Parallel iterative techniques for the solution of elliptic partial differential equations

John Paul Bonomo, Purdue University

Abstract

This dissertation presents a new parallel methodology which can be applied to traditional serial iterative techniques used in solving systems of linear equations. We consider primarily those linear systems which are derived from elliptic PDEs, but our techniques apply to general systems of linear equations as well. Our 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. We present 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 we show 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. We apply pipeline techniques to obtain the Pipeline Successive Overrelaxation and the Generalized Patel-Jordan algorithms for solving 2-D and 3-D elliptic problems. We also apply these techniques in a slightly modified manner to obtain the Pipeline Crank-Nicolson algorithm for parabolic problems. Experimental results for these parallel methods demonstrate that efficiencies of more than 80% can be obtained over a wide range of problem sizes and number of processors when using the optimal values for the pipeline spacing parameters.

Degree

Ph.D.

Advisors

Dyksen, Purdue University.

Subject Area

Computer science

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