Vorträge

In this talk, we discuss non-local operators like elliptic integrodifferential operators of fractional type
\[
Au := p.v. \int_{\mathbb{R}^d} \frac{u(x) - u(y)}{|x-y|^{d+2\alpha}}dy \quad \quad (1)
\]
or the Stokes operator with bounded measurable coefficients $\mu$, formally given by
\[
Au := -div( \mu \nabla u ) + \nabla \phi, \quad div(u) = 0 \; in \; \mathbb{R}^d. \quad \quad (2)
\]
These operators satisfy $L^{2}$-resolvent estimates of the form
\[
|| \lambda ( \lambda + A )^{-1} f ||_{L^2} \leq C || f ||_{L^2} \quad (f \in L^2(\mathbb{R^d}))
\]
for $\lambda$ in some complex sector $\left\{z \in \mathbb{C} \smallsetminus {0} : | arg(z) | < \theta \right\}$. We describe how analogues of such a resolvent estimate can be established in $L^{p}$ by virtue of certain non-local Caccioppoli inequalities. Such estimates build the foundation for many important functional analytic properties of these operators like maximal $L^{q}$-regularity.

More precisely, we establish resolvent estimates in $L^{p}$ for $p$ satisfying
\[
\left|\frac{1}{p} - \frac{1}{2} \right| < \frac{\alpha}{d}
\]
in the case (1) and
\[
\left|\frac{1}{p} - \frac{1}{2}\right| < \frac{1}{d} \quad \quad (3)
\]
in the case (2). This resembles a well-known situation for elliptic systems in divergence form with $L^{\infty}$-coefficients. Here, important estimates like Gaussian upper bounds for the semigroup cease to exist and the $L^{p}$-extrapolation has be concluded by other means. In particular, for elliptic systems one can establish resolvent bounds for numbers p that satisfy (3) and if $d \geq 3$, Davies constructed examples which show that corresponding resolvent bounds do not hold for numbers $1 < p < \infty$ that satisfy
\[
\left|\frac{1}{p} - \frac{1}{2} \right| > \frac{1}{d}.
\]
These elliptic results give an indication that the result for the Stokes operator with $L^{\infty}$-coefficients is optimal as well.

We study integral input-to-state stability of bilinear systems with
unbounded control operators and derive natural sufficient conditions. The
results are applied to a bilinearly controlled Fokker-Planck equation.

Gauss quadrature rules are nowadays not only a powerful tool to compute integrals in many scientific applications, but also a numerical method that most people in the scientific community at least heard of at some point in there life.
Even if they are not the only tool to compute integral numerically, they provide the possibility to integrate any function multiplied by a given weight function (or measure), by estimating the integral of the product using a weighted sum of the function evaluations at given values (nodes).
Classical measures are well known (e.g Legendre, Chebyshev, Laguerre, Hermite), and their associated quadrature rules are well studied and documented in the literature.
While some measures allow to estimate integrals over infinite intervals (e.g Laguerre or Hermite), others also allow to integrate a function with singularities (e.g Chebyshev of the first, third and fourth kind).
However, the use of non-classical measures for specific applications can also be considered, and even this is not often used in the community, many algorithms exist to compute the nodes and weights of those quadrature rules.
In this talk we will give a quick overview of those algorithms, their efficiency, numerical stability, and some current challenge that still need to be solved.
Furthermore, under some conditions, all Gauss quadrature rules share some common properties, in particular when considering a large number of nodes.
We will give a quick overview of those common asymptotic properties, and show how they can be generalized to other applications (e.g barycentric Lagrange interpolation).
While some of those properties have been proven in particular cases, we will present some situations where they have not been proved theoretically yet, or still need to be verified.

Atmospheric motion covers a broad range of time- and spatial scales. Low and high pressure systems can influence us for days or even weeks and they extend up to hundreds of kilometers. In contrast, sound waves pass by in seconds with wavelengths of centimeters to meters. Implicit-explicit (IMEX) time stepping methods can help to avoid drastic limitations on the time step induced by the variety of scales without requiring computationally expensive fully nonlinear implicit solves. I will introduce Spectral Deferred Correction (SDC) methods as a strong competitor to currently used schemes. They allow an easy construction of high order schemes in contrast to e.g IMEX Runge-Kutta methods which require a growing number of coupling conditions with increasing order.

Discrete Schrödinger operators are used to describe systems in theoretical solid-state physics.
In this talk we consider discrete Schödinger operators with both periodic and aperiodic potentials. We analyse spectral properties of these operators and find conditions for the applicability of the so-called "finite section method" that allows us to approximate solutions of systems involving discrete Schrödinger operators.

Solving evolution equations numerically requires discretizing both in time and in space. However, these two problems can be treated seperately. A common approach to spatial discretization relies on solving the weak formulation on finite-dimensional subspaces. On a semigroup level, this corresponds to approximating a semigroup by semigroups on finite-dimensional subspaces. For practical applications, quantifying the convergence speed is essential. This can be achieved by the quantified version of the Trotter-Kato theorem presented in this talk. Rates of strong convergence are obtained on dense subspaces under a joint condition on properties of both the form and the approximating spaces. An outlook to evolution equations with random coefficients and their polynomial chaos approximation will be given as well as a generalization allowing to treat the Dirichlet-to-Neumann operator.

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