von Deylen, Stefan Wilhelm
[Author]
;
Professor Konrad Polthier
[Contributor];
Professor Max Wardetzky
[Contributor]
Numerical Approximation in Riemannian Manifolds by Karcher Means ; Numerische Approximation in Riemannschen Mannigfaltigkeiten mithilfe des Karcher'schen Schwerpunktes
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Doctoral Thesis;
Electronic Thesis;
E-Book
Title:
Numerical Approximation in Riemannian Manifolds by Karcher Means ; Numerische Approximation in Riemannschen Mannigfaltigkeiten mithilfe des Karcher'schen Schwerpunktes
Contributor:
von Deylen, Stefan Wilhelm
[Author]
Published:
Freie Universität Berlin: Refubium (FU Berlin), 2015
Footnote:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Description:
(1) Let $(M,g)$ be a compact Riemannian manifold without boundary, and let $\Delta$ be the $n$-dimensional standard simplex. Following Karcher (1977), we consider, for $n+1$ given points $p_i \in M$, the function \\[ E: M \times \Delta \to \R, (a,\lambda) \mapsto \lambda^0 d^2(a,p_0) + \dots + \lambda^n d^2(a,p_n), \\] where $d$ is the geodesic distance in $M$. If all $p_i$ lie in a sufficiently small geodesic ball, then $x: \lambda \mapsto argmin_a E(a,\lambda)$ is a well-defined mapping $\Delta \to M$ (5.3). We call $s := x(\Delta)$ the Karcher simplex with vertices $p_i$. Suppose $\Delta$ carries a flat Riemannian metric $g^e$ induced by edge lengths $d(p_i,p_j)$. If all edge lengths are small than $h$ and $vol(\Delta,g^e) \geq \alpha h^n$ for some $\alpha > 0$, then we show in 6.17 and 6.23 that \begin{equation} %\tag{A.1a} |(x^*g - g^e)(v,w)| \leq c h^2 |v| |w|, \qquad % |(\nabla^{x^*g} - \nabla^{g^e})_v w| \leq c h |v| |w| \end{equation} with some constant $c$ depending only on the curvature tensor $R$ of $(M,g)$ and $\theta$. With little effort, this gives interpolation estimates for functions $u: s \to \R$ (7.4) and $y: s \to N$ for a second Riemannian manifold $N$ (7.14). Also, following Leibon und Letscher (2000), this simplex construction allows for the definition of a Voronoi decomposition (8.7). Thus we can consider $(M,g)$ to be triangulated in the following. On each simplex, $g$ is approximated by a metric $g^e$ with (A1.a), and weakly differentiable functions $u \in H^1(M,g)$ can be approximated by polynomials $u_h \in P^1(M)$. Via the standard method of surface finite element methods (Dziuk 1988, Holst and Stern 2012), variational problems such as the Poisson problem (10.13, 10.17, 13.14) or the Hodge decomposition (10.15) in $H^1(M,g)$ can be compare to those in $H^1(M,g^e)$ and their Galerkin approximations in $P^1(M)$. Corresponding to the standard surface finite element theory for problems on submanifolds of $\R^m$, also submanifolds $S \subset M$ may be approximated by Karcher simplices. ...