The goal of this book is to provide a general framework for modeling the evolution of boundaries. We present computational techniques for tracking moving interfaces, and give some hint of the flavour and breadth of applications. The work includes examples from physics, chemistry, fluid mechanics, combustion, image processing, material science, fabrication of microelectronic components, computer vision, control theory, and seismology. The intended audience consists of mathematicians, applied scientists, practicing engineers, computer graphic artists, and anyone interested in the evolution of boundaries and interfaces.
The book is divided into four parts. In part I, we begin with a general statement of the problem of a moving interface and discuss the mathematical theory of curve/surface motion, including the growth/decay of total variation, singularity development, entropy conditions, weak solutions, and shocks in the dynamics of moving fronts. This leads to both the time-dependent level set algorithm and the static Hamilton-Jacobi formulation.
Part II presents numerical aspects of the time-dependent level set formulation; after an overview of traditional methods for tracking interfaces, including string methods and cell methods, we present the algorithms and numerical/theoretical analysis of level set methods. Some implementation details are treated, including fast methods, adaptive mesh refinement, and parallel implementations. Finally, we review recent extensions of the level set method, including versions for masking, multiple regions, and triple points.
Part III is devoted to viscosity solutions of Hamilton-Jacobi equations and to a class of fast marching methods for computing solutions to static Hamilton-Jacobi equations. In particular, we discuss upwind, heapsort-based schemes for rapidly solving the eikonal equation, and follow with the construction of approximation schemes for general static Hamilton-Jacobi equations.
The last part IV focuses on applications of both the time-dependent level set method and the fast marching level method to a large collection of problems. The intent is to touch on some problems that have been modeled using level set methods, both to show the breadth of possible applications and to serve as guideposts for further applications.