| In this dissertation, we explore the numerical methods and related issues for a range of themes pertinent to the level set method and eikonal equations. We begin with a chapter containing a brief overview on Hamilton-Jacobi Equations. We attempt to provide general information for the following questions: Where do Hamilton-Jacobi Equations come from? What properties do the solutions possess? and What are the general numerical methods for the construction of solutions? With this background information, we begin to relate the eikonal equation and the level set method to the main themes of this dissertation, and justify its constribution to the field.;We will address the following issues in the remaining chapters: (1) Discontinuous viscosity solutions to a class of fully nonlinear Hamilton-Jacobi Equations. We embed the graph of the solution in a level set function, and study the corresponding curve motion. To keep the curve as the graph of the solution, a singular diffusion term is introduced as the regularization. With this approach, we are able to compute the entropy solutions of conservation laws in their related nonconservative Hamilton-Jacobi forms, as a curve evolution problem in higher dimensions. Lastly in this topic, we apply this approach to study the Wulff flow of multiple-sheets crystal growth. (2) Phase space level set method for tracking hypersurfaces that develop self-intersections during evolutions. This is discussed in the context of geometrical optics. We look at the wavefronts as codimension two objects in phase space as the intersection of several level set functions. The governing PDEs in phase space belong to the class of linear transport equations called Liouville equation. With this approach, we advance the level set method to handle problems with non-simple, open curves. (3) Fast methods for solving first order time-independent Hamilton-Jacobi Equations with convex Hamiltonians. A compact expression for Godunov Hamiltonian is derived. We will demonstrate that, with clever ways of tracing characteristics using this Godunov Hamiltonian, the complexity of solving this class of equations can be much lower than what is reported from the fast marching method. We apply our method to compute distance on manifolds, and also devise a fast and accurate algorithm to compute the Euclidean distance to the given data set. (4) Application of the level set method to the visibility problem. We provide an implicit framework for the visibility problem and derive motion laws for the shadow boundaries corresponding to the view point's motion. This enables us to estimate when a hidden object will appear, and to update the visibility information in an efficient way.;Finally, we point out that the work reported in this thesis comes from the collaboration many people, whom we will mention in the later chapters. |
