Fehlerbearbeitung bei mathematischen Textaufgaben – Sprachliche und strategische Fehlerursachen und ihre Bearbeitung - Prediger, Susanne; Dröse, Jennifer (Lernen und Lernstörungen, 10(2))

High frequency limits for invariant Ruelle densities - Guillarmou, Colin; Hilgert, Joachim; Weich, Tobias (Annales Henri Lebesgue)

Algebraically Independent Generators for the Algebra of Invariant Differential Operators on SLn(R)/SOn(R) - Brennecken, Dominik; Hilgert, Joachim; Ciardo, Lorenzo (Journal of Lie Theory)

Mathematik - Ein Reiseführer 2. Auflage

Higher-rank quantum-classical correspondence - Hilgert, Joachim; Weich, Tobias; Wolf, L.

Philip Ording: 99 Variations on a Proof. Princeton University Press 2019 - Hilgert, Joachim (Mathematische Semesterberichte)

Georg Glaeser (Hrsg.): 77-mal Mathematik für zwischendurch – Unterhaltsame Kuriositäten und unorthodoxe Anwendungen. Springer Spektrum 2020 - Hilgert, Joachim (Mathematische Semesterberichte)

A convergent finite element algorithm for mean curvature flow in higher codimension - Binz, Tim; Kovács, Balázs (arXiv)

A convergent finite element algorithm for generalized mean curvature flows of closed surfaces - Binz, Tim; Kovács, Balázs (IMA Journal of Numerical Analysis)
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.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Error estimates for the Cahn–Hilliard equation with dynamic boundary conditions - Harder, Paula; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract A 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.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Correction to: Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations - Nick, Jörg; Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

A convergent evolving finite element algorithm for Willmore flow of closed surfaces - Kovács, Balázs; Li, Buyang; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data - Hesse, Kerstin; Sloan, Ian H.; Womersley, Robert S. (Journal of Computational and Applied Mathematics)
Keywords: Applied Mathematics; Computational Mathematics

Spectral Theory of the Frame Flow on Hyperbolic 3-Manifolds - Guillarmou, Colin; Küster, Benjamin (Annales Henri Poincaré)
Keywords: Mathematical Physics; Nuclear and High Energy Physics; Statistical and Nonlinear Physics

Mathematics textbooks and curriculum resources as instruments for change - Rezat, Sebastian; Fan, Lianghuo; Pepin, Birgit (ZDM – Mathematics Education)
Abstract In this survey paper we aim to provide an overview of research on mathematics textbooks and, more broadly, curriculum resources as instruments for change related to mathematical content, instructional goals and practices, and student learning of mathematics. In particular, we elaborate on the following themes: (1) The role of curriculum resources as instruments for change from a theoretical perspective; (2) The design of curriculum resources to mediate the implementation of reform ideas and innovative practice; (3) Teachers’ influence on the implementation of change through curriculum resources; (4) Students’ influence on the implementation of change through curriculum resources; and (5) Evidence of curriculum resources yielding changes in student-related factors or variables. We claim that, whilst textbooks and curriculum resources are influential, they alone cannot change teachers’ teaching nor students’ learning practices in times of curricular change. Moreover, more knowledge is needed about features of curriculum resources that support the implementation of change. We contend that curriculum innovations are likely to be successful, if teachers and students are supported to co- and re-design the relevant curriculum trajectories and materials in line with the reform efforts and their own individual needs.
Keywords: General Mathematics; Education

Challenges of making sense of tasks and automated feedback in digital mathematics textbooks - Rezat, Sebastian; Schacht, Florian; Häsel-Weide, Uta (Mathematics Education in the Digital Age. Learning, Practice and Theory)

How automated feedback from a digital mathematics textbook affects primary students’ conceptual development: two case studies - Rezat, Sebastian (ZDM Mathematics Education)
Abstract One of the most prevalent features of digital mathematics textbooks, compared to traditional ones, is the provision of automated feedback on students’ solutions. Since feedback is regarded as an important factor that influences learning, this is often seen as an affordance of digital mathematics textbooks. While there is a large body of mainly quantitative research on the effectiveness of feedback in general, very little is known about how feedback actually affects students’ individual content specific learning processes and conceptual development. A theoretical framework based on Rabardel’s theory of the instrument and Vergnaud’s theory of conceptual fields is developed to study qualitatively how feedback actually functions in the learning process. This framework was applied in a case study of two elementary school students’ learning processes when working on a probability task from a German 3rd grade digital textbook. The analysis allowed detailed reconstruction of how students made sense of the information provided by the feedback and adjusted their behavior accordingly. This in-depth analysis unveiled that feedback does not necessarily foster conceptual development in the desired way, and a correct solution does not always coincide with conceptual understanding. The results point to some obstacles that students face when working individually on tasks from digital mathematics textbooks with automated feedback, and indicate that feedback needs to be developed in design-based research cycles in order to yield the desired effects.
Keywords: General Mathematics; Education

Fachwissen zur Arithmetik bei Grundschullehramtsstudierenden – Entwicklung im ersten Semester und Veränderungen durch eine Lehrinnovation - Hochmuth, Reinhard; Biehler, Rolf; Blum, Werner; Achmetli, Kay; Rode, Jana; Krawitz, Janina; Schukajlow, Stanislaw; Bender, Peter; Haase, Jürgen (Lehrinnovationen in der Hochschulmathematik . Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik)

Integration digitaler Lernmaterialien in die Präsenzlehre am Beispiel des Mathematikvorkurses für Ingenieure an der Universität Paderborn - Fleischmann, Yael; Biehler, Rolf; Gold, Alexander; Mai, Tobias (Lehrinnovationen in der Hochschulmathematik . Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik)

Die Online-Lernmaterialien im Online-Mathematikvorkurs studiVEMINT: Konzeption und Ergebnisse von Nutzer- und Evaluationsstudien - Gold, Alexander; Fleischmann, Yael; Mai, Tobias; Biehler, Rolf; Kempen, Leander (Lehrinnovationen in der Hochschulmathematik . Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik)

Einführung: Lehrinnovationen in der Hochschulmathematik – praxisrelevant – didaktisch fundiert – forschungsbasiert - Biehler, Rolf; Eichler, Andreas; Hochmuth, Reinhard; Rach, Stefanie; Schaper, Niclas (Lehrinnovationen in der Hochschulmathematik . Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik)

Mathematikvorkurse als Brücke in das Studium–Einführung - Biehler, Rolf (Lehrinnovationen in der Hochschulmathematik . Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik)

Introducing students to machine learning with decision trees using CODAP and Jupyter Notebooks - Biehler, Rolf; Fleischer, Yannik (Teaching Statistics)
Keywords: Education; Statistics and Probability

Lehrinnovationen in der Hochschulmathematik

3D-Geometrie–virtuell und real - Biehler, Rolf; Weigand, Hans-Georg (mathematik lehren)

Mathematics in Engineering Education: a Review of the Recent Literature with a View towards Innovative Practices - Pepin, Birgit; Biehler, Rolf; Gueudet, Ghislaine (International Journal of Research in Undergraduate Mathematics Education)
Abstract The aim of the special issue is to bring together important current international research on innovative teaching and learning practices in mathematics in engineering education, and to develop deeper understandings of the characteristics of current teaching and learning practices that can inform the design and implementation of future innovative practice. The focus of this review paper is to provide a state-of-the-art overview of this emerging field at the cross-roads between mathematics and engineering education, in addition to introducing the papers of this special issue. To guide this paper, we posed three review questions: (1) How can current (teaching/learning/study) practices of mathematics in engineering education be characterized with a view towards innovation?; (2) What are the ‘resources’ (cognitive, material, digital, social) used, and what are those that appear also well suited for innovative courses?; (3) What are promising innovative practices in mathematics in engineering education, and what are the implications for curriculum reform? Looking back across the studies we summarized in the review, we conclude that they are lagging behind the more fundamental changes that are happening in engineering education, whilst addressing selected aspects of innovative changes within the current system of engineering education. At the same time, the nine papers of this special issue contribute new perspectives for innovative practices in mathematics in engineering education, for a better understanding of current practices and for future research.
Keywords: Education; Mathematics (miscellaneous)

Design-Based Research in der Hochschullehre am Beispiel der Lehrveranstaltung „Einführung in die Kultur der Mathematik“ - Kempen, Leander; Biehler, Rolf (Lehrinnovationen in der Hochschulmathematik . Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik)

Education and professional development of university mathematics teachers - Winsløw, Carl; Biehler, Rolf; Jaworski, Barbara; Rønning, Frode; Wawro, Megan (Research and Development in University Mathematics Education)

Schatten von 3D-Objekten: Modellierung mit GeoGebra 3D und Anwendungen in der Computergrafik - Höper, Lukas; Malin, Leah; Biehler, Rolf (mathematik lehren)

Faszination 3D-Film: Entwicklung einer 3D-Konstruktion - Hüsing, Sven; Weiser, Niklas; Biehler, Rolf (mathematik lehren)

Consultation Phases in Mathematics Learning and Support Centres - Schürmann, Mirko; Panse, Anja; Shaikh, Zain; Biehler, Rolf; Schaper, Niclas; Liebendörfer, Michael; Hilgert, Joachim (International Journal of Research in Undergraduate Mathematics Education)
Abstract Mathematics Learning Support Centres are becoming more and more common in higher education both internationally and in Germany. Whereas it is clear that their quality largely depends on a functioning interaction in consultations, little is known about how such consultations proceed in detail. On the basis of models from the literature and recorded support sessions (N = 36), we constructed a process model that divides consultations into four ideal–typical phases. In the individual consultations, forward or backward leaps occur, but overall the model seems to describe the data well. A high intercoder reliability shows that it can be applied consistently on real data by different researchers. An analysis of the consultations between students and tutors shows that both mainly work on past attempts or thoughts of the students to solve the exercise or problems and on concrete strategies to solve a problem within the session. In contrast, very little time is dedicated to summarizing and reflecting the solution. The data allows for a more in-depth discussion of what constitutes quality in advising processes and how it might be further explored. Practically, the model may structure support sessions and help in focussing on different goals in different phases.
Keywords: Education; Mathematics (miscellaneous)

The process of actively building a model for a randomization test – insights into learners’ modeling activities based on a case study - Podworny, Susanne; Biehler, Rolf (Mathematical Thinking and Learning)
Keywords: Developmental and Educational Psychology; Education; General Mathematics

A first introduction to data science education in secondary schools: Teaching and learning about data exploration withCODAPusing survey data - Frischemeier, Daniel; Biehler, Rolf; Podworny, Susanne; Budde, Lea (Teaching Statistics)
Keywords: Education; Statistics and Probability

Opportunities and Possibilities of a Network of Mathematical Learning and Support Centres in Germany - Schürmann, Mirko; Liebendörfer, Michael; Gildehaus, Lara; Schaper, Niclas; Hochmuth, Reinhard; Biehler, Rolf; Lankeit, Elisa; Kuklinski, Christiane; Ruge, Johanna (Sigma newsletter)

Was ist eine gute Skizze? Strategiewissen beim mathematischen Modellieren im Bereich der Geometrie fördern - Schönherr, Johanna; Schukajlow, S.; Blomberg, J. (mathematik lehren)

Does strategic knowledge matter? Effects of strategic knowledge about drawing on students’ modeling competencies in the domain of geometry - Schönherr, Johanna; Schukajlow, S.; Blomberg, J.; Leopold, C. (Mathematical Thinking and Learning)

Do emotions and prior performance facilitate the use of the learner-generated drawing strategy? Effects of enjoyment, anxiety, and intramathematical performance on the use of the drawing strategy and modelling performance - Schönherr, Johanna; Blomberg, J.; Schukajlow, S.; Leopold, C. (Contemporary Educational Psychology)

Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation - McLachlan, Robert I; Offen, Christian (Foundations of Computational Mathematics)
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.

Detection of high codimensional bifurcations in variational PDEs - Kreusser, Lisa Maria; McLachlan, Robert I; Offen, Christian (Nonlinearity)

Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach - Flaßkamp, Kathrin; Ober-Blöbaum, Sina; Peitz, Sebastian (Advances in Dynamics, Optimization and Computation)
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.

Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator - Klus, Stefan; Nüske, Feliks; Hamzi, Boumediene (Entropy)
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.

Das Kompetenzzentrum Hochschuldidaktik Mathematik (khdm) - Hochmuth, Reinhard; Liebendörfer, Michael; Biehler, Rolf; Eichler, Andreas (Neues Handbuch Hochschullehre)

Pareto Explorer: a global/local exploration tool for many-objective optimization problems - Schütze, Oliver; Cuate, Oliver; Martín, Adanay; Peitz, Sebastian; Dellnitz, Michael (Engineering Optimization)
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.

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control - Klus, Stefan; Nüske, Feliks; Peitz, Sebastian; Niemann, Jan-Hendrik; Clementi, Cecilia; Schütte, Christof (Physica D: Nonlinear Phenomena)
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.

Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator - Peitz, Sebastian; Klus, Stefan (Lecture Notes in Control and Information Sciences)
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.

Deep model predictive flow control with limited sensor data and online learning - Bieker, Katharina; Peitz, Sebastian; Brunton, Steven L.; Kutz, J. Nathan; Dellnitz, Michael (Theoretical and Computational Fluid Dynamics)
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.

Data-Driven Model Predictive Control using Interpolated Koopman Generators - Peitz, Sebastian; Otto, Samuel E.; Rowley, Clarence W. (SIAM Journal on Applied Dynamical Systems)
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.

Rekonstruktion diagrammatischen Schließens beim Erlernen der Subtraktion negativer Zahlen. Vergleich zweier methodischer Zugänge - Schumacher, Jan; Rezat, Sebastian (Zeichen und Sprache im Mathematikunterricht)
Diagrammatisches Schließen wird im Zusammenhang mit dem Lernen von Mathmematik und ihrer Symbolsprache als wesentliche Theorie der Wissenskonstruktion diskutiert. Dabei wird häufig davon ausgegangen, dass die Wissenskonstruktion im Sinne diagrammatischen Schließens erfolgt. Deskriptive Rekonstruktionen diagrammatischen Schließens bei Lernenden stellen jedoch ein Desiderat der mathematikdidaktischen Forschung dar. Der vorliegende Beitrag befasst sich mit der Fragestellung, wie sich diagrammatisches Schließen bei Lernenden rekonstruieren lässt. Als mögliche Werkzeuge für eine solche Rekonstruktion werden Toulmins Argumentationsschema und Vergnauds Schema-Begriff exemplarisch angewandt, um das diagrammatische Schließen eines Schülerpaars beim Einstieg in die Subtraktion negativer Zahlen zu rekonstruieren. Abschließend wird die tatsächliche Eignung der beiden Ansätze zur Rekonstruktion diagrammatischen Schließens diskutiert.

Deduktion und Abduktion beim diagrammatischen Schließen – das didaktische Potential der Peirceschen Semiotik - Schumacher, Jan

A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors - Gerlach, Raphael; Ziessler, Adrian; Eckhardt, Bruno; Dellnitz, Michael (SIAM Journal on Applied Dynamical Systems)
In this work we present a set-oriented path following method for the computation of relative global attractors of parameter-dependent dynamical systems. We start with an initial approximation of the relative global attractor for a fixed parameter λ0 computed by a set-oriented subdivision method. By using previously obtained approximations of the parameter-dependent relative global attractor we can track it with respect to a one-dimensional parameter λ > λ0 without restarting the whole subdivision procedure. We illustrate the feasibility of the set-oriented path following method by exploring the dynamics in low-dimensional models for shear flows during the transition to turbulence and of large-scale atmospheric regime changes .