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Markus Sprecher

Publications and source records attributed to Markus Sprecher.

4 recordsLinked to original sources
arXiv2018

Projection-Based Finite Elements for Nonlinear Function Spaces

We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise onto the manifold. We show optimal interpolation error bounds with respect to Lebesgue and Sobolev norms. Additionally, we show similar bounds for the test functions, i.e., variations of discrete functions. Combining these results with a nonlinear C\'ea lemma, we prove optimal $L^2$ and $H^1$ discretization error bounds for harmonic maps from a planar domain into a smooth manifold. All these error bounds are also verified numerically.

math.NA↗
arXiv2016

A direct Proof for Quadratic Convergence of the Geometric Newton Method

We consider the problem of numerically computing a critical point of a functional $J\colon M\rightarrow R$ where $M$ is a Riemannian manifold. Due to local quadratic convergence a popular choice to solve this problem is the geometric Newton method. The proofs for quadratic convergence either use computations in a chart or require additional geometric quantities such as parallel translation. In this short note we provide a direct proof for quadratic convergence.

math.GM↗
arXiv2013

Counting Unique-Sink Orientations

Unique-sink orientations (USOs) are an abstract class of orientations of the n-cube graph. We consider some classes of USOs that are of interest in connection with the linear complementarity problem. We summarise old and show new lower and upper bounds on the sizes of some such classes. Furthermore, we provide a characterisation of K-matrices in terms of their corresponding USOs.

math.CO↗