A Technical Survey of Sparse Linear Solvers in Electronic Design Automation

TL;DR

The paper surveys sparse linear solvers in Electronic Design Automation (EDA), addressing the challenge of solving large-scale systems arising from Modified Nodal Analysis and PDE discretizations. It analyzes three solver families—Direct methods (LU/Cholesky), Iterative Krylov methods with preconditioning, and Multigrid (AMG/GMG)—detailing their mathematical foundations, complexity, memory usage, and parallel scalability, as well as practical integration into EDA workflows. The work emphasizes the critical roles of fill-in management, preconditioning, and problem structure (symmetry, indefiniteness, conditioning) in solver selection, especially under frequent matrix updates in transient and multiphysics simulations. It also discusses solver integration in advanced EDA and system simulation tools, including power integrity and electrothermal CFD co-simulations, and highlights the ongoing evolution of solver algorithms, hardware-aware implementations, and hybrid multiphysics strategies, which are essential for next-generation electronic design verification.

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.