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 and resemble W-cycles in the Multigrid method, our algorithm only requires spectral graph sparsifiers and is analogous to V-cycles.

Joint work with Dan Spielman.