I. Introduction
Accurate and efficient analysis of large-scale power grids is crucial for the modern very large-scale integrated (VLSI) circuits design. Large-scale power grid analysis requires solving linear equations with millions or even billions of unknowns, which is computationally challenging due to excessive time and memory consumption. Existing methods for power grid analysis include direct solvers, iterative solvers [1], [2] and other specialized methods such as the hierarchical matrix based method [3] and domain decomposition method (DDM) [4]–[8]. Direct methods, such as Cholesky or LU decomposition [9], [10], exactly solve the simulation problems but require much more memory to produce and store the matrix factors. On the other hand, iterative methods, such as the the Krylov subspace iterative methods [11] or algebraic multigrid (AMG) methods [12], usually have more favorable memory requirements thereby achieving more scalable performance for large problems. Among the most popular iterative methods, the preconditioned conjugate gradient (PCG) or generalized minimal residual (GMRES) algorithms leveraging recent graph spectral sparsification techniques have shown highly scalable performance for large circuit analysis tasks [13]–[16].