Numerically-Stable and Highly-Scalable Parallel LU Factorization for Circuit Simulation

A number of sparse linear systems are solved by sparse LU factorization in a circuit simulation process. The coefficient matrices of these linear systems have the identical structure but different values. Pivoting is usually needed in sparse LU factorization to ensure the numerical stability, which leads to the difficulty of predicting the exact dependencies for scheduling parallel LU factorization. However, the matrix values usually change smoothly in circuit simulation iterations, which provides the potential to "guess" the dependencies. This work proposes a novel parallel LU factorization algorithm with pivoting reduction, but the numerical stability is equivalent to LU factorization with pivoting. The basic idea is to reuse the previous structural and pivoting information as much as possible to perform highly-scalable parallel factorization without pivoting, which is scheduled by the "guessed" dependencies. Once a pivot is found to be too small, the remaining matrix is factorized with pivoting in a pipelined way. Comprehensive experiments including comparisons with state-of-the-art CPU- and GPU-based parallel sparse direct solvers on 66 circuit matrices and real SPICE DC simulations on 4 circuit netlists reveal the superior performance and scalability of the proposed algorithm. The proposed solver is available at https://github.com/chenxm1986/cktso.