Talks

We generalize a classical inequality between the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, planar domains: in 1955, Payne proved that below the k-th eigenvalue of the Dirichlet Laplacian there exist at least k+2 eigenvalues of the Neumann Laplacian, provided the domain is convex. It has, however, been conjectured that this should hold for any domain. Here we show that the statement indeed remains true for all simply connected planar Lipschitz domains. The proof relies on a novel variational principle.

Time-domain simulation of large-scale problems becomes computationally prohibitive if space-parallelization saturates. This is particularly challenging if long time periods are considered, e.g., if the start-up of an electrical machine until steady state is simulated. In this contribution, several parallel-in-time methods are discussed for initial-boundary-value problems and for time-periodic boundary value problems. All those methods are based on a subdivision of the time interval into as many subintervals as computing cores are available. For example, the well-known parareal method works similarly to multiple shooting methods; it solves two types of problems iteratively until convergence is reached: a cheap problem defined on coarse grids is solved sequentially on the whole time-interval to propagate initial conditions (and approximate derivatives) and secondly, high-fidelity problems are solved on the subintervals in parallel. We also discuss Paraexp and Waveform Relaxation methods in the context of real world engineering problems from electrical engineering.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

Physics Informed Neural Networks (PINNs) are becoming increasingly important in solving initial and boundary value problems. In contrast to conventional neural networks, they do not require labelled data for training and can thus be assigned to the field of unsupervised learning [3]. In this work, a PINN is to be trained to learn the equation of motion of a charged particle in an electromagnetic field. It turns out that networks trained using the L-BFGS opimisation algorithm show better convergence behaviour than those trained using the Adam optimisation algorithm commonly used in deep learning. In addition, it turns out that pre-training neural networks on the solution of a numerical method such as the Crank-Nicolson method can significantly speed up the training of PINNS.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

Parallel-in-time algorithms provide an additional layer of concurrency for the numerical integration of models based on time-dependent differential equations. Methods like Parareal, which parallelize across multiple time steps, rely on a computationally cheap and coarse integrator to propagate information forward in time, while a parallelizable expensive fine propagator provides accuracy. Typically, the coarse method is a numerical integrator using lower resolution, reduced order or a simplified model. Our reasearch proposes to use a physics-informed neural network (PINN) instead. We demonstrate for the Black-Scholes equation, a partial differential equation from computational finance, that Parareal with a PINN coarse propagator provides better speedup than a numerical coarse propagator. Training and evaluating a neural network are both tasks whose computing patterns are well suited for GPUs. By contrast, mesh-based algorithms with their low computational intensity struggle to perform well. We show that moving the coarse propagator PINN to a GPU while running the numerical fine propagator on the CPU further improves Parareal's single-node performance. This suggests that integrating machine learning techniques into parallel-in-time integration methods and exploiting their differences in computing patterns might offer a way to better utilize heterogeneous architectures.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

The availability of reliable, high-resolution climate and weather data is important to inform long-term decisions on climate adaptation and mitigation and to guide rapid responses to extreme events. Forecasting models are limited by computational costs and, therefore, often generate coarse-resolution predictions. Two common ways to decrease computational efforts with DL are downscaling, the increase of the resolution directly on the predicted climate variables, and emulation, the replacement of model parts to achieve faster runs initially. Here, we look at several downscaling tasks and an aerosol emulation problem. While deep learning shows promising results it may not obey simple physical constraints, such as mass conservation or mass positivity. We tackle this by investigating both soft and hard constraining methodologies in different setups, showing that incorporating hard constraints can be beneficial for both downscaling and emulation problems.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

The theory of evolutionary equations, afforded by Rainer Picard (Dresden) et al., provides a well-posedness theorem applicable to a vast amount of linear PDEs including heat, wave and Maxwell's equations as well as equations including fractional derivatives and integrals. In this talk, I will discuss this well-posedness theorem in the autonomous case. I will show how to impose initial and boundary conditions on such evolutionary equations, and I will present a possible evolutionary approach to control theory for PDEs.