Generalized Jacobian-Free Nonlinear Solvers Using Krylov Methods

Table of Links

Abstract and 1. Introduction

2. Mathematical Description and 2.1. Numerical Algorithms for Nonlinear Equations

2.2. Globalization Strategies

2.3. Sensitivity Analysis

2.4. Matrix Coloring & Sparse Automatic Differentiation

3. Special Capabilities

3.1. Composable Building Blocks

3.2. Smart PolyAlgortihm Defaults

3.3. Non-Allocating Static Algorithms inside GPU Kernels

3.4. Automatic Sparsity Exploitation

3.5. Generalized Jacobian-Free Nonlinear Solvers using Krylov Methods

4. Results and 4.1. Robustness on 23 Test Problems

4.2. Initializing the Doyle-Fuller-Newman (DFN) Battery Model

4.3. Large Ill-Conditioned Nonlinear Brusselator System

5. Conclusion and References

us to use Least Squares Krylov Methods like LSMR efficiently. For certain Krylov Methods to converge, it is imperative to use Linear Preconditioning, which often requires a materialized Jacobian. In such cases, we provide an external control – concrete_jac – that overrides the default choice between materialized Jacobian and JacobianOperator and forces a concrete materialized Jacobian if set to true. In Subsection 4.3, we demonstrate the use of Jacobian-Free Newton and Dogleg Methods with GMRES [51] and preconditioning from IncompleteLU.jl and AlgebraicMultigrid.jl. We show that for large-scale systems, Krylov Methods [Figure 11] significantly outperform other methods [Figure 10]. Additionally, all our sparse Jacobian tooling is compatible with the Krylov Solvers, allowing us to generate cheaper sparse Jacobians for the preconditioning.

[11] https://invenia.github.io/blog/2019/11/06/julialang-features-part-2/