Chris Rackauckas - how compiler smarts can help improve the performance of numerical methods

NonlinearSolve.jl integrates compiler optimizations with numerical methods to improve performance of solving nonlinear systems in Julia

Achieved 20-100x speedups compared to traditional tools like SciPy, MATLAB and R by using:

• Automatic sparsity pattern detection
• Compiler-based optimizations
• Poly-algorithms that adaptively switch between methods
• GPU acceleration support

Key features:

• Seamless integration with Julia’s differential equations, optimization and machine learning ecosystems
• Automatic detection of optimal solving strategies
• Support for multiple hardware targets (CUDA, AMD, Intel, Apple GPUs)
• Automatic differentiation and symbolic computation capabilities
• Robust benchmarking system for continuous performance testing

Novel approach to handling nonlinear systems:

• Automatic simplification of equations when possible
• Smart sparsity pattern detection without user input
• Efficient handling of large systems through Krylov methods
• Built-in support for batch processing multiple problems

Real-world applications:

• Chemical reaction systems
• Black hole dynamics modeling
• Deep equilibrium networks
• Model auto-completion and equation discovery
• Scientific machine learning with small datasets

Design philosophy emphasizes:

• Common interfaces across different types of solvers
• Automatic feature detection over manual configuration
• Integration with existing scientific computing workflows
• Performance optimization through compiler analysis
• Extensibility and compatibility with existing Fortran/C codes