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 3.2. Smart PolyAlgortihm Defaults NonlinearSolve.jl comes with a set of well-benchmarked defaults designed to be fast and robust in the average case. For Nonlinear Problems with Intervals, we default to Interpolate, Truncate, and Project (ITP) [48]. For more advanced problems, we default to a polyalgorithm that selects the internal solvers based on the problem specification. Our selection happens twofold, first at the nonlinear solver level [Figure 4] and next at the linear solver level[10] [Figure 5]. 3.2.1. Default Nonlinear Solver Selection. By default, we use a polyalgorithm that balances speed and robustness. We start with Quasi-Newton Algorithms – Broyden [49] and Klement [50]. If these fail, we use a modified version of Broyden, 3.2.1. Default Nonlinear Solver Selection. using a true Jacobian initialization. Finally, if Quasi-Newton Methods fail, we use . Newton Raphson with Line Search and finally fall back to Trust Region Methods. Additionally, if a custom Jacobian function is provided or the system is small (≤ 25 states), we avoid using Quasi-Newton Methods and directly use the first-order solvers. We have designed our default to be robust, allowing it to solve all 23 test problems [Subsection 4.1]. Additionally, as seen in Figure 9, our default solver successfully solves new problems. Our default strongly contrasts with other software like Sundials and MINPACK, which can only solve 15 (22 with line search) and 17 (possibly 18, depending on settings) problems, respectively, out of the 23 problems. 3.2.2. Default Linear Solver Selection. For ill-conditioned Jacobians in the linear system (𝐴𝑥 = 𝑏), we default to QR for regular cases or SVD for extremely ill-conditioned cases. We run a polyalgorithm for well-conditioned systems to determine the best linear solver. If 𝐴 is a matrix-free object, we use GMRES [51] from Krylov.jl [52] and solve the linear system without ever materializing the Jacobian. 3.2.2. Default Linear Solver Selection. For other cases, if 𝐴 is reasonably small, we use the native Julia Implementation . of LU Factorization [53] from RecursiveFactorization.jl [54]. We use the LU factorization routines from OpenBLAS or MKL for larger systems if present on the user’s system. 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 [10] Default Linear Solver selection happens in LinearSolve.jl. 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.

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

NonlinearSolve.jl: Smart Defaults for Robust and Efficient Nonlinear Solving

About Author

Comments

TOPICS

THIS ARTICLE WAS FEATURED IN

Related Stories

Automatic Sparsity Detection for Nonlinear Equations: What You Need to Know

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

Mathematical Description and Numerical Algorithms for Nonlinear Equations

Globalization Strategies: What Do They Allow Us to Do?

Sensitivity Analysis Explained: What Exactly Is It?

Matrix Coloring & Sparse Automatic Differentiation: The Jacob Computation

Automatic Sparsity Detection for Nonlinear Equations: What You Need to Know

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

Mathematical Description and Numerical Algorithms for Nonlinear Equations

Globalization Strategies: What Do They Allow Us to Do?

Sensitivity Analysis Explained: What Exactly Is It?

Matrix Coloring & Sparse Automatic Differentiation: The Jacob Computation

Light-Mode

Classic

Newspaper

Minty

Dark-Mode

Neon Noir

Minty

HN StartUps