Johannes Hertrich — Page 3

arXiv2019

Infinity-Laplacians on Scalar- and Vector-Valued Functions and Optimal Lipschitz Extensions on Graphs

Extending functions from boundary values plays an important role in various applications. In this thesis we consider discrete and continuous formulations of the problem based on $p$-Laplacians, in particular for $p=\infty$ and tight Lipschitz extensions. The thesis gives an overview of the existing theory and provides some novel results on the approximation of tight Lipschitz extensions for vector-valued functions.

math.NA↗

arXiv2019

Minimal Lipschitz and $\infty$-Harmonic Extensions of Vector-Valued Functions on Finite Graphs

This paper deals with extensions of vector-valued functions on finite graphs fulfilling distinguished minimality properties. We show that so-called lex and L-lex minimal extensions are actually the same and call them minimal Lipschitz extensions. Then we prove that the solution of the graph $p$-Laplacians converge to these extensions as $p\to \infty$. Furthermore, we examine the relation between minimal Lipschitz extensions and iterated weighted midrange filters and address their connection to $\infty$-Laplacians for scalar-valued functions. A convergence proof for an iterative algorithm proposed by Elmoataz et al.~(2014) for finding the zero of the $\infty$-Laplacian is given. Finally, we present applications in image inpainting.

math.NA↗