Abstract: Part 1: First, I will describe an adaptive moving mesh method for solving space-fractional partial differential equations of fractional order between 1 and 2. The fractional Laplacian in the PDE model is defined in terms of the Riesz-derivative. The approach extends the so-called L2 method to the non-uniform mesh case. The spatial mesh generation makes use of a moving mesh PDE, MMPDE5 with additional filtering. Numerical experiments are given for the space-fractional Gray-Scott reaction-diffusion model. They reveal a rich set of different patterns, showing interesting and surprising differences in behaviour, compared to the well-known integer order case. The adaptive method detects self-replication patterns, travelling waves, and chaotic solutions, along with two remarkable evolution processes depending on the fractional order: from self-replication to standing waves and from travelling waves back to self-replication.

Part 2: Secondly, I will address a PDE model with a half-Laplacian operator. The analysis of this model relies on on the relationship between the Hilbert transform and the half-Laplacian. A doubling-splitting method is proposed, which results in a backward wave equation (BWE). Next, a second-order parallel boundary value method is applied over a large time scale, showing that the method is convergent and stable, even in ill-posed cases. Two special cases are discussed: an advection-dominated PDE and the space-fractional Schrödinger equation. It is shown that the solution to the BWE is equivalent to the one of the original PDE, both analytically and numerically. As an additional surprising result, we find traveling wave solutions for the linear fractional-order Schrödinger equation.

We consider the task of learning dynamical systems from data in a system-agnostic framework. This presentation is divided into two parts. First, we utilize a simulation-based approach to investigate algorithms for learning chaotic Ordinary Differential Equations (ODEs). We demonstrate that for noise-free data and low-dimensional systems, this task is effectively solved, as polynomial regression-based methods can achieve machine-precision forecasts. However, we show that observational noise remains a significant challenge for most algorithms. In the second part, we address this challenge by developing nonparametric statistical theory for learning ODEs from noisy observations. Specifically, we establish minimax optimal error rates for two contrasting observational models: the Stubble model, consisting of many short trajectories, and the Snake model, consisting of a single long trajectory. We conclude by discussing challenges at the intersection of dynamical systems and statistical learning.

Clouds are important features of the atmosphere, determining the energy budget by interacting with incoming solar radiation and outgoing thermal radiation. For pure ice clouds, the net impact of the different radiative effects is still unknown, and there is no generally accepted theory of clouds in terms of a closed system of partial differential equations or similar.

In this talk, I will present a simple but physically consistent ice cloud model which is a 3D nonlinear ODE system (depending on several parameters). This model constitutes a nonlinear oscillator with two Hopf bifurcations in the relevant parameter regime. Apart from the equilibrium points and bifurcations, limits cycles and scaling behaviours of the system for varying parameters can be determined numerically. Finally, the model shows very good agreement with measurement data, indicating that the main physics is captured and such a simple model might be a helpful tool for investigating ice clouds.

This joint work with Peter Spichtinger.

Let \(p\) be a real polynomial in $n$ variables of even degree \(2d\). A fundamental computational task, with applications in optimization and real algebraic geometry, is to decide whether \(p\) can be written as a sum of squares of polynomials. That is, whether there exist polynomials \(q_1, \ldots, q_m\) such that \(p = q_1^2 + q_2^2 + \cdots + q_m^2\). In this talk, I will discuss the computational complexity of this question.

(based on joint work with Nikolas Gärtner and Victor Magron)

Suppose we are given some data, and we hypothesize a structural causal model to describe them: how can we narrow the set of causal graphs compatible with our observations? The theory of identifiability aims to answer this question. We show that, in the case of additive noise models, the score function of the data contains all the information about the causal graph. However, this requires strong and, crucially, hard-to-verify modeling assumptions, like additivity of the noise. When direct experiments to infer causality are not feasible, this raises the question: how can we move past these restrictions? Borrowing ideas from independent component analysis, we show how multiple environments (read: non i.i.d. data) can overcome these limitations: for structural causal models with arbitrary causal mechanisms, data from only three environments uniquely identify the causal graph from the Jacobian of the score function. Thus, non-i.i.d.-ness turns from a curse into a blessing for causal discovery.

Creating physician rosters is a challenging task due to varying shift structures, qualifications, and department- or hospital-specific regulations. These variations mean that department-specific tools often fail to generalize across hospital settings. To address this, we developed a flexible mixed-integer programming (MIP) model capable of representing different roster structures, and we embedded it into an adaptable web application with an advanced graphical user interface (GUI), allowing physicians to specify preferences and hospital staff to configure the MIP model to their roster requirements without any mathematical or technical background.

The practical implementation of such a system is essential for ensuring long-term acceptance in clinical environments. A sustainable solution must be easy to use, accessible, and well integrated into existing IT workflows. This talk presents the implementation process of our physician rostering framework and highlights the key design decisions that support its usability in practice. Particular emphasis is placed on interface elements that must be operated frequently during the rostering process, as well as on features that allow departments to tailor the system to their specific requirements. We conclude by discussing challenges during integration and showing how the application has been successfully deployed in a hospital department.