Deterministic Random Walk: A New Preconditioner for Power Grid Analysis

Iterative linear equation solvers rely on high-quality preconditioners to achieve fast convergence. For sparse symmetric systems arising from large power grid analysis problems, however, preconditioners generated by traditional incomplete Cholesky factorization are usually of low quality, resulting in slow convergence. On the other hand, preconditioners generated by random walks are quite effective to reduce the number of iterations, though requiring considerable amount of time to compute in a stochastic manner. We propose in this paper a new preconditioning technique for power grid analysis, named deterministic random walk that combines the advantages of the above two approaches. Our proposed algorithm computes the preconditioners in a deterministic manner to reduce computation time, while achieving similar quality as stochastic random walk preconditioning by modifying fill-ins to compensate dropped entries. We have proved that for such compensation scheme, our algorithm will always succeed, which otherwise cannot be guaranteed by traditional incomplete factorizations. We demonstrate that by incorporating our proposed preconditioner, a conjugate gradient solver is able to outperform a state-of-the-art algebraic multigrid preconditioned solver for dc analysis, and is very efficient for transient simulation on public IBM power grid benchmarks.