Eigel, Martin Schneider, Reinhold Trunschke, Philipp

We consider best approximation problems in a nonlinear subset ℳ of a Banach space of functions (𝒱,∥•∥). The norm is assumed to be a generalization of the L 2-norm for which only a weighted Monte Carlo estimate ∥•∥n can be computed. The objective is to obtain an approximation v ∈ ℳ of an unknown function u ∈ 𝒱 by minimizing the empirical norm ∥u − v...

Trunschke, Philipp

This thesis considers the problem of approximating low-rank tensors from data and its use for the non-intrusive solution of high-dimensional parametric partial differential equations (PDEs) and stochastic differential equations (SDEs). High-dimensional here refers to the large number of variables on which the solution depends. The looming curse of ...

Zur, Jan

In this doctoral thesis, we consider the zeros of harmonic mappings in the complex plane. Our study was originally motivated by the theory of gravitational lensing in astrophysics, where special cases of such functions and their zeros play an important role. However, in this work we focus on much more general functions. Our results range from purel...

García Ramos, Luis Alberto

In this dissertation we study various preconditioning methods based on the complex shifted Laplacian for the Helmholtz equation. This equation describes time-harmonic solutions to the wave equation and appears in various important engineering and scientific applications. First, we introduce a polynomial preconditioner which combines the shifted Lap...

Farchmin, Nando

This thesis concerns the combination of dependable error control and data based approximation to derive non-intrusive and reliable algorithms for uncertainty quantification in forward and inverse problems. In particular Bayesian inverse problems subject to high-dimensional parametric forward models driven by partial differential equations are the m...

Götte, Michael

We introduce the concept of block sparse tensor trains to the mathematical community, which is known to physicists as sector decomposed matrix product states. We generalize it to a class of Laplace-like operators and show that it is well-behaved with the tangent space projection on the tensor train manifold. We discuss known tensor network methods ...

Saverin, Joseph

The ability to make predictions about the flow in the wake of a lift-generating body has a range of important applications. One example is the wake of an aircraft, where this flow significantly affects the proximity with which a trailing aircraft can take off, cruise or land. Another application is the placement and operation of wind turbines in a ...

Sallandt, Leon Jasper

We consider high-dimensional, non-linear functional equations. These functional equations are mostly the Bellman equation known from optimal control or related fields. Within this framework we deal with the occurring non-linearity using fixed-point iterations, for the most part the Policy Iteration algorithm, reducing them to a series of linear pro...

Timme, Sascha

Numerical nonlinear algebra is concerned with the development of numerical methods to solve problems in nonlinear algebra. The main computational task is the solution of systems of polynomial equations. In this thesis, we focus on the numerical solution of polynomial systems using homotopy continuation methods. We apply techniques from numerical an...

Zimmer, Christoph

This thesis is devoted to the application and analysis of time integration schemes for differential-algebraic equations (DAEs) stated in (abstract) Banach spaces. The existence, uniqueness, and regularity of solutions of these so-called operator DAEs are analyzed with the help of temporal discretization methods. The convergence behavior of the time...