Talks

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).

In my introductory talk I will introduce myself and present the results of my master thesis.
The objective of my master thesis was to numerically solve a model, simulating the transportation of a tracer bolus through blood flow in the liver. A good model is important, especially in the field of cancer research, because tumor perfusion and other vascular properties are important parameters of cancer’s response to therapy. Good perfusion imaging allows an accurate model of the tumor’s vascular state and perfusion. With this model, critical determinants in the tumor’s progression and its response to therapy can be derived.

For the implementation I used a weighted essentially non-oscillatory (WENO) solver and tested it for accuracy, especially for its ability to solve the advection equation with space dependent velocity. WENO schemes have gained a lot of influence in numerical solutions of hyperbolic problems. The main advantage of WENO schemes and the reason they are so heavily used is their capability to achieve arbitrarily high-order formal accuracy in smooth regions while still maintaining stable and, most of all, non-oscillatory and sharp discontinuity transitions. The essential idea behind the scheme lies in the stencil choosing procedure.