Vorträge

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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In many problems, like the discretized Stokes or Navier-Stokes equation, linear systems of saddle point type arise. Since the condition number for such problems can grow unbounded, as the number of unknowns grows, good preconditioners are key for solving such problems fast.

In this talk I will introduce some general preconditioning techniques for saddle point problems, and how to apply them to the discretized Stokes and Navier-Stokes equation

In this talk, we examine linearized kinetic BGK equation in 1D velocity dimension. It is closely related to the Maxwell-Boltzmann equation for gas dynamics. The equation that we are interested is obtained by linearization of the equation around Maxwellian. We discuss the kinetic and macroscopic equations and the boundary and coupling conditions for this equation.

Furthermore, we will drive coupling conditions for macroscopic equations on different network and compare the solutions with Maxwell and half-moment approximations. Moreover, the macroscopic equations on the network with the different Knudsen numbers are numerically compared with each other.

Positive definite kernel functions are powerful tools, which can be used to solve a variety of mathematical problems. One possible application of kernel-based methods is the reconstruction of images from scattered Radon data, which is described in [1]. More precisely, the authors introduced weighted kernel functions to solve the reconstruction problem via generalized interpolation. Although the reconstruction method was quite competitive in comparison to standard Fourier-based methods, a detailed discussion on well-posedness and stability was mainly missing.

In this talk, I will explain the basics of kernel-based generalized interpolation and discuss the well-posedness of the proposed reconstruction method. Like most kernel-based methods, the reconstruction method also suffers from bad condition numbers. I will show how to apply well-known stabilization methods from standard Lagrangian interpolation to the generalized case to improve the stability significantly.


[1] S. De Marchi, A. Iske, G. Santin. Image reconstruction from scattered Radon data by weighted kernel functions. Calcolo 55, 2018.

In my inaugural talk I would like to introduce myself and present the topic of my master thesis. To this end, I will first provide a brief introduction to empirical processes and long-range dependence. Afterwards, we consider the problem of testing the equality of two non-parametric regression functions. Finally, we provide a goodness of fit test for the error distribution.

The characterisation of the dynamics of a small inertial particle in a viscous fluid is a problem that dates to Stokes[1], back in 1851. Since his first attempt, many have tried and several formulas have been obtained for different types of flows, as well
as more general cases; however, the scientific community did not agree in a general formula until 1983, when M. Maxey and J. Riley[2] obtained a formula from first principles. This formula includes an integro-differential term, called the Basset History term, which
requires information for the whole history of the particle dynamics and creates difficulties in the numerical implementation due to fast increasing storage requeriments.
In the last decade, the Maxey-Riley formula has drawn the interest of many mathematicians and so, local and global existence and uniqueness of mild solutions have been proved ([3] & [4]). Nevertheless, a method to bypass the history term and obtain the trajectory
of the particle remained unknown until the publication of an accurate solution method by S.Ganga Prasath et al (2019) [5].
In this presentation I will analyse the Maxey Riley equation and will identify the core ideas within S. Ganga Prasath's method to solve the Maxey Riley equation as well as its implementation for certain fluid flows.

[1] Stokes, G. G. (1851). On the effect of the internal friction of fluids on the motion of pendulums.
[2] Maxey, M. R., & Riley, J. J. (1983). Equation of motion for a small rigid sphere in a nonuniform flow. The Physics of Fluids, 26(4), 883-889.
[3] Farazmand, M., & Haller, G. (2015). The Maxey–Riley equation: Existence, uniqueness and regularity of solutions. Nonlinear Analysis: Real World Applications, 22, 98-106.
[4] Langlois, G. P., Farazmand, M., & Haller, G. (2015). Asymptotic dynamics of inertial particles with memory. Journal of nonlinear science, 25(6), 1225-1255.
[5] Prasath, S. G., Vasan, V., & Govindarajan, R. (2019). Accurate solution method for the Maxey–Riley equation, and the effects of Basset history. Journal of Fluid Mechanics, 868, 428-460.