TY - JOUR AB - Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML. AU - Dellnitz, Michael AU - Hüllermeier, Eyke AU - Lücke, Marvin AU - Ober-Blöbaum, Sina AU - Offen, Christian AU - Peitz, Sebastian AU - Pfannschmidt, Karlson ID - 21600 IS - 2 JF - SIAM Journal on Scientific Computing TI - Efficient time stepping for numerical integration using reinforcement learning VL - 45 ER - TY - CONF AU - Schwerin, Imke AU - Häsel-Weide, Uta ED - Novotna, J. ED - Moraova, H. ID - 46757 T2 - International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education TI - Second grader´s understanding of doubling and halfing in various representations ER - TY - CHAP AU - Schmidt, Rebekka AU - Tenberge, Claudia AU - Häsel-Weide, Uta ED - Vöing, N. ED - Schmidt, R. ED - Neiske, I. ID - 46758 T2 - Aktive Teilhabe fördern – ICM und Student Engagement in der Hochschullehre TI - Lehre in Zeiten von Digitalisierung und Inklusion - Beispiele aus drei Fächern ER - TY - CHAP AU - Biehler, Rolf AU - Engel, Joachim AU - Frischemeier, Daniel ED - Bruder, Regina ED - Büchter, A. ED - Gasteiger, H. ED - Schmidt-Thieme, B. ED - Weigand, HG. ID - 47416 SN - 9783662666036 T2 - Handbuch der Mathematikdidaktik TI - Stochastik: Leitidee Daten und Zufall ER - TY - GEN AB - In this proceeding we consider a translation invariant Nelson type model in two spatial dimensions modeling a scalar relativistic particle in interaction with a massive radiation field. As is well-known, the corresponding Hamiltonian can be defined with the help of an energy renormalization. First, we review a Feynman-Kac formula for the semigroup generated by this Hamiltonian proven by the authors in a recent preprint (where several matter particles and exterior potentials are treated as well). After that, we employ a few technical key relations and estimates obtained in our preprint to present an otherwise self-contained derivation of new Feynman-Kac formulas for the fiber Hamiltonians attached to fixed total momenta of the translation invariant system. We conclude by inferring an alternative derivation of the Feynman-Kac formula for the full translation invariant Hamiltonian. AU - Hinrichs, Benjamin AU - Matte, Oliver ID - 47534 T2 - arXiv:2309.09005 TI - Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions ER - TY - CHAP AU - Biehler, Rolf AU - Frischemeier, Daniel AU - Gould, Ronald AU - Pfannkuch, Maxine ED - Pepin, Birgit ED - Gueudet, Ghislaine ED - Choppin, Jeffrey ID - 48042 T2 - Handbook of Digital Resources in Mathematics Education TI - Impacts of Digitalization on Content and Goals of Statistics Education ER - TY - CHAP AU - Leiss, Dominik AU - Gerlach, Kerstin AU - Wessel, Lena AU - Schmidt-Thieme, Barbara ID - 48319 SN - 9783662666036 T2 - Handbuch der Mathematikdidaktik TI - Sprache und Mathematiklernen ER - TY - CHAP AU - Kortmeyer, Jörg AU - Biehler, Rolf ED - Dreyfus, T. ED - Gonzalez-Martin, A. S. ED - Nardi, E. ED - Monaghan, J. ED - Thompson, P. W. ID - 48480 T2 - The Learning and Teaching of Calculus Across Disciplines – Proceedings of the Second Calculus Conference TI - The use of integrals for accumulation and mean values in basic electrical engineering courses ER - TY - JOUR AU - Biehler, Rolf AU - Krüger, Katja ID - 49049 IS - 115 JF - GDM-Mitteilungen TI - Von Glückstal (Ukraine) nach Frankfurt: Problemlösen, Algorithmen, Stochastik – Nachruf auf Arthur Engel ER - TY - CONF AU - Wallner, Melina AU - Häsel-Weide, Uta ED - Novotna, J. ED - Moraova, H. ID - 46756 T2 - International Symposium in Elementary Mathematics Teaching. Proceedings: New Directions in Elementary Mathematics Education TI - Conceptual understanding of third grades for axial symmetry ER -