Vorträge

Accurate flow simulations remain a challenging task. In this talk we discuss the use of deep neural networks for augmenting classical finite element simulations in fluid-dynamics.
We first investigate the classical DFG-benchmark in 3D. We extend these settings to higher Reynolds numbers. At high Reynolds numbers, accurate simulations in 3D settings become increasingly difficult, and the classical methods reach their limits. To address this issue, we discuss approaches to connect the finite element method with neural networks. We propose the Deep Neural Network Multigrid Solver, which combines a geometric multigrid solver with a deep neural network to overcome limitations of classical methods. This approach uses classical simulation techniques where their strengths are eminent, such as the efficient representation of a coarse, large-scale flow field. Neural networks are used when a full resolution of the effects does not seem possible or efficient. While our method is tightly embedded in a geometric multigrid framework, it remains flexible, allowing for the coarse grid problem to be addressed with various solvers and enabling local enrichment without a global fine grid.
We demonstrate the efficiency, generalizability, and scalability of our proposed approach using 3D simulations. Our focus is on issues of stability, generalizability.
In the second part of the talk we give an outlook on future developments towards an extension to space-time multigrid methods within the framework of space-time finite element methods. We discuss the efficient implementation of space-time multigrid methods using the matrix-free framework provided by the deal.ii finite element library and demonstrate its effectiveness for the heat and acoustic wave equation.

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Radial basis function finite difference (RBF-FD) discretization has recently emerged as an al-
ternative to classical finite difference or finite element discretization of (systems) of partial
differential equations. We focus on the construction of preconditioners for the iterative solution
of the resulting linear systems of equa- tions. In RBF-FD, a higher discretization accuracy may
be obtained by increasing the stencil size. This, however, leads to a less sparse and often also
worse conditioned stiffness matrix which are both challenges for subsequent iterative solvers. We
propose to construct preconditioners based on stiffness matrices resulting from RBF-FD dis-
cretization with smaller stencil sizes compared to the one for the actual system to be solved. In
our numerical results, we focus on RBF-FD discretizations based on polyharmonic splines (PHS)
with polynomial augmentation. We illustrate the performance of smaller stencil preconditioners
in the solution of the three-dimensional convection-diffusion equation.

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I will be introducing myself formally and presenting my master's thesis, which concerns the study of numerical schemes for solving singularly perturbed differential equations, focusing on the Haar wavelet method.

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This talk focuses on the mathematical models underlying reinforcement learning in artificial intelligence, particularly the reward functions in Markov Decision Processes. I argue that ethical principles related to well-being, safety, and equality are inherently reflected in these mathematical models. Building on this foundation, I then demonstrate how ethics can inform computational mathematics, while also addressing the challenges one encounters in this domain. Specifically, I discuss how the mathematical models behind reinforcement learning may rely on a distorted representation of ethics with respect to the determinacy and commensurability of ethical values.

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In Numerical Weather Prediction and Climate modelling, computational efficiency and numerical accuracy are paramount. This work aims to implement time-parallel Spectral Deferred Correction (SDC) methods in LFRic-Atmosphere, the Met Office’s next-generation atmospheric model, designed to exploit the new supercomputers with improved scalability; the use of a quasi-uniform cubed-sphere mesh is integral to this, as is the underlying lowest-order compatible finite element spatial discretisation. LFRic-Atmosphere has an iterative semi-implicit time stepping structure with a Method of Lines finite-transport scheme using an explicit Runge-Kutta time discretisation. Time parallel SDC offers increased temporal accuracy with small computation cost, this could be utilised over the whole time discretisation, or to target a specific time discretised component.
I will present two approaches in this talk. The first approach is using serial SDC as the time discretisation of LFRic-Atmosphere’s finite-volume transport scheme. The second approach is using a serial IMEX SDC time stepper to compare to the semi-implicit time stepping structure in LFRic-Atmosphere. My initial work has explored both using the shallow water equations, I will present results from the standard shallow water test-cases.

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The Maxey-Riley Equation (MRE) models the motion of a finite-sized, spherical particle moving in a fluid. Applications using the MRE are, for example, the study of the spread of Coronavirus particles in a room, the formation of clouds and the so-called marine snow. The MRE is a second-order, implicit integro-differential equation with a singular kernel at initial time. For over 35 years, researchers used approximations and numerical schemes with high storage requirements or ignored the integral term, although its impact can be relevant. A major break-through was reached in 2019, when Prasath et al. mapped the MRE to a time-dependent Robin-type boundary condition of the 1D Heat equation, thus removing the requirement to store the full history. They provided an implicit integral form of the solution by using the so-called Fokas method that could be later solved with a numerical scheme and a nonlinear solver. While Prasath et al.’s method can deliver numerical solutions accurately, the need to evaluate nested integrals makes it computationally costly and it becomes impractical for computing trajectories of a large number of particles. In this talk, we present a new fourth order finite differences scheme and compare its accuracy and performance with Prasath et al’s method as well as other existing schemes. We then apply our method for the calculation of Lagrangian Coherent Structures, a large scale fluid structure, and point out for which cases, the approximations on the MRE have a considerable influence on these structures and the use of the full MRE models is relevant.

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