A Subgraph Preconditioning Algorithm for Large Linear Systems

In this paper, a preconditioning algorithm for sparse, symmetric, diagonally dominant (SDD) linear systems is proposed by using combinatorial techniques. Firstly, we construct preconditioners by finding a subgraph based on the relationship between SDD matrices and undirected graphs. The subgraph is built by augmenting a low-stretch spanning tree with some extra high stretch edges. Then we implement the algorithm for building a low stretch tree and give a parallel implementation of the subgraph preconditioning algorithm based on PETSc software. Finally, numerical experiments arising from both elliptic PDEs and Laplacian systems of network graphs are tested to evaluate the performance of our algorithm. Numerical experiments show that preconditioners constructed by our algorithm are more efficient than incomplete Choleskey factorization preconditioners and Vaidya's preconditioners. Besides, based on the tree structure, our preconditioners perform respectable parallel scalability.