Talks

We present some recent developments for the numerical simulation of
transport-dominated problems such as compressible fluid flows and
nonlinear dispersive wave equations. We begin with a brief review
of modern entropy-stable semidiscretizations of hyperbolic conservation
laws and use the method of lines to obtain efficient, fully discrete
numerical methods. Next, we introduce means to preserve the entropy
structures also under time discretization. Therefore, we present the
relaxation approach, a recent technique developed as small modifications
of standard time integration schemes such as Runge-Kutta or linear
multistep methods, which is designed to preserve the conservation or
dissipation of important functionals of the solution. This can be an
entropy in the case of compressible fluid flows, the energy of
Hamiltonian problems, or another nonlinear invariant.

We applied the Parallel-In-Time algorithm Parareal to the ocean-circulation model FESOM2 to demonstrate its applicability to complex problems in climate research. The talk is intended to give an overview of the technical challenges that can be expected when attempting to parallelize state-of-the-art simulation software in time. With the convergence results presented the talk concludes with a discussion of whether and how an efficient application of Parareal could be achieved.

We study the interplay between moment and tail inequalities in the concentration of measure phenomenon. A motivating example are so-called higher order concentration bounds, where functions are addressed which have unbounded first order derivatives (or differences) but whose derivatives of some higher order are bounded. A variety of different situations is considered like (classical) Euclidean spaces, discrete situations, functions of independent random variables and the Poisson space. A special emphasis is put on pointing out the parallels and common ground throughout all these cases.

Since we now have implemented an efficient solver for the Maxey-Riley equation, we can generate a lot of trajectory data. This data could be used to train a neural network, which can predict the trajectories given a starting postion. Because the dynamics are governed by an integro-differential equation, the future path of a trajectory depends on the whole past. This characteristic could be handled using recurrent neural networks.
I will show how the network performs on different velocity fields. There are cases where the model does a great job in the prediction, but there are still many problems to discuss and it will be interesting to hear some thoughts.
Finally I will show a network architecture that combines a Verlet integrator with a recurrent neural network.