Materials-simulation-skills nonlinear-solvers

install

source · Clone the upstream repo

git clone https://github.com/HeshamFS/materials-simulation-skills

Claude Code · Install into ~/.claude/skills/

T=$(mktemp -d) && git clone --depth=1 https://github.com/HeshamFS/materials-simulation-skills "$T" && mkdir -p ~/.claude/skills && cp -r "$T/skills/core-numerical/nonlinear-solvers" ~/.claude/skills/heshamfs-materials-simulation-skills-nonlinear-solvers && rm -rf "$T"

manifest: skills/core-numerical/nonlinear-solvers/SKILL.md

source content

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 3.8+
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

Script Outputs (JSON Fields)

Script	Key Outputs
`scripts/solver_selector.py`	`recommended` , `alternatives` , `notes`
`scripts/convergence_analyzer.py`	`converged` , `convergence_type` , `estimated_rate` , `diagnosis`
`scripts/jacobian_diagnostics.py`	`condition_number` , `jacobian_quality` , `rank_deficient`
`scripts/globalization_advisor.py`	`strategy` , `line_search_type` , `trust_region_type` , `parameters`
`scripts/residual_monitor.py`	`patterns_detected` , `alerts` , `recommendations`
`scripts/step_quality.py`	`ratio` , `step_quality` , `accept_step` , `trust_radius_action`

Workflow

Characterize problem - Identify type, size, Jacobian availability
Select solver - Run
```
scripts/solver_selector.py
```
Choose globalization - Run
```
scripts/globalization_advisor.py
```
Analyze Jacobian - If available, run
```
scripts/jacobian_diagnostics.py
```
Monitor residuals - During solve, use
```
scripts/residual_monitor.py
```
Analyze convergence - Run
```
scripts/convergence_analyzer.py
```
Evaluate steps - For trust region, use
```
scripts/step_quality.py
```

Conversational Workflow Example

User: My Newton solver for a phase-field simulation is converging very slowly. After 50 iterations, the residual only dropped from 1 to 0.1.

Agent workflow:

Analyze convergence:

python3 scripts/convergence_analyzer.py --residuals 1,0.8,0.6,0.5,0.4,0.3,0.2,0.15,0.12,0.1 --json

Check globalization strategy:

python3 scripts/globalization_advisor.py --problem-type root-finding --jacobian-quality ill-conditioned --previous-failures 0 --json

Recommend: Switch to trust region with Levenberg-Marquardt regularization, or use Newton-Krylov with better preconditioning.

Pre-Solve Checklist

Confirm problem type (root-finding, optimization, least-squares)
Assess Jacobian availability and cost
Check initial guess quality
Set appropriate tolerances
Choose globalization strategy
Prepare to monitor convergence

CLI Examples

# Select solver for large unconstrained optimization
python3 scripts/solver_selector.py --size 50000 --smooth --memory-limited --json

# Analyze convergence from residual history
python3 scripts/convergence_analyzer.py --residuals 1,0.1,0.01,0.001,0.0001 --tolerance 1e-6 --json

# Diagnose Jacobian quality
python3 scripts/jacobian_diagnostics.py --matrix jacobian.txt --json

# Get globalization recommendation
python3 scripts/globalization_advisor.py --problem-type optimization --jacobian-quality good --json

# Monitor residual patterns
python3 scripts/residual_monitor.py --residuals 1,0.8,0.9,0.7,0.75,0.6 --target-tolerance 1e-8 --json

# Evaluate step quality for trust region
python3 scripts/step_quality.py --predicted-reduction 0.5 --actual-reduction 0.4 --step-norm 0.8 --gradient-norm 1.0 --trust-radius 1.0 --json

Error Handling

Error	Cause	Resolution
`problem_size must be positive`	Invalid size	Check problem dimension
`constraint_type must be one of...`	Unknown constraint	Use: none, bound, equality, inequality
`residuals must be non-negative`	Invalid residual data	Check residual computation
`Matrix file not found`	Invalid path	Verify Jacobian file exists

Interpretation Guidance

Convergence Type

Type	Meaning	Action
quadratic	Optimal Newton	Continue, near solution
superlinear	Quasi-Newton working	Monitor for stagnation
linear	Acceptable	May improve with preconditioner
sublinear	Too slow	Change method or formulation
stagnated	No progress	Check Jacobian, preconditioner
diverged	Increasing residual	Add globalization, check Jacobian

Jacobian Quality

Quality	Condition Number	Action
good	< 10⁶	Standard Newton works
moderately-conditioned	10⁶ - 10¹⁰	Consider scaling
ill-conditioned	> 10¹⁰	Use regularization
near-singular	∞	Reformulate or use LM

Step Quality (Trust Region)

Ratio ρ	Quality	Trust Radius
ρ < 0	very_poor	Shrink aggressively
ρ < 0.25	marginal	Shrink
0.25 ≤ ρ < 0.75	good	Maintain
ρ ≥ 0.75	excellent	Expand if at boundary

Security

Input Validation

```
--size
```
(problem size) is validated as a positive integer, bounded at 10 billion
```
--residuals
```
are validated as finite non-negative numbers, capped at 100,000 entries
```
--tolerance
```
and
```
--target-tolerance
```
are validated as finite positive numbers
```
--problem-type
```
and
```
--constraint-type
```
are validated against fixed allowlists
```
--jacobian-quality
```
is validated against a fixed allowlist (
```
good
```
,
```
ill-conditioned
```
, etc.)

Step quality parameters (

predicted-reduction

actual-reduction

step-norm

gradient-norm

trust-radius

) are validated as finite numbers

File Access

```
jacobian_diagnostics.py
```
reads a single matrix file specified by
```
--matrix
```
; no directory traversal beyond the given path
Matrix files are size-limited and loaded with
```
allow_pickle=False
```
to prevent code execution
All other scripts read no external files; inputs are provided via CLI arguments
Scripts write only to stdout (JSON output)

Tool Restrictions

Read: Used to inspect script source, references, and user configuration files

Bash: Used to execute the six Python analysis scripts (

solver_selector.py

convergence_analyzer.py

jacobian_diagnostics.py

globalization_advisor.py

residual_monitor.py

step_quality.py

) with explicit argument lists

Write: Used to save analysis results or solver recommendations; writes are scoped to the user's working directory
Grep/Glob: Used to locate relevant files and search references

Safety Measures

No
```
eval()
```
,
```
exec()
```
, or dynamic code generation
All subprocess calls use explicit argument lists (no
```
shell=True
```
)
Matrix dimension limits prevent memory exhaustion when loading Jacobian files
Residual history analysis operates on bounded-length numeric arrays only

Limitations

No global convergence guarantee: All methods may fail for pathological problems
Jacobian accuracy: Finite-difference Jacobian may be inaccurate near discontinuities
Large dense problems: May require specialized solvers not covered here
Constrained optimization: Complex constraints need SQP or interior point methods

References

```
references/solver_decision_tree.md
```
- Problem-based solver selection
```
references/method_catalog.md
```
- Method details and parameters
```
references/convergence_diagnostics.md
```
- Diagnosing convergence issues
```
references/globalization_strategies.md
```
- Line search and trust region

Version History

v1.0.0 : Initial release with 6 analysis scripts