Vorträge

This talk will be devoted to several topics in classical and modern control theory. Classical stabilization techniques for linear and nonlinear control systems as well as modern attempts to linearize nonlinear systems will constitute the core for this presentation. In particular, the first part will consist of a brief summary of my Master's Thesis on the foundations of mathematical control theory in finite dimension. During the second part, we will catch a glimpse into modern control theory involving the Koopman operator focussing on advances and difficulties. The third part will be on my current PhD topic "Time-Optimal Control of Linear Systems in Non-Reflexive Banach Spaces". The official introduction to my person will also not come too short.

This talk will start with a high-level overview on how machine learning can be used to improve weather and climate predictions. Afterwards, the talk will provide more detail on recent developments of machine learned weather forecast models and how they compare to conventional models and numerical methods.

We study the propagation of electromagnetic waves in heterogeneous structures. The governing equations for this problem are Maxwell's equations with highly oscillatory parameters. We use an analytic homogenization result, which yields an effective Maxwell system that involves additional dispersive effects.

The Finite Element Heterogeneous Multiscale Method (FE-HMM) is used to discretize in space, and we provide a semi-discrete error estimate. The rigorous error analysis in space is supplemented by a standard time discretization combined with a recursive approximation of the convolution that relies on the assumption that the convolution kernel is an exponential function. Eventually, we present numerical experiments both for the microscopic and the macroscopic scale.

We consider stochastic PDEs driven by an additive or multiplicative Gaussian noise of the form
$$
\begin{cases} \mathrm{d} u &=(A u + F(t,u))\,\mathrm{d} t + G(t,u) \,\mathrm{d} W~~~\text{ on } [0,T],\\ u(0) &= u_0 \in L^p(\Omega;X)
\end{cases}
$$
on a Hilbert space $X$. Here, $A$ is the generator of a contractive $C_0$-semigroup $(S(t))_{t\geq 0}$, $W$ is a cylindrical Brownian motion, $F$ and $G$ are globally Lipschitz and of linear growth, $p \in [2,\infty)$, and $u_0$ is the initial data.
Our aim is to obtain strong convergence rates for a temporal discretisation scheme of the form $U_0 = u_0$,
$$
U_j = R_k U_{j-1} + k R_k F(t_{j-1},U_{j-1})+ R_k G(t_{j-1},U_{j-1}) \Delta W^{j},~j=1,\ldots,N_k
$$
with time step $k>0$, Wiener increments $\Delta W^j$, and contractive time discretisation scheme $R:[0,\infty) \to \mathcal{L}(X)$ approximating $S$ to order $\alpha \in (0,\frac{1}{2}]$ on a subspace $Y\subseteq X$. Among others, this setting covers the splitting scheme, the implicit Euler, and the Crank-Nicholson method.

Assuming additional structure of $F$ and $G$ as well as $Y$, we obtain the following bound for the pathwise uniform strong error
$$
\left(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|u(t_j) - U_j\|_X^p \right)^{1/p}
\le C(1+\|u_0\|_{L^p(\Omega;Y)}) \left(\log\left(\frac{T}{k}\right)\right)k^{\alpha}.
$$
In particular, this implies that the convergence rate of the uniform strong error is given by the order of the scheme up to a logarithmic correction factor. This factor can be avoided for the splitting scheme.

This is joint work with Mark Veraar and Jan van Neerven (TU Delft).