Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Mathematical Description and 2.1. Numerical Algorithms for Nonlinear Equations 2. Mathematical Description and 2.1. Numerical Algorithms for Nonlinear Equations 2.2. Globalization Strategies 2.2. Globalization Strategies 2.3. Sensitivity Analysis 2.3. Sensitivity Analysis 2.4. Matrix Coloring & Sparse Automatic Differentiation 2.4. Matrix Coloring & Sparse Automatic Differentiation 3. Special Capabilities 3. Special Capabilities 3.1. Composable Building Blocks 3.1. Composable Building Blocks 3.2. Smart PolyAlgortihm Defaults 3.2. Smart PolyAlgortihm Defaults 3.3. Non-Allocating Static Algorithms inside GPU Kernels 3.3. Non-Allocating Static Algorithms inside GPU Kernels 3.4. Automatic Sparsity Exploitation 3.4. Automatic Sparsity Exploitation 3.5. Generalized Jacobian-Free Nonlinear Solvers using Krylov Methods 3.5. Generalized Jacobian-Free Nonlinear Solvers using Krylov Methods 4. Results and 4.1. Robustness on 23 Test Problems 4. Results and 4.1. Robustness on 23 Test Problems 4.2. Initializing the Doyle-Fuller-Newman (DFN) Battery Model 4.2. Initializing the Doyle-Fuller-Newman (DFN) Battery Model 4.3. Large Ill-Conditioned Nonlinear Brusselator System 4.3. Large Ill-Conditioned Nonlinear Brusselator System 5. Conclusion and References 5. Conclusion and References 3.1. Composable Building Blocks NonlinearSolve.jl stands out for its modular design, allowing users to combine various building blocks to create custom solver algorithms. This section delves into the flexibility and power of NonlinearSolve.jl’s architecture, focusing on its core components: Jacobian Computation Algorithm, Globalization Strategy, and Descent Algorithm. Jacobian Computation: NonlinearSolve.jl supports a range of methods for Jacobian computation. Firstly, we support various automatic differentiation and finite differencing backends to compute the exact Jacobian. Additionally, to support Quasi-Newton methods, we provide different abstractions to iteratively approximate the Jacobian.


Globalization Strategy: This module facilitates the global convergence of the local solvers [Subsection 2.2] and can seamlessly integrate with the other blocks.


Descent Algorithm: These determine the direction vector given the Jacobian (or Higher-Order Derivatives) and the current value. Jacobian Computation: NonlinearSolve.jl supports a range of methods for Jacobian computation. Firstly, we support various automatic differentiation and finite differencing backends to compute the exact Jacobian. Additionally, to support Quasi-Newton methods, we provide different abstractions to iteratively approximate the Jacobian. Jacobian Computation: NonlinearSolve.jl supports a range of methods for Jacobian computation. Firstly, we support various automatic differentiation and finite differencing backends to compute the exact Jacobian. Additionally, to support Quasi-Newton methods, we provide different abstractions to iteratively approximate the Jacobian. Globalization Strategy: This module facilitates the global convergence of the local solvers [Subsection 2.2] and can seamlessly integrate with the other blocks. Globalization Strategy: This module facilitates the global convergence of the local solvers [Subsection 2.2] and can seamlessly integrate with the other blocks. Descent Algorithm: These determine the direction vector given the Jacobian (or Higher-Order Derivatives) and the current value. Descent Algorithm: These determine the direction vector given the Jacobian (or Higher-Order Derivatives) and the current value. These building blocks provide a modular way to construct algorithms like: The modular architecture of NonlinearSolve.jl not only supports a broad range of algorithms but also empowers users to craft sophisticated solver combinations. Among the possibilities, users can integrate Pseudo-Transient methods with Globalization strategies or pair Quasi-Newton Methods with Globalization techniques. This flexibility opens the door to creating tailored solutions that can more effectively navigate complex problem landscapes, highlighting NonlinearSolve.jl’s capability to adapt and optimize across a diverse set of nonlinear equation challenges. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Authors:
(1) AVIK PAL, CSAIL MIT, Cambridge, MA;
(2) FLEMMING HOLTORF;
(3) AXEL LARSSON;
(4) TORKEL LOMAN;
(5) UTKARSH;
(6) FRANK SCHÄFER;
(7) QINGYU QU;
(8) ALAN EDELMAN;
(9) CHRIS RACKAUCKAS, CSAIL MIT, Cambridge, MA. Authors: Authors: (1) AVIK PAL, CSAIL MIT, Cambridge, MA; (2) FLEMMING HOLTORF; (3) AXEL LARSSON; (4) TORKEL LOMAN; (5) UTKARSH; (6) FRANK SCHÄFER; (7) QINGYU QU; (8) ALAN EDELMAN; (9) CHRIS RACKAUCKAS, CSAIL MIT, Cambridge, MA.

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