Type of Document Dissertation Author Tugurlan, Maria Cristina Author's Email Address ctugur@math.lsu.edu URN etd-09152008-143521 Title Fast Marching Methods - Parallel Implementation and Analysis Degree Doctor of Philosophy (Ph.D.) Department Mathematics Advisory Committee

Advisor Name Title 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 AbstractFast 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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