Parallel iterative techniques for the solution of elliptic partial differential equations
This dissertation presents a new parallel methodology which can be applied to traditional serial iterative techniques used in solving systems of linear equations. The author considers primarily those linear systems which are derived from elliptic PDEs, but his techniques apply to general systems of linear equations as well. His methodology transforms a serial method to a pipelined parallel version in which consecutive iterations are applied in pipeline-like fashion throughout the linear system while maintaining a minimum spacing between each iteration. This spacing is controlled by the pipeline spacing parameter. The resulting parallel methods contain desirable characteristics from previously investigated synchronous and asynchronous iterative methods. First, pipelined algorithms maintain the order in which unknowns are updated in the serial algorithms and therefore maintain the same convergence rates. Secondly, these algorithms maintain a small level of communication throughout its execution. This amount of communication overhead can be controlled by the user by adjusting the value of the pipeline spacing parameter. In this manner the algorithm can be fine-tuned for maximum efficiency. He presents theoretical time complexity work which accurately models the algorithms behavior and leads to the determination of the optimal value of the pipeline spacing parameter. Furthermore he shows that as the size of the problem increases, the limit of the speedup obtained by pipeline methods is the optimal speedup of p for p processors, given some modest constraints on the number of processors used relative to the problem size. He applies pipeline techniques to obtain the Pipeline Successive Overrelaxation and the Generalized Patel-Jordan algorithms for solving 2-D and 3-D elliptic problems. He also applies these techniques in a slightly modified manner to obtain the Pipeline Crank-Nicolson algorithm for parabolic problems.
- Research Organization:
- Purdue Univ., Lafayette, IN (USA)
- OSTI ID:
- 5896772
- Country of Publication:
- United States
- Language:
- English
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