Optimal a priori discretization error bounds for geodesic finite elements

We prove optimal bounds for the discretization error of geodesic finite elements for variational partial differential equations for functions that map into a nonlinear space. For this, we first generalize the well-known Céa lemma to nonlinear function spaces. In a second step, we prove optimal interpolation error estimates for pointwise interpolation by geodesic finite elements of arbitrary order. These two results are both of independent interest. Together they yield optimal a priori error estimates for a large class of manifold-valued variational problems. We measure the discretization error both intrinsically using an H1-type Finsler norm and with the H1-norm using embeddings of the codomain in a linear space. To measure the regularity of the solution, we propose a nonstandard smoothness descriptor for manifold-valued functions, which bounds additional terms not captured by Sobolev norms. As an application, we obtain optimal a priori error estimates for discretizations of smooth harmonic maps using geodesic finite elements, yielding the first high-order scheme for this problem

The Square Root Velocity Framework for Curves in a Homogeneous Space

In this paper we study the shape space of curves with values in a homogeneous space $M = G/K$, where $G$ is a Lie group and $K$ is a compact Lie subgroup. We generalize the square root velocity framework to obtain a reparametrization invariant metric on the space of curves in $M$. By identifying curves in $M$ with their horizontal lifts in $G$, geodesics then can be computed. We can also mod out by reparametrizations and by rigid motions of $M$. In each of these quotient spaces, we can compute Karcher means, geodesics, and perform principal component analysis. We present numerical examples including the analysis of a set of hurricane paths.Comment: To appear in 3rd International Workshop on Diff-CVML Workshop, CVPR 201

On the complexity of optimal homotopies

In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves $\gamma_1$ and $\gamma_2$ on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between $\gamma_1$ and $\gamma_2$ where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems. We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fr\'echet distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting

Harmonic maps into singular spaces and Euclidean buildings.

by Lam Kwan Hang.Thesis (M.Phil.)--Chinese University of Hong Kong, 2001.Includes bibliographical references (leaves 75-76).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.5Chapter 2 --- Maps into locally compact Riemannian complex --- p.8Chapter 2.1 --- Exitsence of energy minimizing maps --- p.8Chapter 2.2 --- Length minimizing curves --- p.10Chapter 3 --- Harmonic maps into nonpositively curved spaces --- p.13Chapter 3.1 --- Nonpositively curved spaces --- p.13Chapter 3.2 --- Properties of the distance function --- p.16Chapter 4 --- Basic properties of harmonic maps into NPC spaces --- p.21Chapter 4.1 --- Monotonicity formula --- p.21Chapter 4.2 --- Approximately differentiable maps --- p.24Chapter 4.3 --- Local properties of harmonic maps --- p.28Chapter 5 --- Existence and uniqueness of harmonic maps in a ho- motopy class --- p.33Chapter 5.1 --- Convexity properties of the energy functional --- p.33Chapter 5.2 --- Existence and Uniqueness Theorem --- p.37Chapter 6 --- Homogeneous approximating maps --- p.40Chapter 6.1 --- Regular homogeneous map --- p.40Chapter 6.2 --- Homogeneous approximating map --- p.46Chapter 7 --- More results on regularity --- p.52Chapter 7.1 --- Intrinsically differentiable maps --- p.52Chapter 7.2 --- Good homogeneous approximating map --- p.62Chapter 8 --- Harmonic maps into building-like complexes --- p.65Chapter 8.1 --- F-connected complex --- p.65Chapter 8.2 --- Regularity and the Bochner technique --- p.66Bibliography --- p.7

Length bounds for the conjugacy search problem in relatively hyperbolic groups, limit groups and residually free groups

In this thesis we prove conjugacy length bounds for several classes of groups. We use geometric and algebraic methods to show that there is a polynomial conjugacy length bound for relatively hyperbolic groups, a linear multiple conjugacy length bound for limit groups, and a polynomial multiple conjugacy length bound for nitely presented residually free groups

Counting geodesics of given commutator length

Let $\Sigma$ be a closed hyperbolic surface. We study, for fixed $g$, the asymptotics of the number of those periodic geodesics in $\Sigma$ having at most length $L$ and which can be written as the product of $g$ commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in $\Sigma$. In the appendix we use the same strategy to give a proof of Huber's geometric prime number theorem.Comment: 57 pages, 6 figure

Nonlinear Data: Theory and Algorithms

Techniques and concepts from differential geometry are used in many parts of applied mathematics today. However, there is no joint community for users of such techniques. The workshop on Nonlinear Data assembled researchers from fields like numerical linear algebra, partial differential equations, and data analysis to explore differential geometry techniques, share knowledge, and learn about new ideas and applications