Parallel Directionally Split Solver Based on Reformulation of Pipelined Thomas Algorithm

A very efficient direct solver, known as the Thomas algorithm, is frequently used for the solution of band matrix systems that typically appear in models of mechanics. The processor idle time is a reason for the poor parallelization efficiency of the solvers based on the pipelined parallel Thomas algorithms. In this research an efficient parallel algorithm for 3-D directionally split problems is developed. The proposed algorithm is based on a reformulated version of the pipelined Thomas algorithm that starts the backward step computations immediately after the completion of the forward step computations for the first portion of lines. This algorithm has data available for other computational tasks while processors are idle from the Thomas algorithm. The proposed 3-D directionally split solver is based on the static scheduling of processors where local and non-local, data-dependent and data-independent computations are scheduled while processors are idle. A theoretical model of parallelization efficiency is used to define optimal parameters of the algorithm, to show an asymptotic parallelization penalty and to obtain an optimal cover of a global domain with subdomains. It is shown by computational experiments and by the theoretical model that the proposed algorithm reduces the parallelization penalty about two times over the basic algorithm for the range of the number of processors (subdomains) considered and the number of grid nodes per subdomain. Key words, parallel computing, parallelization model, directionally split methods, pipelined Thomas Algorithm, banded matrices, ADI and FS mcthods Subject classification. Computer Science