Talks

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.

Lower bounds for variances are often needed to derive central limit theorems. In this talk, we establish a specific lower bound for the variance of a Poisson functional that uses the difference operator of Malliavin calculus.
Poisson functionals, i.e. random variables that depend on a Poisson process, are widely used in stochastic geometry. In this talk, we show how to apply our lower variance bound to statistics of spatial random graphs, the $L^p$ surface area of random polytopes and the total edge length of hyperbolic radial spanning trees. This talk is based on joint work with M. Schulte.

Spectral Deferred Correction (SDC) methods are iterative solvers for collocation discretization of ordinary differential equations, but each iterate can also be interpreted as particular Runge-Kutta (RK) scheme. In contrast to fixed RK schemes, viewing SDC as a fixed point iteration allows combining them with various kinds of deliberate perturbations resulting from mesh adaptivity or algebraic adaptivity in PDEs, lossy compression in parallel-in-time solvers, or inexact computations in scale-separated long time integrations, for improved performance. It also fosters a deeper understanding of SDC approximation error behavior, and the construction of more efficient preconditioners. In the talk, we will touch several of these aspects, and provide a - necessarily incomplete - overview of the astonishing flexibility of SDC methods.

The talk discusses the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows.
Here, gradient forces appearing in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their
$L^2$-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions
of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples.

This is joint work with Christian Merdon (WIAS)

Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential equations. Commonly used concepts of approximation methods are well-posed and convergent, by provable approximation orders. On the down side, however, these methods often suffer from the curse of dimensionality, which limits their approximation behavior. Nonlinear approximation methods, such as neural networks, were shown to be very efficient approximating high-dimensional functions. We investigate nonlinear approximation methods that are constructed by composing standard basis sets with normalizing flows. Such models yield richer approximation spaces while maintaining the density properties of the initial basis set, as we show. We investigate such approximation schemes for solving molecular Schrödinger equations and provide linear and nonlinear convergence analysis.

Zoomlink:
https://tuhh.zoom.us/j/84729171896?pwd=ODArbForaUxMM3Q3VTJsNG1kaVNYQT09