Multigrid algorithms on parallel processing systems
This thesis considers the parallelization of multigrid algorithms for the solution of elliptic partial differential equations. Multigrid algorithms are among the fastest methods for a wide variety of problems, and are now used in many scientific disciplines. Structurally, the algorithm proceeds by iterating on a hierarchy of consecutively coarser and coarser grids. While essential to rapid convergence, these coarser grids make it more difficult to efficiently parallelize the algorithm on massively parallel machines. That is, the presence of fewer points (and hence less computational work) often corresponds to numerous inactive processors during coarse grid computations. It is therefore the intent of this thesis to analyze the effects of the multigrid hierarchy on parallel performance and to propose and analyze new highly parallel multigrid-like algorithms. A variety of multigrid-like algorithms are presented. Each of these algorithms accelerates the convergence of the basic multigrid method by solving additional subproblems in parallel with coarse grid processing. The key to the success of each method lies in its ability to decompose the original problem into subtasks whose solutions can be constructively combined. The primary emphasis of this thesis is the extraction of two fundamental decomposition principles used to form subproblems. One of these principles relies on the cancellation of error terms between multiple coarse grid solutions. The other principle centers on creating noninterfering subproblems using approximate A-orthogonal subspaces. In addition to analysis, the author discusses some of the practical tradeoffs associated with a few of these new algorithms such as numerical behavior, programmability, and applicability to complex problems. At the present time, it appears that both the problem to be solved as well as the machine to be utilized will determine which approach is most appropriate.
- Research Organization:
- Stanford Univ., CA (USA)
- OSTI ID:
- 6155968
- Resource Relation:
- Other Information: Thesis (Ph.D)
- Country of Publication:
- United States
- Language:
- English
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