@article{30861, abstract = {{AbstractWe consider the problem of maximization of metabolite production in bacterial cells formulated as a dynamical optimal control problem (DOCP). According to Pontryagin’s maximum principle, optimal solutions are concatenations of singular and bang arcs and exhibit the chattering or Fuller phenomenon, which is problematic for applications. To avoid chattering, we introduce a reduced model which is still biologically relevant and retains the important structural features of the original problem. Using a combination of analytical and numerical methods, we show that the singular arc is dominant in the studied DOCPs and exhibits the turnpike property. This property is further used in order to design simple and realistic suboptimal control strategies.}}, author = {{Caillau, Jean-Baptiste and Djema, Walid and Gouzé, Jean-Luc and Maslovskaya, Sofya and Pomet, Jean-Baptiste}}, issn = {{0022-3239}}, journal = {{Journal of Optimization Theory and Applications}}, keywords = {{Applied Mathematics, Management Science and Operations Research, Control and Optimization}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Turnpike Property in Optimal Microbial Metabolite Production}}}, doi = {{10.1007/s10957-022-02023-0}}, year = {{2022}}, } @article{29673, abstract = {{Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this paper, we compare the ground-state transformation and Nelson's stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schrödinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schrödinger equation. Our findings open up a new avenue towards solving Schrödinger's equation using recently developed tools from data science.}}, author = {{Klus, Stefan and Nüske, Feliks and Peitz, Sebastian}}, journal = {{Journal of Physics A: Mathematical and Theoretical}}, number = {{31}}, pages = {{314002}}, publisher = {{IOP Publishing Ltd.}}, title = {{{Koopman analysis of quantum systems}}}, doi = {{10.1088/1751-8121/ac7d22}}, volume = {{55}}, year = {{2022}}, } @unpublished{34618, abstract = {{In this article, we show how second-order derivative information can be incorporated into gradient sampling methods for nonsmooth optimization. The second-order information we consider is essentially the set of coefficients of all second-order Taylor expansions of the objective in a closed ball around a given point. Based on this concept, we define a model of the objective as the maximum of these Taylor expansions. Iteratively minimizing this model (constrained to the closed ball) results in a simple descent method, for which we prove convergence to minimal points in case the objective is convex. To obtain an implementable method, we construct an approximation scheme for the second-order information based on sampling objective values, gradients and Hessian matrices at finitely many points. Using a set of test problems, we compare the resulting method to five other available solvers. Considering the number of function evaluations, the results suggest that the method we propose is superior to the standard gradient sampling method, and competitive compared to other methods.}}, author = {{Gebken, Bennet}}, booktitle = {{arXiv:2210.04579}}, title = {{{Using second-order information in gradient sampling methods for nonsmooth optimization}}}, year = {{2022}}, } @phdthesis{31556, abstract = {{Mehrzieloptimierung behandelt Probleme, bei denen mehrere skalare Zielfunktionen simultan optimiert werden sollen. Ein Punkt ist in diesem Fall optimal, wenn es keinen anderen Punkt gibt, der mindestens genauso gut ist in allen Zielfunktionen und besser in mindestens einer Zielfunktion. Ein notwendiges Optimalitätskriterium lässt sich über Ableitungsinformationen erster Ordnung der Zielfunktionen herleiten. Die Menge der Punkte, die dieses notwendige Kriterium erfüllen, wird als Pareto-kritische Menge bezeichnet. Diese Arbeit enthält neue Resultate über Pareto-kritische Mengen für glatte und nicht-glatte Mehrzieloptimierungsprobleme, sowohl was deren Berechnung betrifft als auch deren Struktur. Im glatten Fall erfolgt die Berechnung über ein Fortsetzungsverfahren, im nichtglatten Fall über ein Abstiegsverfahren. Anschließend wird die Struktur des Randes der Pareto-kritischen Menge analysiert, welcher aus Pareto-kritischen Mengen kleinerer Subprobleme besteht. Schlussendlich werden inverse Probleme betrachtet, bei denen zu einer gegebenen Datenmenge ein Zielfunktionsvektor gefunden werden soll, für den die Datenpunkte kritisch sind.}}, author = {{Gebken, Bennet}}, title = {{{Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization}}}, doi = {{10.17619/UNIPB/1-1327}}, year = {{2022}}, } @unpublished{33150, abstract = {{In this article, we build on previous work to present an optimization algorithm for nonlinearly constrained multi-objective optimization problems. The algorithm combines a surrogate-assisted derivative-free trust-region approach with the filter method known from single-objective optimization. Instead of the true objective and constraint functions, so-called fully linear models are employed and we show how to deal with the gradient inexactness in the composite step setting, adapted from single-objective optimization as well. Under standard assumptions, we prove convergence of a subset of iterates to a quasi-stationary point and if constraint qualifications hold, then the limit point is also a KKT-point of the multi-objective problem.}}, author = {{Berkemeier, Manuel Bastian and Peitz, Sebastian}}, booktitle = {{arXiv:2208.12094}}, title = {{{Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients}}}, year = {{2022}}, } @article{20731, abstract = {{We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the ℓ1-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the ℓ1-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the ℓ1-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach.}}, author = {{Bieker, Katharina and Gebken, Bennet and Peitz, Sebastian}}, journal = {{IEEE Transactions on Pattern Analysis and Machine Intelligence}}, number = {{11}}, pages = {{7797--7808}}, publisher = {{IEEE}}, title = {{{On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation}}}, doi = {{10.1109/TPAMI.2021.3114962}}, volume = {{44}}, year = {{2022}}, } @inproceedings{30733, abstract = {{Hamilton-Jacobi reachability methods for safety-critical control have been well studied, but the safety guarantees derived rely on the accuracy of the numerical computation. Thus, it is crucial to understand and account for any inaccuracies that occur due to uncertainty in the underlying dynamics and environment as well as the induced numerical errors. To this end, we propose a framework for modeling the error of the value function inherent in Hamilton-Jacobi reachability using a Gaussian process. The derived safety controller can be used in conjuncture with arbitrary controllers to provide a safe hybrid control law. The marginal likelihood of the Gaussian process then provides a confidence metric used to determine switches between a least restrictive controller and a safety controller. We test both the prediction as well as the correction capabilities of the presented method in a classical pursuit-evasion example.}}, author = {{Vertovec, Nikolaus and Ober-Blöbaum, Sina and Margellos, Kostas}}, location = {{London}}, pages = {{1870--1875}}, title = {{{Verification of safety critical control policies using kernel methods}}}, year = {{2022}}, } @article{45970, abstract = {{ We introduce a new phase field model for tumor growth where viscoelastic effects are taken into account. The model is derived from basic thermodynamical principles and consists of a convected Cahn–Hilliard equation with source terms for the tumor cells and a convected reaction–diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behavior of tumors, are taken into account. The model is completed by a viscoelastic system consisting of the Navier–Stokes equation for the hydrodynamic quantities, and a general constitutive equation with stress relaxation for the left Cauchy–Green tensor associated with the elastic part of the total mechanical response of the viscoelastic material. For a specific choice of the elastic energy density and with an additional dissipative term accounting for stress diffusion, we prove existence of global-in-time weak solutions of the viscoelastic model for tumor growth in two space dimensions [Formula: see text] by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. [Formula: see text], is required. Moreover, in arbitrary dimensions [Formula: see text], we show stability and existence of solutions for the fully-discrete finite element scheme, where positive definiteness of the discrete Cauchy–Green tensor is proved with a regularization technique that was first introduced by Barrett and Boyaval [Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837]. After that, we improve the regularity results in arbitrary dimensions [Formula: see text] and in two dimensions [Formula: see text], where a CFL condition is required. Then, in two dimensions [Formula: see text], we pass to the limit in the discretization parameters and show that subsequences of discrete solutions converge to a global-in-time weak solution. Finally, we present numerical results in two dimensions [Formula: see text]. }}, author = {{Garcke, Harald and Kovács, Balázs and Trautwein, Dennis}}, issn = {{0218-2025}}, journal = {{Mathematical Models and Methods in Applied Sciences}}, keywords = {{Applied Mathematics, Modeling and Simulation}}, number = {{13}}, pages = {{2673--2758}}, publisher = {{World Scientific Pub Co Pte Ltd}}, title = {{{Viscoelastic Cahn–Hilliard models for tumor growth}}}, doi = {{10.1142/s0218202522500634}}, volume = {{32}}, year = {{2022}}, } @article{45969, abstract = {{AbstractAn evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction–diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the$$H^1$$H1norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.}}, author = {{Elliott, Charles M. and Garcke, Harald and Kovács, Balázs}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{4}}, pages = {{873--925}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces}}}, doi = {{10.1007/s00211-022-01301-3}}, volume = {{151}}, year = {{2022}}, } @article{45963, abstract = {{AbstractThe scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell’s equations. The time-dependent boundary integral equation is discretized with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge–Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments.}}, author = {{Nick, Jörg and Kovács, Balázs and Lubich, Christian}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{4}}, pages = {{1123--1164}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Time-dependent electromagnetic scattering from thin layers}}}, doi = {{10.1007/s00211-022-01277-0}}, volume = {{150}}, year = {{2022}}, } @article{45964, abstract = {{Abstract Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability and convergence of time discretizations of nonlinear parabolic equations on an evolving surface, with linearly implicit backward differentiation formulae characteristic trajectories of the surface, for general locally Lipschitz nonlinearities. The discrete maximal $L^p$-regularity is used to prove the boundedness and stability of numerical solutions in the $L^\infty (0,T;W^{1,\infty })$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^\infty (0,T;W^{1,\infty })$ norm is obtained by combining the stability analysis with the consistency estimates.}}, author = {{Kovács, Balázs and Li, Buyang}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems}}}, doi = {{10.1093/imanum/drac033}}, year = {{2022}}, } @article{45966, abstract = {{Abstract This paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $\tau \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.}}, author = {{Altmann, Robert and Kovács, Balázs and Zimmer, Christoph}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, number = {{2}}, pages = {{950--975}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions}}}, doi = {{10.1093/imanum/drac002}}, volume = {{43}}, year = {{2022}}, } @article{45968, abstract = {{Abstract We derive a numerical method, based on operator splitting, to abstract parabolic semilinear boundary coupled systems. The method decouples the linear components that describe the coupling and the dynamics in the abstract bulk- and surface-spaces, and treats the nonlinear terms similarly to an exponential integrator. The convergence proof is based on estimates for a recursive formulation of the error, using the parabolic smoothing property of analytic semigroups, and a careful comparison of the exact and approximate flows. This analysis also requires a deep understanding of the effects of the Dirichlet operator (the abstract version of the harmonic extension operator), which is essential for the stable coupling in our method. Numerical experiments, including problems with dynamic boundary conditions, reporting on convergence rates are presented.}}, author = {{Csomós, Petra and Farkas, Bálint and Kovács, Balázs}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Error estimates for a splitting integrator for abstract semilinear boundary coupled systems}}}, doi = {{10.1093/imanum/drac079}}, year = {{2022}}, } @article{45958, abstract = {{AbstractIn this paper, we consider a non-linear fourth-order evolution equation of Cahn–Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix–vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.}}, author = {{Beschle, Cedric Aaron and Kovács, Balázs}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{1}}, pages = {{1--48}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces}}}, doi = {{10.1007/s00211-022-01280-5}}, volume = {{151}}, year = {{2022}}, } @article{45956, abstract = {{Abstract The full Maxwell equations in the unbounded three-dimensional space coupled to the Landau–Lifshitz–Gilbert equation serve as a well-tested model for ferromagnetic materials. We propose a weak formulation of the coupled system based on the boundary integral formulation of the exterior Maxwell equations. We show existence and partial uniqueness of a weak solution and propose a new numerical algorithm based on finite elements and boundary elements as spatial discretization with backward Euler and convolution quadrature for the time domain. This is the first numerical algorithm which is able to deal with the coupled system of Landau–Lifshitz–Gilbert equation and full Maxwell’s equations without any simplifications like quasi-static approximations (e.g. eddy current model) and without restrictions on the shape of the domain (e.g. convexity). We show well-posedness and convergence of the numerical algorithm under minimal assumptions on the regularity of the solution. This is particularly important as there are few regularity results available and one generally expects the solution to be non-smooth. Numerical experiments illustrate and expand on the theoretical results.}}, author = {{Bohn, Jan and Feischl, Michael and Kovács, Balázs}}, issn = {{1609-4840}}, journal = {{Computational Methods in Applied Mathematics}}, keywords = {{Applied Mathematics, Computational Mathematics, Numerical Analysis}}, number = {{1}}, pages = {{19--48}}, publisher = {{Walter de Gruyter GmbH}}, title = {{{FEM-BEM Coupling for the Maxwell–Landau–Lifshitz–Gilbert Equations via Convolution Quadrature: Weak Form and Numerical Approximation}}}, doi = {{10.1515/cmam-2022-0145}}, volume = {{23}}, year = {{2022}}, } @article{55280, author = {{Elsholtz, Ch. and Klahn, B. and Technau, Marc}}, journal = {{Acta Arith.}}, number = {{3}}, pages = {{251–263}}, title = {{{On polynomials with roots modulo almost all primes}}}, doi = {{10.4064/aa220407-9-7}}, volume = {{205:3}}, year = {{2022}}, } @article{34839, abstract = {{We describe the relations among the ℓ-torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent in some sense. Altogether, the three conjectures are equivalent for the class of solvable groups. We then prove the ℓ-torsion conjecture for ℓ-groups and the other two conjectures for nilpotent groups.}}, author = {{Klüners, Jürgen and Wang, Jiuya}}, issn = {{0002-9939}}, journal = {{Proceedings of the American Mathematical Society}}, keywords = {{Applied Mathematics, General Mathematics}}, number = {{7}}, pages = {{2793--2805}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{ℓ-torsion bounds for the class group of number fields with an ℓ-group as Galois group}}}, doi = {{10.1090/proc/15882}}, volume = {{150}}, year = {{2022}}, } @article{34835, abstract = {{We prove an upper bound for the asymptotics of counting functions of number fields with nilpotent Galois groups. }}, author = {{Klüners, Jürgen}}, issn = {{0065-1036}}, journal = {{Acta Arithmetica}}, keywords = {{Algebra and Number Theory}}, number = {{2}}, pages = {{165--184}}, publisher = {{Institute of Mathematics, Polish Academy of Sciences}}, title = {{{The asymptotics of nilpotent Galois groups}}}, doi = {{10.4064/aa211207-16-5}}, volume = {{204}}, year = {{2022}}, } @article{44624, author = {{Faulwasser, Timm and Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Schaller, Manuel and Worthmann, Karl}}, journal = {{Mathematics of Control, Signals, and Systems}}, pages = {{759--788}}, publisher = {{Springer}}, title = {{{Manifold turnpikes, trims, and symmetries}}}, volume = {{34}}, year = {{2022}}, } @article{33741, abstract = {{There are many concepts of signed graph coloring which are defined by assigning colors to the vertices of the graphs. These concepts usually differ in the number of self-inverse colors used. We introduce a unifying concept for this kind of coloring by assigning elements from symmetric sets to the vertices of the signed graphs. In the first part of the paper, we study colorings with elements from symmetric sets where the number of self-inverse elements is fixed. We prove a Brooks’-type theorem and upper bounds for the corresponding chromatic numbers in terms of the chromatic number of the underlying graph. These results are used in the second part where we introduce the symset-chromatic number χsym(G,σ) of a signed graph (G,σ). We show that the symset-chromatic number gives the minimum partition of a signed graph into independent sets and non-bipartite antibalanced subgraphs. In particular, χsym(G,σ)≤χ(G). In the final section we show that these colorings can also be formalized as DP-colorings.}}, author = {{Cappello, Chiara and Steffen, Eckhard}}, issn = {{0218-0006}}, journal = {{Annals of Combinatorics}}, keywords = {{Discrete Mathematics and Combinatorics}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Symmetric Set Coloring of Signed Graphs}}}, doi = {{10.1007/s00026-022-00593-4}}, year = {{2022}}, }