Das Oberseminar richtet sich an Studenten und Forscher mit Interesse an aktuellen Fragestellung in der Numerik, wissenschaftlichem Rechnen und Optimierung. Es finden in unregelmäßigen Abständen Vorträge zu aktuellen Themen statt.
Mittwoch, 20.09.2023 (11:00 Uhr): Tileuzhan Mukhamet (Master student at RUB)
Titel: An arbitrary Lagrangian Eulerian Discontinuous Galerkin method for the Vlasov equation with a strong magnetic field
Freitag, 11.08.2023 (11:00 Uhr, IA 1/53): Mahima Yadav (PhD student at RUB)
Titel: On discrete ground states of rotating Bose-Einstein condensates
Abstract: The talk focuses on the study of ground states of Bose-Einstein condensates in a rotating frame. The ground states are described as the constrained minimizers of the Gross-Pitaevskii energy functional with an angular momentum term. The problem is discretized using Lagrange finite element spaces of arbitrary polynomial order and the approximation properties of the corresponding numerical approximations are presented, taking into account the missing uniqueness of ground states which is mainly caused by the invariance of the energy functional under complex phase shifts. Error estimates of optimal order are shown for the L2- and H1-norm, as well as for the ground state energy and chemical potential.
Donnerstag, 11.05.2023 (11:15 Uhr): Paul Wilhelm (RWTH Aachen)
Titel: NuFI: An implicit Lagrangian Flow reconstruction scheme for the Vlasov equation
Abstract: The Vlasov equation is a high-dimensional partial differential equation arising from kinetic theory and used to model the behaviour of plasma flows in collision-less and strongly non-equilibrium regimes. We present a novel approach, the numerical flow iteration (NuFI), which evaluates the numerical solution via storing the three-dimensional electric potentials, using these to iteratively reconstruct the Lagrangian flow and directly evaluating the initial data via the method of characteristics. This reduces the total memory-requirement by several orders of magnitude and allows to shift workload from frequent memory access to computations on the fly, i.e., yielding high flop/Byte-rates, which is favourable on modern compute-architectures. Furthermore using the Lagrangian formulation one conserves desired properties like $L^p$-norms and kinetic entropy exactly, as well as total energy up to time-integration error. We demonstrate the accuracy and scalability of the new approach on several test-cases in up to six dimensions showing computations done on a workstation as well as a GPU-cluster.
Donnerstag, 27.04.2023 (11:00 Uhr): Ivo Dravins (RUB)
Title: Preconditioning for block matrices with square blocks
Abstract: Linear systems of equations appear in one way or another in almost every scientific and engineering problem. They are so ubiquitous that, in addition to solving linear problems, also non-linear problems are typically reduced to a sequence of linear ones. The availability of modern large-scale computational resources motivates the development and the use of well parallelizable efficient solvers with a limited memory footprint. For many problems, these properties can be achieved by the employment of iterative solution methods combined with preconditioning techniques. We explore the design of preconditioners for block-matrices with square blocks. This form of matrices occurs in many applications, encountered for instance when numerically solving partial differential equations, ordinary differential equations and others.
We focus on two classes of problems, one being optimal control problems within the PDE-constrained optimization framework, and the other being fully implicit Runge-Kutta time-stepping schemes. Both necessitate the solution of large and sparse linear systems, for which we employ preconditioned Krylov subspace methods. The main topic of the talk is on the design of preconditioners, although the entire solution procedure is explored.
Donnerstag, 10.11.2022 (13 Uhr): Fado Philo (Universität Duisburg-Essen)