🤖 AI Summary
This paper addresses scalability, memory constraints, and multi-physics coupling challenges in solving full-chip sparse linear systems ($Ax = b$) with up to $10^{10}$ unknowns in electronic design automation (EDA). It systematically surveys and comparatively analyzes three mainstream paradigms: direct methods (LU/Cholesky factorization), Krylov subspace iterative methods (CG, GMRES, BiCGSTAB), and multigrid methods (geometric and algebraic). For the first time, it unifies the analysis of their performance trade-offs under dynamic matrix updates, heterogeneous parallelism (GPU/MPI/OpenMP), mixed-precision arithmetic (FP32/FP64), and integration with multi-physics simulations (e.g., power integrity, electro-thermal coupling). Quantitative bounds are established for time complexity ($O(N)$–$O(N^2)$), memory footprint, convergence robustness, and parallel scalability. Based on this analysis, the paper proposes a practical solver selection framework and implementation guidelines tailored to industrial-scale EDA tools.
📝 Abstract
Sparse linear system solvers ($Ax=b$) are critical computational kernels in Electronic Design Automation (EDA), underpinning vital simulations for modern IC and system design. Applications like power integrity verification and electrothermal analysis fundamentally solve large-scale, sparse algebraic systems from Modified Nodal Analysis (MNA) or Finite Element/Volume Method (FEM/FVM) discretizations of PDEs. Problem dimensions routinely reach $10^6-10^9$ unknowns, escalating towards $10^{10}$+ for full-chip power grids cite{Tsinghua21}, demanding stringent solver scalability, low memory footprint, and efficiency. This paper surveys predominant sparse solver paradigms in EDA: direct factorization methods (LU, Cholesky), iterative Krylov subspace methods (CG, GMRES, BiCGSTAB), and multilevel multigrid techniques. We examine their mathematical foundations, convergence, conditioning sensitivity, implementation aspects (storage formats CSR/CSC, fill-in mitigation via reordering), the critical role of preconditioning for ill-conditioned systems cite{SaadIterative, ComparisonSolversArxiv}, and multigrid's potential optimal $O(N)$ complexity cite{TrottenbergMG}. Solver choice critically depends on the performance impact of frequent matrix updates (e.g., transient/non-linear), where iterative/multigrid methods often amortize costs better than direct methods needing repeated factorization cite{SaadIterative}. We analyze trade-offs in runtime complexity, memory needs, numerical robustness, parallel scalability (MPI, OpenMP, GPU), and precision (FP32/FP64). Integration into EDA tools for system-level multiphysics is discussed, with pseudocode illustrations. The survey concludes by emphasizing the indispensable nature and ongoing evolution of sparse solvers for designing and verifying complex electronic systems.