Sparsified Cholesky Solvers for SDD linear systems
Abstract
We show that Laplacian and symmetric diagonally dominant (SDD) matrices can be well approximated by linearsized sparse Cholesky factorizations. We show that these matrices have constantfactor approximations of the form $L L^{T}$, where $L$ is a lowertriangular matrix with a number of nonzero entries linear in its dimension. Furthermore linear systems in $L$ and $L^{T}$ can be solved in $O (n)$ work and $O(\log{n}\log^2\log{n})$ depth, where $n$ is the dimension of the matrix. We present nearly linear time algorithms that construct solvers that are almost this efficient. In doing so, we give the first nearlylinear work routine for constructing spectral vertex sparsifiersthat is, spectral approximations of Schur complements of Laplacian matrices.
 Publication:

arXiv eprints
 Pub Date:
 June 2015
 arXiv:
 arXiv:1506.08204
 Bibcode:
 2015arXiv150608204T
 Keywords:

 Computer Science  Data Structures and Algorithms