Optimizing Tridiagonal Solvers for the Alternating Direction Method on Boolean Cube Multiprocessors

• Sets of tridiagonal systems occur in many applications. Fast Poisson solvers and Alternate Direction Methods make use of tridiagonal system solvers. Network-based multiprocessors provide a cost-effective alternative to traditional supercomputer architectures. The complexity of concurrent algorithms for the solution of multiple tridiagonal systems on Boolean-cube-configured multiprocessors with distributed memory are investigated. Variations of odd-even cyclic reduction, parallel cyclic reduction, and algorithms making use of data transposition with or without substructuring and local elimination, or pipelined elimination, are considered. A simple performance model is used for algorithm comparison, and the validity of the model is verified on an Intel iPSC/ 1. For many combinations of machine and system parameters, pipelined elimination, or equation transposition with or without substructuring is optimum. Hybrid algorithms that at any stage choose the best algorithm among the considered ones for the remainder of the problem are presented. It is shown that the optimum partitioning of a set of independent tridiagonal systems among a set of processors yields the embarrassingly parallel case. If the systems originate from a lattice and solutions are computed in alternating directions, then to first order the aspect ratio of a computational lattice shall be the same as that of the lattice forming the base for the equations. The experiments presented here demonstrate the importance of combining in the communication system for architectures with a relatively high communications start-up time.

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NATURVETENSKAP  -- Data- och informationsvetenskap (hsv//swe)

NATURAL SCIENCES  -- Computer and Information Sciences (hsv//eng)

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