NonlinearSolve.jl: High-Performance and Robust Solvers for Systems of Nonlinear Equations in Julia

2024/03/25 by Avik Pal, Pal, Avik, Flemming Holtorf +20 · 3 voices · 5 citations
Computer Science · Physics and Astronomy · #Parallel Computing and Optimization Techniques #Polynomial and algebraic computation #Quantum chaos and dynamical systems #math.NA

paper · pdf · doi:10.48550/arxiv.2403.16341

openalex publication_date 2024/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Efficiently solving nonlinear equations underpins numerous scientific and engineering disciplines, yet scaling these solutions for challenging system models remains a challenge. This paper presents NonlinearSolve.jl -- a suite of high-performance open-source nonlinear equation solvers implemented natively in the Julia programming language. NonlinearSolve.jl distinguishes itself by offering a unified API that accommodates a diverse range of solver specifications alongside features such as automatic algorithm selection based on runtime analysis, support for GPU-accelerated computation through static array kernels, and the utilization of sparse automatic differentiation and Jacobian-free Krylov methods for large-scale problem-solving. Through rigorous comparison with established tools such as PETSc SNES, Sundials KINSOL, and MINPACK, NonlinearSolve.jl demonstrates robustness and efficiency, achieving significant advancements in solving nonlinear equations while being implemented in a high-level programming language. The capabilities of NonlinearSolve.jl unlock new potentials in modeling and simulation across various domains, making it a valuable addition to the computational toolkit of researchers and practitioners alike.

Citations

• Automated translation and accelerated solving of differential equations on multiple GPU platforms
• Enabling New Flexibility in the SUNDIALS Suite of Nonlinear and Differential/Algebraic Equation Solvers
• Don't Unroll Adjoint: Differentiating SSA-Form Programs
• Recent advances in trust region algorithms
• Julia: A Fresh Approach to Numerical Computing
• Julia: A Fresh Approach to Numerical Computing
• Algorithm 851
• SUNDIALS
• Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell

Cited by

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• Conserving mass, momentum, and energy for the Benjamin-Bona-Mahony, Korteweg-de Vries, and nonlinear Schrödinger equations
• High-order mass- and energy-conserving methods for the nonlinear Schrödinger equation and its hyperbolization
• On properties of hydraulic equilibria in district heating networks
• The influence of grain crushing and pore collapse on the formation of faults

Discussions

• Nonlinearsolve.jl: Fast and Robust Solvers for Nonlinear Equations in Julia [hn, 156 points, 35 comments]
• Solving f(x)=0 is just Newton's method, right? Well the #julialang nonlinear solvers have had lots of innovations in this very common numerical problems. Our nonlinear solvers demonstrate robustness w [bsky, 11 points, 0 comments]
• https://bsky.app/profile/rusirius.bsky.social/post/3komis2utlf23 [bsky, 0 points, 0 comments]

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