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

##### Publisher

The University of Arizona.##### Rights

Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction, presentation (such as public display or performance) of protected items is prohibited except with permission of the author.##### 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.##### Type

Electronic Dissertationtext

##### Degree Name

Ph.D.##### Degree Level

doctoral##### Degree Program

Graduate CollegeMathematics