@article{19945, abstract = {{Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations, …) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration.}}, author = {{McLachlan, Robert I and Offen, Christian and Tapley, Benjamin K}}, issn = {{2158-2505}}, journal = {{Journal of Computational Dynamics}}, number = {{1}}, pages = {{111--130}}, publisher = {{American Institute of Mathematical Sciences (AIMS)}}, title = {{{Symplectic integration of PDEs using Clebsch variables}}}, doi = {{10.3934/jcd.2019005}}, volume = {{6}}, year = {{2019}}, } @article{21944, author = {{Nüske, Feliks and Boninsegna, Lorenzo and Clementi, Cecilia}}, issn = {{0021-9606}}, journal = {{The Journal of Chemical Physics}}, title = {{{Coarse-graining molecular systems by spectral matching}}}, doi = {{10.1063/1.5100131}}, year = {{2019}}, } @article{16709, author = {{Sahai, Tuhin and Ziessler, Adrian and Klus, Stefan and Dellnitz, Michael}}, issn = {{0924-090X}}, journal = {{Nonlinear Dynamics}}, title = {{{Continuous relaxations for the traveling salesman problem}}}, doi = {{10.1007/s11071-019-05092-5}}, year = {{2019}}, } @article{10593, abstract = {{We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. Using a recent convergence result for the numerical approximation via Extended Dynamic Mode Decomposition (EDMD), we show that the value of the K-ROM based objective function converges in measure to the value of the full objective function. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier–Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.}}, author = {{Peitz, Sebastian and Klus, Stefan}}, issn = {{0005-1098}}, journal = {{Automatica}}, pages = {{184--191}}, title = {{{Koopman operator-based model reduction for switched-system control of PDEs}}}, doi = {{10.1016/j.automatica.2019.05.016}}, volume = {{106}}, year = {{2019}}, } @article{10595, abstract = {{In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems where only a subset of the set of objective functions is taken into account. If the Pareto critical set is completely described by its boundary (e.g., if we have more objective functions than dimensions in decision space), then this can be used to efficiently solve the MOP by solving a number of MOPs with fewer objective functions. If this is not the case, the results can still give insight into the structure of the Pareto critical set.}}, author = {{Gebken, Bennet and Peitz, Sebastian and Dellnitz, Michael}}, issn = {{0925-5001}}, journal = {{Journal of Global Optimization}}, number = {{4}}, pages = {{891--913}}, title = {{{On the hierarchical structure of Pareto critical sets}}}, doi = {{10.1007/s10898-019-00737-6}}, volume = {{73}}, year = {{2019}}, } @inproceedings{10597, abstract = {{In comparison to classical control approaches in the field of electrical drives like the field-oriented control (FOC), model predictive control (MPC) approaches are able to provide a higher control performance. This refers to shorter settling times, lower overshoots, and a better decoupling of control variables in case of multi-variable controls. However, this can only be achieved if the used prediction model covers the actual behavior of the plant sufficiently well. In case of model deviations, the performance utilizing MPC remains below its potential. This results in effects like increased current ripple or steady state setpoint deviations. In order to achieve a high control performance, it is therefore necessary to adapt the model to the real plant behavior. When using an online system identification, a less accurate model is sufficient for commissioning of the drive system. In this paper, the combination of a finite-control-set MPC (FCS-MPC) with a system identification is proposed. The method does not require high-frequency signal injection, but uses the measured values already required for the FCS-MPC. An evaluation of the least squares-based identification on a laboratory test bench showed that the model accuracy and thus the control performance could be improved by an online update of the prediction models.}}, author = {{Hanke, Soren and Peitz, Sebastian and Wallscheid, Oliver and Böcker, Joachim and Dellnitz, Michael}}, booktitle = {{2019 IEEE International Symposium on Predictive Control of Electrical Drives and Power Electronics (PRECEDE)}}, isbn = {{9781538694145}}, title = {{{Finite-Control-Set Model Predictive Control for a Permanent Magnet Synchronous Motor Application with Online Least Squares System Identification}}}, doi = {{10.1109/precede.2019.8753313}}, year = {{2019}}, } @inproceedings{29867, author = {{Faulwasser, Tim and Flaßkamp, K. and Ober-Blöbaum, Sina and Worthmann, Karl}}, pages = {{490--495}}, title = {{{Towards velocity turnpikes in optimal control of mechanical systems}}}, volume = {{52(16)}}, year = {{2019}}, } @article{16708, abstract = {{ In this work we extend the novel framework developed by Dellnitz, Hessel-von Molo, and Ziessler to the computation of finite dimensional unstable manifolds of infinite dimensional dynamical systems. To this end, we adapt a set-oriented continuation technique developed by Dellnitz and Hohmann for the computation of such objects of finite dimensional systems with the results obtained in the work of Dellnitz, Hessel-von Molo, and Ziessler. We show how to implement this approach for the analysis of partial differential equations and illustrate its feasibility by computing unstable manifolds of the one-dimensional Kuramoto--Sivashinsky equation as well as for the Mackey--Glass delay differential equation. }}, author = {{Ziessler, Adrian and Dellnitz, Michael and Gerlach, Raphael}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, number = {{3}}, pages = {{1265--1292}}, title = {{{The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques}}}, doi = {{10.1137/18m1204395}}, volume = {{18}}, year = {{2019}}, } @article{34917, abstract = {{We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space (V,q) with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of (V,q). This yields a good algorithm to enumerate a system of representatives of proper isometry classes of lattices in genera of maximal lattices in (V,q).}}, author = {{Kirschmer, Markus and Nebe, Gabriele}}, issn = {{1793-0421}}, journal = {{International Journal of Number Theory}}, keywords = {{Algebra and Number Theory}}, number = {{02}}, pages = {{309--325}}, publisher = {{World Scientific Pub Co Pte Lt}}, title = {{{Quaternary quadratic lattices over number fields}}}, doi = {{10.1142/s1793042119500131}}, volume = {{15}}, year = {{2019}}, } @article{34916, abstract = {{We describe the powers of irreducible polynomials occurring as characteristic polynomials of automorphisms of even unimodular lattices over number fields. This generalizes results of Gross & McMullen and Bayer-Fluckiger & Taelman.}}, author = {{Kirschmer, Markus}}, issn = {{0022-314X}}, journal = {{Journal of Number Theory}}, keywords = {{Algebra and Number Theory}}, pages = {{121--134}}, publisher = {{Elsevier BV}}, title = {{{Automorphisms of even unimodular lattices over number fields}}}, doi = {{10.1016/j.jnt.2018.08.004}}, volume = {{197}}, year = {{2019}}, } @article{45948, author = {{Kovács, Balázs and Li, Buyang and Lubich, Christian}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{4}}, pages = {{797--853}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{A convergent evolving finite element algorithm for mean curvature flow of closed surfaces}}}, doi = {{10.1007/s00211-019-01074-2}}, volume = {{143}}, year = {{2019}}, } @article{55284, author = {{Elsholtz, Ch. and Technau, Marc and Technau, N.}}, journal = {{Mathematika}}, number = {{4}}, pages = {{990–1009}}, title = {{{The maximal order of iterated multiplicative functions}}}, doi = {{10.1112/S0025579319000214}}, volume = {{64}}, year = {{2019}}, } @article{55285, author = {{Technau, Marc}}, journal = {{Notes Number Theory Discrete Math.}}, number = {{2}}, pages = {{127–135}}, title = {{{Generalised Beatty sets}}}, doi = {{10.7546/nntdm.2019.25.2.127-135}}, volume = {{25}}, year = {{2019}}, } @article{34915, abstract = {{We describe the determinants of the automorphism groups of Hermitian lattices over local fields. Using a result of G. Shimura, this yields an explicit method to compute the special genera in a given genus of Hermitian lattices over a number field.}}, author = {{Kirschmer, Markus}}, issn = {{0003-889X}}, journal = {{Archiv der Mathematik}}, keywords = {{General Mathematics}}, number = {{4}}, pages = {{337--347}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Determinant groups of Hermitian lattices over local fields}}}, doi = {{10.1007/s00013-019-01348-z}}, volume = {{113}}, year = {{2019}}, } @article{21, abstract = {{We address the general mathematical problem of computing the inverse p-th root of a given matrix in an efficient way. A new method to construct iteration functions that allow calculating arbitrary p-th roots and their inverses of symmetric positive definite matrices is presented. We show that the order of convergence is at least quadratic and that adaptively adjusting a parameter q always leads to an even faster convergence. In this way, a better performance than with previously known iteration schemes is achieved. The efficiency of the iterative functions is demonstrated for various matrices with different densities, condition numbers and spectral radii.}}, author = {{Richters, Dorothee and Lass, Michael and Walther, Andrea and Plessl, Christian and Kühne, Thomas}}, journal = {{Communications in Computational Physics}}, number = {{2}}, pages = {{564--585}}, publisher = {{Global Science Press}}, title = {{{A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices}}}, doi = {{10.4208/cicp.OA-2018-0053}}, volume = {{25}}, year = {{2019}}, } @unpublished{16711, abstract = {{Embedding techniques allow the approximations of finite dimensional attractors and manifolds of infinite dimensional dynamical systems via subdivision and continuation methods. These approximations give a topological one-to-one image of the original set. In order to additionally reveal their geometry we use diffusion mapst o find intrinsic coordinates. We illustrate our results on the unstable manifold of the one-dimensional Kuramoto--Sivashinsky equation, as well as for the attractor of the Mackey-Glass delay differential equation.}}, author = {{Gerlach, Raphael and Koltai, Péter and Dellnitz, Michael}}, booktitle = {{arXiv:1902.08824}}, title = {{{Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems}}}, year = {{2019}}, } @misc{8482, author = {{Jurgelucks, Benjamin and Schulze, Veronika and Feldmann, Nadine and Claes, Leander}}, title = {{{Arbitrary sensitivity for inverse problems in piezoelectricity}}}, year = {{2019}}, } @article{19935, abstract = {{A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples. }}, author = {{McLachlan, Robert I and Offen, Christian}}, issn = {{0951-7715}}, journal = {{Nonlinearity}}, pages = {{2895--2927}}, title = {{{Bifurcation of solutions to Hamiltonian boundary value problems}}}, doi = {{10.1088/1361-6544/aab630}}, year = {{2018}}, } @article{19937, abstract = {{Symplectic integrators can be excellent for Hamiltonian initial value problems. Reasons for this include their preservation of invariant sets like tori, good energy behaviour, nonexistence of attractors, and good behaviour of statistical properties. These all refer to {\em long-time} behaviour. They are directly connected to the dynamical behaviour of symplectic maps φ:M→M' on the phase space under iteration. Boundary value problems, in contrast, are posed for fixed (and often quite short) times. Symplecticity manifests as a symplectic map φ:M→M' which is not iterated. Is there any point, therefore, for a symplectic integrator to be used on a Hamiltonian boundary value problem? In this paper we announce results that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not.}}, author = {{McLachlan, Robert I and Offen, Christian}}, issn = {{1017-1398}}, journal = {{Numerical Algorithms}}, pages = {{1219--1233}}, title = {{{Symplectic integration of boundary value problems}}}, doi = {{10.1007/s11075-018-0599-7}}, year = {{2018}}, } @unpublished{21634, abstract = {{Predictive control of power electronic systems always requires a suitable model of the plant. Using typical physics-based white box models, a trade-off between model complexity (i.e. accuracy) and computational burden has to be made. This is a challenging task with a lot of constraints, since the model order is directly linked to the number of system states. Even though white-box models show suitable performance in most cases, parasitic real-world effects often cannot be modeled satisfactorily with an expedient computational load. Hence, a Koopman operator-based model reduction technique is presented which directly links the control action to the system's outputs in a black-box fashion. The Koopman operator is a linear but infinite-dimensional operator describing the dynamics of observables of nonlinear autonomous dynamical systems which can be nicely applied to the switching principle of power electronic devices. Following this data-driven approach, the model order and the number of system states are decoupled which allows us to consider more complex systems. Extensive experimental tests with an automotive-type permanent magnet synchronous motor fed by an IGBT 2-level inverter prove the feasibility of the proposed modeling technique in a finite-set model predictive control application.}}, author = {{Hanke, Sören and Peitz, Sebastian and Wallscheid, Oliver and Klus, Stefan and Böcker, Joachim and Dellnitz, Michael}}, booktitle = {{arXiv:1804.00854}}, title = {{{Koopman Operator-Based Finite-Control-Set Model Predictive Control for Electrical Drives}}}, year = {{2018}}, }