Improving PL Approximations of Harmonic Maps and a Penalty-Function Approach to Harmonic Maps

Abstract

In this dissertation, we consider two problems in the approximation of harmonic maps between empirically-known manifolds embedded in Euclidean spaces. First, we suppose that the domain manifold $\mathcal{M}$ is known by a point cloud $M_s$ and that $M_s$ has been used to reconstruct a $PL$ surface $\tau$ approximating $\mathcal{M}$. Then, assuming that we have a $PL$ map $f^\tau: \tau\to \mathbb{R}^k_\mathcal{N}$ that approximates a harmonic map $\mathcal{M}\to \mathcal{N}$, we ask how we can improve the approximation by using finer triangulations. In particular, we ask where we should add a single vertex to the domain triangulation $\tau$ given a single added vertex in the image triangulation $\mathscr{T}=f^\tau(\tau)$. We prove that a unique optimal solution exists when $\mathcal{M}\subset\mathbb{R}^2$. We find this optimal solution explicitly for several special cases and constrain it in general via an inverse problem. We then consider the problem of how to approximate a harmonic map $\mathcal{M}\to \mathcal{N}$ when the target manifold $\mathcal{N}\subset \mathbb{R}^k$ is known only by the point cloud $N_t$. We explicitly construct a $\mathscr{C}^1$ penalty functional $\mathscr{P}_t$ which is determined by $N_t$ and such that, if $f: \mathcal{M}\to \mathbb{R}^k$ and $f(\mathcal{M})$ is near $\mathcal{N}$, then $\mathscr{P}_t(f)$ is small. We then investigate how approximate harmonic maps can be found by locally minimizing functionals of the form $\mathscr{G}_t(\lambda, f)=\mathscr{E}(f)+\lambda\mathscr{P}_t(f)$ where $\mathscr{E}$ is the Dirichlet energy and $\lambda$ is a positive parameter. This work provides a foundation for extending the methods of \cite{ChenStruwe1989} to sampled manifolds.