@article{45967, author = {{Binz, Tim and Kovács, Balázs}}, journal = {{arXiv}}, title = {{{A convergent finite element algorithm for mean curvature flow in higher codimension}}}, year = {{2021}}, } @article{45962, abstract = {{Abstract An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow and powers of mean and inverse mean curvature flow. Error estimates are proved for semidiscretizations and full discretizations for the generalized flow. The algorithm proposed and studied here combines evolving surface finite elements, whose nodes determine the discrete surface, and linearly implicit backward difference formulae for time integration. The numerical method is based on a system coupling the surface evolution to nonlinear second-order parabolic evolution equations for the normal velocity and normal vector. A convergence proof is presented in the case of finite elements of polynomial degree at least 2 and backward difference formulae of orders 2 to 5. The error analysis combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed surface position, velocity, normal vector, normal velocity and therefore for the mean curvature. The stability analysis is performed in the matrix–vector formulation and is independent of geometric arguments, which only enter the consistency analysis. Numerical experiments are presented to illustrate the convergence results and also to report on monotone quantities, e.g. Hawking mass for inverse mean curvature flow, and complemented by experiments for nonconvex surfaces.}}, author = {{Binz, Tim and Kovács, Balázs}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, number = {{3}}, pages = {{2545--2588}}, publisher = {{Oxford University Press (OUP)}}, title = {{{A convergent finite element algorithm for generalized mean curvature flows of closed surfaces}}}, doi = {{10.1093/imanum/drab043}}, volume = {{42}}, year = {{2021}}, } @article{45957, abstract = {{AbstractA proof of convergence is given for a bulk–surface finite element semidiscretisation of the Cahn–Hilliard equation with Cahn–Hilliard-type dynamic boundary conditions in a smooth domain. The semidiscretisation is studied in an abstract weak formulation as a second-order system. Optimal-order uniform-in-time error estimates are shown in the $L^2$- and $H^1$-norms. The error estimates are based on a consistency and stability analysis. The proof of stability is performed in an abstract framework, based on energy estimates exploiting the anti-symmetric structure of the second-order system. Numerical experiments illustrate the theoretical results.}}, author = {{Harder, Paula and Kovács, Balázs}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, number = {{3}}, pages = {{2589--2620}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Error estimates for the Cahn–Hilliard equation with dynamic boundary conditions}}}, doi = {{10.1093/imanum/drab045}}, volume = {{42}}, year = {{2021}}, } @article{45961, author = {{Nick, Jörg and Kovács, Balázs and Lubich, Christian}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{4}}, pages = {{997--1000}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Correction to: Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations}}}, doi = {{10.1007/s00211-021-01196-6}}, volume = {{147}}, year = {{2021}}, } @article{45959, author = {{Kovács, Balázs and Li, Buyang and Lubich, Christian}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{3}}, pages = {{595--643}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{A convergent evolving finite element algorithm for Willmore flow of closed surfaces}}}, doi = {{10.1007/s00211-021-01238-z}}, volume = {{149}}, year = {{2021}}, } @article{55290, author = {{Baier, St. and Mazumder, D. and Technau, Marc}}, journal = {{Unif. Distrib. Theory}}, number = {{2}}, pages = {{1–48}}, title = {{{On the distribution of αp modulo one in quadratic number fields}}}, doi = {{10.2478/udt-2021-0006}}, volume = {{16}}, year = {{2021}}, } @article{55289, author = {{Technau, Marc and Zafeiropoulos, A.}}, journal = {{Acta Arith.}}, number = {{1}}, pages = {{93–104}}, title = {{{Metric results on summatory arithmetic functions on Beatty sets}}}, doi = {{10.4064/aa200128-10-6}}, volume = {{197}}, year = {{2021}}, } @article{34840, abstract = {{In this paper we obtain a complete list of imaginary n-quadratic fields with class groups of exponent 3 and 5 under ERH for every positive integer n where an n-quadratic field is a number field of degree 2ⁿ represented as the composite of n quadratic fields. }}, author = {{Klüners, Jürgen and Komatsu, Toru}}, issn = {{0025-5718}}, journal = {{Mathematics of Computation}}, keywords = {{Applied Mathematics, Computational Mathematics, Algebra and Number Theory}}, number = {{329}}, pages = {{1483--1497}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{Imaginary multiquadratic number fields with class group of exponent $3$ and $5$}}}, doi = {{10.1090/mcom/3609}}, volume = {{90}}, year = {{2021}}, } @article{34912, abstract = {{Let E be an ordinary elliptic curve over a finite field and g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E⁹ . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E³ and of the Igusa modular form in dimension 4. We illustrate our algorithms with examples of curves with many rational points over finite fields. }}, author = {{Kirschmer, Markus and Narbonne, Fabien and Ritzenthaler, Christophe and Robert, Damien}}, issn = {{0025-5718}}, journal = {{Mathematics of Computation}}, keywords = {{Applied Mathematics, Computational Mathematics, Algebra and Number Theory}}, number = {{333}}, pages = {{401--449}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{Spanning the isogeny class of a power of an elliptic curve}}}, doi = {{10.1090/mcom/3672}}, volume = {{91}}, year = {{2021}}, } @article{19938, abstract = {{We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error. }}, author = {{McLachlan, Robert I and Offen, Christian}}, journal = {{Foundations of Computational Mathematics}}, number = {{6}}, pages = {{1363--1400}}, title = {{{Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation}}}, doi = {{10.1007/s10208-020-09454-z}}, volume = {{20}}, year = {{2020}}, } @article{19939, author = {{Kreusser, Lisa Maria and McLachlan, Robert I and Offen, Christian}}, issn = {{0951-7715}}, journal = {{Nonlinearity}}, number = {{5}}, pages = {{2335--2363}}, title = {{{Detection of high codimensional bifurcations in variational PDEs}}}, doi = {{10.1088/1361-6544/ab7293}}, volume = {{33}}, year = {{2020}}, } @inbook{17411, abstract = {{Many dynamical systems possess symmetries, e.g. rotational and translational invariances of mechanical systems. These can be beneficially exploited in the design of numerical optimal control methods. We present a model predictive control scheme which is based on a library of precomputed motion primitives. The primitives are equivalence classes w.r.t. the symmetry of the optimal control problems. Trim primitives as relative equilibria w.r.t. this symmetry, play a crucial role in the algorithm. The approach is illustrated using an academic mobile robot example.}}, author = {{Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Peitz, Sebastian}}, booktitle = {{Advances in Dynamics, Optimization and Computation}}, editor = {{Junge, Oliver and Schütze, Oliver and Froyland, Gary and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin}}, isbn = {{9783030512637}}, issn = {{2198-4182}}, publisher = {{Springer}}, title = {{{Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach}}}, doi = {{10.1007/978-3-030-51264-4_9}}, year = {{2020}}, } @article{21819, abstract = {{Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.}}, author = {{Klus, Stefan and Nüske, Feliks and Hamzi, Boumediene}}, issn = {{1099-4300}}, journal = {{Entropy}}, title = {{{Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator}}}, doi = {{10.3390/e22070722}}, year = {{2020}}, } @article{10596, abstract = {{Multi-objective optimization is an active field of research that has many applications. Owing to its success and because decision-making processes are becoming more and more complex, there is a recent trend for incorporating many objectives into such problems. The challenge with such problems, however, is that the dimensions of the solution sets—the so-called Pareto sets and fronts—grow with the number of objectives. It is thus no longer possible to compute or to approximate the entire solution set of a given problem that contains many (e.g. more than three) objectives. On the other hand, the computation of single solutions (e.g. via scalarization methods) leads to unsatisfying results in many cases, even if user preferences are incorporated. In this article, the Pareto Explorer tool is presented—a global/local exploration tool for the treatment of many-objective optimization problems (MaOPs). In the first step, a solution of the problem is computed via a global search algorithm that ideally already includes user preferences. In the second step, a local search along the Pareto set/front of the given MaOP is performed in user specified directions. For this, several continuation-like procedures are proposed that can incorporate preferences defined in decision, objective, or in weight space. The applicability and usefulness of Pareto Explorer is demonstrated on benchmark problems as well as on an application from industrial laundry design.}}, author = {{Schütze, Oliver and Cuate, Oliver and Martín, Adanay and Peitz, Sebastian and Dellnitz, Michael}}, issn = {{0305-215X}}, journal = {{Engineering Optimization}}, number = {{5}}, pages = {{832--855}}, title = {{{Pareto Explorer: a global/local exploration tool for many-objective optimization problems}}}, doi = {{10.1080/0305215x.2019.1617286}}, volume = {{52}}, year = {{2020}}, } @article{16288, abstract = {{We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.}}, author = {{Klus, Stefan and Nüske, Feliks and Peitz, Sebastian and Niemann, Jan-Hendrik and Clementi, Cecilia and Schütte, Christof}}, issn = {{0167-2789}}, journal = {{Physica D: Nonlinear Phenomena}}, title = {{{Data-driven approximation of the Koopman generator: Model reduction, system identification, and control}}}, doi = {{10.1016/j.physd.2020.132416}}, volume = {{406}}, year = {{2020}}, } @inbook{16289, abstract = {{In the development of model predictive controllers for PDE-constrained problems, the use of reduced order models is essential to enable real-time applicability. Besides local linearization approaches, proper orthogonal decomposition (POD) has been most widely used in the past in order to derive such models. Due to the huge advances concerning both theory as well as the numerical approximation, a very promising alternative based on the Koopman operator has recently emerged. In this chapter, we present two control strategies for model predictive control of nonlinear PDEs using data-efficient approximations of the Koopman operator. In the first one, the dynamic control system is replaced by a small number of autonomous systems with different yet constant inputs. The control problem is consequently transformed into a switching problem. In the second approach, a bilinear surrogate model is obtained via a convex combination of these autonomous systems. Using a recent convergence result for extended dynamic mode decomposition (EDMD), convergence of the reduced objective function can be shown. We study the properties of these two strategies with respect to solution quality, data requirements, and complexity of the resulting optimization problem using the 1-dimensional Burgers equation and the 2-dimensional Navier–Stokes equations as examples. Finally, an extension for online adaptivity is presented.}}, author = {{Peitz, Sebastian and Klus, Stefan}}, booktitle = {{Lecture Notes in Control and Information Sciences}}, isbn = {{9783030357122}}, issn = {{0170-8643}}, pages = {{257--282}}, publisher = {{Springer}}, title = {{{Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator}}}, doi = {{10.1007/978-3-030-35713-9_10}}, volume = {{484}}, year = {{2020}}, } @article{16290, abstract = {{The control of complex systems is of critical importance in many branches of science, engineering, and industry, many of which are governed by nonlinear partial differential equations. Controlling an unsteady fluid flow is particularly important, as flow control is a key enabler for technologies in energy (e.g., wind, tidal, and combustion), transportation (e.g., planes, trains, and automobiles), security (e.g., tracking airborne contamination), and health (e.g., artificial hearts and artificial respiration). However, the high-dimensional, nonlinear, and multi-scale dynamics make real-time feedback control infeasible. Fortunately, these high- dimensional systems exhibit dominant, low-dimensional patterns of activity that can be exploited for effective control in the sense that knowledge of the entire state of a system is not required. Advances in machine learning have the potential to revolutionize flow control given its ability to extract principled, low-rank feature spaces characterizing such complex systems.We present a novel deep learning modelpredictive control framework that exploits low-rank features of the flow in order to achieve considerable improvements to control performance. Instead of predicting the entire fluid state, we use a recurrent neural network (RNN) to accurately predict the control relevant quantities of the system, which are then embedded into an MPC framework to construct a feedback loop. In order to lower the data requirements and to improve the prediction accuracy and thus the control performance, incoming sensor data are used to update the RNN online. The results are validated using varying fluid flow examples of increasing complexity.}}, author = {{Bieker, Katharina and Peitz, Sebastian and Brunton, Steven L. and Kutz, J. Nathan and Dellnitz, Michael}}, issn = {{0935-4964}}, journal = {{Theoretical and Computational Fluid Dynamics}}, pages = {{577–591}}, title = {{{Deep model predictive flow control with limited sensor data and online learning}}}, doi = {{10.1007/s00162-020-00520-4}}, volume = {{34}}, year = {{2020}}, } @article{16309, abstract = {{In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via Dynamic Mode Decomposition, a quantization approach was recently proposed in [Peitz & Klus, Automatica 106, 2019]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operator-based reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls. We show that when using the Koopman generator, this relaxation --- realized by linear interpolation between two operators --- does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball.}}, author = {{Peitz, Sebastian and Otto, Samuel E. and Rowley, Clarence W.}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, number = {{3}}, pages = {{2162--2193}}, title = {{{Data-Driven Model Predictive Control using Interpolated Koopman Generators}}}, doi = {{10.1137/20M1325678}}, volume = {{19}}, year = {{2020}}, } @article{29399, author = {{Limebeer, D. J. N. and Ober-Blöbaum, Sina and Farshi, F. H.}}, journal = {{IEEE Transactions on Automatic Control}}, pages = {{1381--1396}}, title = {{{Variational integrators for dissipative systems}}}, volume = {{65(4)}}, year = {{2020}}, } @article{16297, abstract = {{In real-world problems, uncertainties (e.g., errors in the measurement, precision errors) often lead to poor performance of numerical algorithms when not explicitly taken into account. This is also the case for control problems, where optimal solutions can degrade in quality or even become infeasible. Thus, there is the need to design methods that can handle uncertainty. In this work, we consider nonlinear multi-objective optimal control problems with uncertainty on the initial conditions, and in particular their incorporation into a feedback loop via model predictive control (MPC). In multi-objective optimal control, an optimal compromise between multiple conflicting criteria has to be found. For such problems, not much has been reported in terms of uncertainties. To address this problem class, we design an offline/online framework to compute an approximation of efficient control strategies. This approach is closely related to explicit MPC for nonlinear systems, where the potentially expensive optimization problem is solved in an offline phase in order to enable fast solutions in the online phase. In order to reduce the numerical cost of the offline phase, we exploit symmetries in the control problems. Furthermore, in order to ensure optimality of the solutions, we include an additional online optimization step, which is considerably cheaper than the original multi-objective optimization problem. We test our framework on a car maneuvering problem where safety and speed are the objectives. The multi-objective framework allows for online adaptations of the desired objective. Alternatively, an automatic scalarizing procedure yields very efficient feedback controls. Our results show that the method is capable of designing driving strategies that deal better with uncertainties in the initial conditions, which translates into potentially safer and faster driving strategies.}}, author = {{Hernández Castellanos, Carlos Ignacio and Ober-Blöbaum, Sina and Peitz, Sebastian}}, journal = {{International Journal of Robust and Nonlinear Control}}, pages = {{7593--7618}}, title = {{{Explicit Multi-objective Model Predictive Control for Nonlinear Systems Under Uncertainty}}}, doi = {{10.1002/rnc.5197}}, volume = {{30(17)}}, year = {{2020}}, }