Generalized Jacobian-Free Nonlinear Solvers Using Krylov Methods

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