexly constrained multiobjective optimization problems. It is targeted at heterogeneous and expensive problems, i.e., problems that have at least one objective function that is computationally expensive. The method is derivative-free in the sense that neither need derivative information be available for the expensive objectives nor are gradients approximated using repeated function evaluations as is the case in finite-difference methods. Instead, a multiobjective trust region approach is used that works similarly to its well-known scalar pendants. Local surrogate models constructed from evaluation data of the true objective functions are employed to compute possible descent directions. In contrast to existing multiobjective trust region algorithms, these surrogates are not polynomial but carefully constructed radial basis function networks. This has the important advantage that the number of data points scales linearly with the parameter space dimension. The local models qualify as fully linear and the corresponding general scalar framework is adapted for problems with multiple objectives. Convergence to Pareto critical points is proven and numerical examples illustrate our findings.}}, author = {{Berkemeier, Manuel Bastian and Peitz, Sebastian}}, issn = {{2297-8747}}, journal = {{Mathematical and Computational Applications}}, number = {{2}}, title = {{{Derivative-Free Multiobjective Trust Region Descent Method Using Radial Basis Function Surrogate Models}}}, doi = {{10.3390/mca26020031}}, volume = {{26}}, year = {{2021}}, } @article{21820, abstract = {{The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.}}, author = {{Nüske, Feliks and Koltai, Péter and Boninsegna, Lorenzo and Clementi, Cecilia}}, issn = {{1099-4300}}, journal = {{Entropy}}, title = {{{Spectral Properties of Effective Dynamics from Conditional Expectations}}}, doi = {{10.3390/e23020134}}, year = {{2021}}, } @article{16867, abstract = {{In this article, we present an efficient descent method for locally Lipschitz continuous multiobjective optimization problems (MOPs). The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth MOPs with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy a necessary condition for Pareto optimality. Using a set of test problems, we compare our method to the multiobjective proximal bundle method by M\"akel\"a. The results indicate that our method is competitive while being easier to implement. While the number of objective function evaluations is larger, the overall number of subgradient evaluations is lower. Finally, we show that our method can be combined with a subdivision algorithm to compute entire Pareto sets of nonsmooth MOPs.}}, author = {{Gebken, Bennet and Peitz, Sebastian}}, journal = {{Journal of Optimization Theory and Applications}}, pages = {{696--723}}, title = {{{An efficient descent method for locally Lipschitz multiobjective optimization problems}}}, doi = {{10.1007/s10957-020-01803-w}}, volume = {{188}}, year = {{2021}}, } @article{16295, abstract = {{It is a challenging task to identify the objectives on which a certain decision was based, in particular if several, potentially conflicting criteria are equally important and a continuous set of optimal compromise decisions exists. This task can be understood as the inverse problem of multiobjective optimization, where the goal is to find the objective function vector of a given Pareto set. To this end, we present a method to construct the objective function vector of an unconstrained multiobjective optimization problem (MOP) such that the Pareto critical set contains a given set of data points with prescribed KKT multipliers. If such an MOP can not be found, then the method instead produces an MOP whose Pareto critical set is at least close to the data points. The key idea is to consider the objective function vector in the multiobjective KKT conditions as variable and then search for the objectives that minimize the Euclidean norm of the resulting system of equations. By expressing the objectives in a finite-dimensional basis, we transform this problem into a homogeneous, linear system of equations that can be solved efficiently. Potential applications of this approach include the identification of objectives (both from clean and noisy data) and the construction of surrogate models for expensive MOPs.}}, author = {{Gebken, Bennet and Peitz, Sebastian}}, journal = {{Journal of Global Optimization}}, pages = {{3--29}}, publisher = {{Springer}}, title = {{{Inverse multiobjective optimization: Inferring decision criteria from data}}}, doi = {{10.1007/s10898-020-00983-z}}, volume = {{80}}, year = {{2021}}, } @phdthesis{32057, abstract = {{Ein zentraler Aspekt bei der Untersuchung dynamischer Systeme ist die Analyse ihrer invarianten Mengen wie des globalen Attraktors und (in)stabiler Mannigfaltigkeiten. Insbesondere wenn das zugrunde liegende System von einem Parameter abhängt, ist es entscheidend, sie im Bezug auf diesen Parameter effizient zu verfolgen. Für die Berechnung invarianter Mengen stützen wir uns für ihre Approximation auf numerische Algorithmen. Typischerweise können diese Methoden jedoch nur auf endlich-dimensionale dynamische Systeme angewendet werden. In dieser Arbeit präsentieren wir daher einen numerischen Rahmen für die globale dynamische Analyse unendlich-dimensionaler Systeme. Wir werden Einbettungstechniken verwenden, um das core dynamical system (CDS) zu definieren, welches ein dynamisch äquivalentes endlich-dimensionales System ist.Das CDS wird dann verwendet, um eingebettete invariante Mengen, also eins-zu-eins Bilder, mittels Mengen-orientierten numerischen Methoden zu approximieren. Bei der Konstruktion des CDS ist es entscheidend, eine geeignete Beobachtungsabbildung auszuwählen und die geeignete inverse Abbildung zu entwerfen. Dazu werden wir geeignete numerische Implementierungen des CDS für DDEs und PDEs vorstellen. Für eine nachfolgende geometrische Analyse der eingebetteten invarianten Menge betrachten wir eine Lerntechnik namens diffusion maps, die ihre intrinsische Geometrie enthüllt sowie ihre Dimension schätzt. Schließlich wenden wir unsere entwickelten numerischen Methoden an einigen bekannten unendlich-dimensionale dynamischen Systeme an, wie die Mackey-Glass-Gleichung, die Kuramoto-Sivashinsky-Gleichung und die Navier-Stokes-Gleichung.}}, author = {{Gerlach, Raphael}}, title = {{{The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems}}}, doi = {{10.17619/UNIPB/1-1278}}, year = {{2021}}, } @article{34042, author = {{Li, Jiaao and Ma, Yulai and Miao, Zhengke and Shi, Yongtang and Wang, Weifan and Zhang, Cun-Quan}}, issn = {{0095-8956}}, journal = {{Journal of Combinatorial Theory, Series B}}, keywords = {{Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Theoretical Computer Science}}, pages = {{61--80}}, publisher = {{Elsevier BV}}, title = {{{Nowhere-zero 3-flows in toroidal graphs}}}, doi = {{10.1016/j.jctb.2021.11.001}}, volume = {{153}}, year = {{2021}}, } @inproceedings{29421, author = {{Ober-Blöbaum, Sina and Vermeeren, M.}}, booktitle = {{7th IIFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC}}, pages = {{327--333}}, title = {{{Superconvergence of galerkin variational integrators}}}, volume = {{54(19)}}, year = {{2021}}, } @article{16294, abstract = {{Model predictive control is a prominent approach to construct a feedback control loop for dynamical systems. Due to real-time constraints, the major challenge in MPC is to solve model-based optimal control problems in a very short amount of time. For linear-quadratic problems, Bemporad et al. have proposed an explicit formulation where the underlying optimization problems are solved a priori in an offline phase. In this article, we present an extension of this concept in two significant ways. We consider nonlinear problems and - more importantly - problems with multiple conflicting objective functions. In the offline phase, we build a library of Pareto optimal solutions from which we then obtain a valid compromise solution in the online phase according to a decision maker's preference. Since the standard multi-parametric programming approach is no longer valid in this situation, we instead use interpolation between different entries of the library. To reduce the number of problems that have to be solved in the offline phase, we exploit symmetries in the dynamical system and the corresponding multiobjective optimal control problem. The results are verified using two different examples from autonomous driving.}}, author = {{Ober-Blöbaum, Sina and Peitz, Sebastian}}, journal = {{International Journal of Robust and Nonlinear Control}}, pages = {{380--403}}, title = {{{Explicit multiobjective model predictive control for nonlinear systems with symmetries}}}, doi = {{10.1002/rnc.5281}}, volume = {{31(2)}}, year = {{2021}}, } @article{29543, author = {{Djema, Walid and Giraldi, Laetitia and Maslovskaya, Sofya and Bernard, Olivier}}, issn = {{0005-1098}}, journal = {{Automatica}}, keywords = {{Electrical and Electronic Engineering, Control and Systems Engineering}}, publisher = {{Elsevier BV}}, title = {{{Turnpike features in optimal selection of species represented by quota models}}}, doi = {{10.1016/j.automatica.2021.109804}}, volume = {{132}}, year = {{2021}}, } @article{32810, author = {{Li, Jiaao and Ma, Yulai and Shi, Yongtang and Wang, Weifan and Wu, Yezhou}}, issn = {{0195-6698}}, journal = {{European Journal of Combinatorics}}, keywords = {{Discrete Mathematics and Combinatorics}}, publisher = {{Elsevier BV}}, title = {{{On 3-flow-critical graphs}}}, doi = {{10.1016/j.ejc.2021.103451}}, volume = {{100}}, year = {{2021}}, } @inproceedings{22894, abstract = {{The first order optimality conditions of optimal control problems (OCPs) can be regarded as boundary value problems for Hamiltonian systems. Variational or symplectic discretisation methods are classically known for their excellent long term behaviour. As boundary value problems are posed on intervals of fixed, moderate length, it is not immediately clear whether methods can profit from structure preservation in this context. When parameters are present, solutions can undergo bifurcations, for instance, two solutions can merge and annihilate one another as parameters are varied. We will show that generic bifurcations of an OCP are preserved under discretisation when the OCP is either directly discretised to a discrete OCP (direct method) or translated into a Hamiltonian boundary value problem using first order necessary conditions of optimality which is then solved using a symplectic integrator (indirect method). Moreover, certain bifurcations break when a non-symplectic scheme is used. The general phenomenon is illustrated on the example of a cut locus of an ellipsoid.}}, author = {{Offen, Christian and Ober-Blöbaum, Sina}}, issn = {{2405-8963}}, keywords = {{optimal control, catastrophe theory, bifurcations, variational methods, symplectic integrators}}, location = {{Berlin, Germany}}, pages = {{334--339}}, title = {{{Bifurcation preserving discretisations of optimal control problems}}}, doi = {{https://doi.org/10.1016/j.ifacol.2021.11.099}}, volume = {{54(19)}}, year = {{2021}}, } @inproceedings{21572, author = {{Ridderbusch, Steffen and Offen, Christian and Ober-Blöbaum, Sina and Goulart, Paul}}, booktitle = {{2021 60th IEEE Conference on Decision and Control (CDC)}}, location = {{Austin, TX, USA}}, pages = {{2896}}, publisher = {{IEEE}}, title = {{{Learning ODE Models with Qualitative Structure Using Gaussian Processes }}}, doi = {{10.1109/CDC45484.2021.9683426}}, year = {{2021}}, } @inproceedings{21592, abstract = {{We propose a reachability approach for infinite and finite horizon multi-objective optimization problems for low-thrust spacecraft trajectory design. The main advantage of the proposed method is that the Pareto front can be efficiently constructed from the zero level set of the solution to a Hamilton-Jacobi-Bellman equation. We demonstrate the proposed method by applying it to a low-thrust spacecraft trajectory design problem. By deriving the analytic expression for the Hamiltonian and the optimal control policy, we are able to efficiently compute the backward reachable set and reconstruct the optimal trajectories. Furthermore, we show that any reconstructed trajectory will be guaranteed to be weakly Pareto optimal. The proposed method can be used as a benchmark for future research of applying reachability analysis to low-thrust spacecraft trajectory design.}}, author = {{Vertovec, Nikolaus and Ober-Blöbaum, Sina and Margellos, Kostas}}, location = {{Rotterdam, the Netherlands}}, pages = {{1975--1980}}, title = {{{Multi-objective minimum time optimal control for low-thrust trajectory design}}}, year = {{2021}}, } @inproceedings{29868, author = {{Jiménez, F. and Ober-Blöbaum, Sina}}, booktitle = {{Nichtlineare Sci 31}}, title = {{{Fractional Damping Through Restricted Calculus of Variations}}}, volume = {{46}}, year = {{2021}}, }