Nonlinear Solvers

Select and configure nonlinear solvers for root-finding f(x)=0, optimization min F(x), and least-squares problems — choose among Newton, Newton-Krylov, quasi-Newton (BFGS, L-BFGS), Broyden, Anderson acceleration, and Levenberg-Marquardt methods, configure line search or trust-region globalization, diagnose convergence rate (quadratic, linear, stagnated), and assess Jacobian quality and conditioning. Use when a Newton solver converges slowly or diverges, choosing between line search and trust region, debugging nonlinear iteration failures in FEM or phase-field codes, or selecting a solver for large-scale unconstrained optimization, even if the user only says "my Newton iterations aren't converging."

Science & Simulation|v1|Updated 7/2/2026|GitHub source

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# Nonlinear Solvers

## Goal

Provide a universal workflow to select a nonlinear solver, configure globalization strategies, and diagnose convergence for root-finding, optimization, and least-squares problems.

## Requirements

- Python 3.10+
- NumPy (for Jacobian diagnostics)
- SciPy (optional, for advanced analysis)

## Inputs to Gather

| Input | Description | Example |
|-------|-------------|---------|
| Problem type | Root-finding, optimization, least-squares | `root-finding` |
| Problem size | Number of unknowns | `n = 10000` |
| Jacobian availability | Analytic, finite-diff, unavailable | `analytic` |
| Jacobian cost | Cheap or expensive to compute | `expensive` |
| Constraints | None, bounds, equality, inequality | `none` |
| Smoothness | Is objective/residual smooth? | `yes` |
| Residual history | Sequence of residual norms | `1,0.1,0.01,...` |

## Decision Guidance

### Solver Selection Flowchart

```
Is Jacobian available and cheap?
├── YES → Problem size?
│   ├── Small (n < 1000) → Newton (full)
│   └── Large (n ≥ 1000) → Newton-Krylov
└── NO → Is objective smooth?
    ├── YES → Memory limited?
    │   ├── YES → L-BFGS or Broyden
    │   └── NO → BFGS
    └── NO → Anderson acceleration or Picard
```

### Quick Reference

| Problem Type | First Choice | Alternative | Globalization |
|--------------|--------------|-------------|---------------|
| Small root-finding | Newton | Broyden | Line search |
| Large root-finding | Newton-Krylov | Anderson | Trust region |
| Optimization | L-BFGS | BFGS | Wolfe line search |
| Least-squares | Levenberg-Marquardt | Gauss-Newton | Trust region |
| Bound constrained | L-BFGS-B | Trust-region reflective | Projected |

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