What Are the Special Capabilities of NonlinearSolve.jl?

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. Special Capabilities In the previous section, we provided an overview of solving nonlinear equations. Here, we will focus on specific capabilities of our package NonlinearSolve.jl. Let’s cover the typical workflow of using NonlinearSolve.jl. First, we define the nonlinear equations. We will use a mutating function to compute our residual value from 𝑢 and 𝑝 and store it in F. In Julia, mutating function names are appended with a . ! as a convention. Next, we construct a NonlinearFunction – a wrapper over f! that can store additional information like user-defined Jacobian functions and sparsity patterns. 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.