pGRASS-Solver: A Parallel Iterative Solver for Scalable Power Grid Analysis Based on Graph Spectral Sparsification

Due to the rapid advance of the integrated circuit technology, power grid analysis usually imposes a severe computational challenge, where linear equations with millions or even billions of unknowns need to be solved. Recent graph spectral sparsification techniques have shown promising performance in accelerating power grid analysis. However, previous graph sparsification based iterative solvers are restricted by difficulty of parallelization. Existing graph sparsification algorithms are implemented under the assumption of serial computing, while factorization and backward/forward substitution of the spar-sifier's Laplacian matrix are also hard to parallelize. On the other hand, partition based iterative methods which can be easily parallelized lack a direct control of the relative condition number of the preconditioner and consume more memory. In this work, we propose a novel parallel iterative solver for scalable power grid analysis by integrating graph sparsification techniques and partition based methods. We first propose a practically-efficient parallel graph sparsification algorithm. Then, domain decomposition method is leveraged to solve the sparsifier's Laplacian matrix. An efficient graph sparsification based parallel preconditioner is obtained, which not only leads to fast convergence but also enjoys ease of parallelization. Extensive experiments are carried out to demonstrate the superior efficiency of the proposed solver for large-scale power grid analysis, showing 5.2X speedup averagely over the state-of-the-art parallel iterative solver. Moreover, it solves a real-world power grid matrix with 0.36 billion nodes and 8.7 billion nonzeros within 23 minutes on a 16-core machine, which is 9.5X faster than the best result of sequential graph sparsification based solver.

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].