@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{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}}, } @misc{33273, abstract = {{Dieses Lernangebot widmet sich der linearen Algebra als dem Teil der Mathematik, der neben der Optimierung und der Stochastik die Grundlage für praktisch alle Entwicklungen im Bereich Künstliche Intelligenz (KI) darstellt. Das Fach ist jedoch für Anfänger meist ungewohnt abstrakt und wird daher oft als besonders schwierig und unanschaulich empfunden. In diesem Kurs wird das Erlernen mathematischer Kenntnisse in linearer Algebra verknüpft mit dem aktuellen und faszinierenden Anwendungsfeld der künstlichen neuronalen Netze (KNN). Daraus ergeben sich in natürlicher Weise Anwendungsbeispiele, an denen die wesentlichen Konzepte der linearen Algebra erklärt werden können. Behandelte Themen sind: Der Vektorraum der reellen Zahlentupel, reelle Vektorräume allgemein Lineare Abbildungen Matrizen Koordinaten und darstellende Matrizen Lineare Gleichungssysteme, Gaußalgorithmus Determinante Ein Ausblick auf nichtlineare Techniken, die für neuronale Netzwerke relevant sind.}}, author = {{Schramm, Thomas and Gasser, Ingenuin and Schwenker, Sören and Seiler, Ruedi and Lohse, Alexander and Zobel, Kay}}, publisher = {{Hamburg Open Online University}}, title = {{{Linear Algebra driven by Data Science}}}, year = {{2020}}, } @misc{33272, author = {{Nijholt, Eddie and Rink, Bob and Schwenker, Sören}}, booktitle = {{DSWeb}}, number = {{April}}, title = {{{Generalised Symmetry in Network Dynamics}}}, year = {{2020}}, } @article{33262, abstract = {{The authors of Berg et al. [J. Algebra 348 (2011) 446–461] provide an algorithm for finding a complete system of primitive orthogonal idempotents for CM, where M is any finite R-trivial monoid. Their method relies on a technical result stating that R-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an R-trivial monoid may be realized by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where L-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We first show that our algorithm works for ZM, after which we prove that it also works for RM where R is an arbitrary ring with a known complete system of primitive orthogonal idempotents. In particular, our algorithm works if R is any field. In this respect our result constitutes a considerable generalization of the results in Berg et al. [J. Algebra 348 (2011) 446–461]. Moreover, the system of idempotents for RM is obtained from the one our algorithm yields for ZM in a straightforward manner. In other words, for any finite R-trivial monoid M our algorithm only has to be performed for ZM, after which a system of idempotents follows for any ring with a given system of idempotents.}}, author = {{Nijholt, Eddie and Rink, Bob and Schwenker, Sören}}, issn = {{0219-4988}}, journal = {{Journal of Algebra and Its Applications}}, keywords = {{Applied Mathematics, Algebra and Number Theory}}, number = {{12}}, publisher = {{World Scientific Pub Co Pte Ltd}}, title = {{{A new algorithm for computing idempotents of ℛ-trivial monoids}}}, doi = {{10.1142/s0219498821502273}}, volume = {{20}}, year = {{2020}}, } @article{33263, abstract = {{Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. We show that many such properties can be encoded by means of quiver representations. These properties include classical symmetry, hidden symmetry, and feedforward structure, as well as subnetwork and quotient relations in network dynamical systems. A quiver equivariant dynamical system consists of a collection of dynamical systems with maps between them that send solutions to solutions. We prove that such quiver structures are preserved under Lyapunov--Schmidt reduction, center manifold reduction, and normal form reduction.}}, author = {{Nijholt, Eddie and Rink, Bob W. and Schwenker, Sören}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, keywords = {{Modeling and Simulation, Analysis}}, number = {{4}}, pages = {{2428--2468}}, publisher = {{Society for Industrial & Applied Mathematics (SIAM)}}, title = {{{Quiver Representations and Dimension Reduction in Dynamical Systems}}}, doi = {{10.1137/20m1345670}}, volume = {{19}}, year = {{2020}}, } @article{45953, abstract = {{Abstract $L^2$ norm error estimates of semi- and full discretizations of wave equations with dynamic boundary conditions, using bulk–surface finite elements and Runge–Kutta methods, are studied. The analysis rests on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed, which fit into the abstract framework. For problems with velocity terms or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than 2. These can also be observed in the presented numerical experiments.}}, author = {{Hipp, David and Kovács, Balázs}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, number = {{1}}, pages = {{638--728}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates}}}, doi = {{10.1093/imanum/drz073}}, volume = {{41}}, year = {{2020}}, } @article{45955, author = {{Akrivis, Georgios and Feischl, Michael and Kovács, Balázs and Lubich, Christian}}, issn = {{0025-5718}}, journal = {{Mathematics of Computation}}, keywords = {{Applied Mathematics, Computational Mathematics, Algebra and Number Theory}}, number = {{329}}, pages = {{995--1038}}, publisher = {{American Mathematical Society (AMS)}}, title = {{{Higher-order linearly implicit full discretization of the Landau–Lifshitz–Gilbert equation}}}, doi = {{10.1090/mcom/3597}}, volume = {{90}}, year = {{2020}}, } @article{45952, author = {{Kovács, Balázs and Li, Buyang and Lubich, Christian}}, issn = {{1463-9963}}, journal = {{Interfaces and Free Boundaries}}, keywords = {{Applied Mathematics}}, number = {{4}}, pages = {{443--464}}, publisher = {{European Mathematical Society - EMS - Publishing House GmbH}}, title = {{{A convergent algorithm for forced mean curvature flow driven by diffusion on the surface}}}, doi = {{10.4171/ifb/446}}, volume = {{22}}, year = {{2020}}, } @phdthesis{33265, abstract = {{This thesis deals with the investigation of dynamical properties – in particular generic synchrony breaking bifurcations – that are inherent to the structure of a semigroup network as well the numerous algebraic structures that are related to these types of networks. Most notably we investigate the interplay between network dynamics and monoid representation theory as induced by the fundamental network construction in terms of hidden symmetry as introduced by RINK and SANDERS. After providing a brief survey of the field of network dynamics in Part I, we thoroughly introduce the formalism of semigroup networks, the customized dynamical systems theory, and the necessary background from monoid representation theory in Chapters 3 and 4. The remainder of Part II investigates generic synchrony breaking bifurcations and contains three major results. The first is Theorem 5.11, which shows that generic symmetry breaking steady state bifurcations in monoid equivariant dynamics occur along absolutely indecomposable subrepresentations – a natural generalization of the corresponding statement for group equivariant dynamics. Then Theorem 7.12 relates the decomposition of a representation given by a network with high-dimensional internal phase spaces to that induced by the same network with one-dimensional internal phase spaces. This result is used to show that there is a smallest dimension of internal dynamics in which all generic l-parameter bifurcations of a fundamental network can be observed (Theorem 7.24). In Part III, we employ the machinery that was summarized and further developed in Part II to feedforward networks. We propose a general definition of this structural feature of a network and show that it can equivalently be characterized in different algebraic notions in Theorem 8.35. These are then exploited to fully classify the corresponding monoid representation for any feedforward network and to classify generic synchrony breaking steady state bifurcations with one- or highdimensional internal dynamics.}}, author = {{Schwenker, Sören}}, publisher = {{Universität Hamburg}}, title = {{{Genericity in Network Dynamics}}}, 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}}, }