We present the first parallel algorithm for solving systems of linear equations in symmetric, diagonally dominant (SDD) matrices that runs in polylogarithmic time and nearly-linear work. The heart of our algorithm is a construction of a sparse approximate inverse chain for the input matrix: a sequence of sparse matrices whose product approximates its inverse. Whereas other fast algorithms for solving systems of equations in SDD matrices exploit low-stretch spanning trees, our algorithm only requires spectral graph sparsifiers.
@article{arxiv.1311.3286,
title = {An Efficient Parallel Solver for SDD Linear Systems},
author = {Richard Peng and Daniel A. Spielman},
journal= {arXiv preprint arXiv:1311.3286},
year = {2013}
}