A parallel two-level hybrid method for tridiagonal systems and its application to fast Poisson solvers

Abstract—A new method, namely, the Parallel Two-Level Hybrid (PTH) method, is developed to solve tridiagonal systems on parallel computers. PTH has two levels of parallelism. The first level is based on algorithms developed from the Sherman-Morrison modification formula, and the second level can choose different parallel tridiagonal solvers for different applications. By choosing different outer and inner solvers and by controlling its two-level partition, PTH can deliver better performance for different applications on different machine ensembles and problem sizes. In an extreme case, the two levels of parallelism can be merged into one, and PTH can be the best algorithm otherwise available. Theoretical analyses and numerical experiments indicate that PTH is significantly better than existing methods on massively parallel computers. For instance, using PTH in a fast Poisson solver results in a 2-folds speedup compared to a conventional parallel Poisson solver on a 512 nodes IBM machine. When only the tridiagonal solver is considered, PTH is over 10 times faster than the currently used implementation. Index Terms—Parallel processing, scalable computing, tridiagonal systems, Poisson solver.