### Title page for ETD etd-09152008-143521

Type of Document Dissertation
Author Tugurlan, Maria Cristina
URN etd-09152008-143521
Title Fast Marching Methods - Parallel Implementation and Analysis
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Blaise Bourdin Committee Chair
Ambar Sengupta Committee Member
Jimmie Lawson Committee Member
Jing Wang Committee Member
Robert Lipton Committee Member
Robert Perlis Committee Member
Jerry Trahan Dean's Representative
Keywords
• Fast Marching Methods
• Fast Sweeping
• upwind scheme
• parallel implementation
• MPI
• PETSc
• convergence analysis
• scalability analysis
Date of Defense 2008-08-29
Availability unrestricted
Abstract
Fast Marching represents a very efficient technique for solving front propagation problems, which can be formulated as partial differential equations with Dirichlet boundary conditions, called Eikonal equation: $F(x)|\nabla T(x)|=1$, for $x \in \Omega$ and $T(x)=0$ for $x \in \Gamma$, where $\Omega$ is a domain in $\mathbb{R}^n$, $\Gamma$ is the initial position of a curve evolving with normal velocity F>0.

Fast Marching Methods are a necessary step in Level Set Methods, which are widely used today in scientific computing. The classical Fast Marching Methods, based on finite differences, are typically sequential. Parallelizing Fast Marching Methods is a step forward for employing the Level Set Methods on supercomputers.

The efficiency of the parallel Fast Marching implementation depends on the required amount of communication between sub-domains and on algorithm ability to preserve the upwind structure of the numerical scheme during execution. To address these problems, I develop several parallel strategies which allow fast convergence. The strengths of these approaches are illustrated on a series of benchmarks which include the study of the convergence, the error estimates, and the proof of the monotonicity and stability of the algorithms.

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