2015/05/14 by Stefan W. von Deylen, von Deylen, Stefan · 1 citation
Mathematics · Computer Science · Medicine · #Geometric Analysis and Curvature Flows #Topological and Geometric Data Analysis #Advanced Neuroimaging Techniques and Applications
paper · doi:10.48550/arxiv.1505.03710
(1) For a compact Riemannian manifold without boundary (M,g) containing n+1 points pi and the n-dimensional standard simplex Δ, the miniser of E: M × Δ→ \mathbf R, (a,λ) ↦ λ0 d2(a,p0) + … + λn d2(a,pn) is considered as point with "barycentric coordinates" λi within the so-called Karcher simplex (or Riemannian simplex or geodesic finite element) defined by vertices pi. In the small, existence and uniqueness is well-known. Now suppose Δ carries a flat Riemannian metric ge induced by edge lengths d(pi,pj), where d is the geodesic distance in M. If all edge lengths are small than h and vol(Δ,ge) ≥ αhn for some α> 0, then we can show that |(x^*g - ge)(v,w)| ≤ c h2 |v| |w|, |(∇x^*g - ∇ge)v w| ≤ c h |v| |w| with some constant c depending only on the curvature tensor R of (M,g) and α. From this we derive several estimates for Finite Element calculations in which (M,g) is replaced by a piecewise flat realised simplicial complex. (2) Let M be the geometric realisation of a simplicial complex K. The simplicial cohomology (Ck(K), ∂^*) has been interpreted as "discrete outer calculus" (DEC) in the literature. We define spaces P-1Ωk ⊂ L^∞Ωk and outer differentials and give an isometric cochain map Ck → P-1Ωk. This reduces the computation of variational problems in discrete outer calculus to variational problems in a trial space of non-conforming differential forms. We investigate the approximation properties of P-1Ωk in H1Ωk and compare the solutions to variational problems in both spaces.