Decomposition method for block-tridiagonal matrix systems

TL;DR

This paper introduces a decomposition method enabling parallel solution of block-tridiagonal systems, with analytical speedup estimates and practical validation showing up to tenfold reduction in computation time.

Contribution

The paper presents a novel decomposition approach for parallel solving of block-tridiagonal matrices, including analytical speedup analysis and application to boundary value problems.

Findings

Analytical speedup estimates match practical results.

Maximum speedup occurs at an optimal number of processors.

Application reduces solution time by up to 10 times.

Abstract

The decomposition method which makes the parallel solution of the block-tridiagonal matrix systems possible is presented. The performance of the method is analytically estimated based on the number of elementary multiplicative operations for its parallel and serial parts. The computational speedup with respect to the conventional sequential Thomas algorithm is assessed for various types of the application of the method. It is observed that the maximum of the analytical speedup for a given number of blocks on the diagonal is achieved at some finite number of parallel processors. The values of the parameters required to reach the maximum computational speedup are obtained. The benchmark calculations show a good agreement of analytical estimations of the computational speedup and practically achieved results. The application of the method is illustrated by employing the decomposition method to the matrix system originated from a boundary value problem for the two-dimensional integro-differential Faddeev equations. The block-tridiagonal structure of the matrix arises from the proper discretization scheme including the finite-differences over the first coordinate and spline approximation over the second one. The application of the decomposition method for parallelization of solving the matrix system reduces the overall time of calculation up to 10 times.