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nonlinear-solvers

📁 heshamfs/materials-simulation-skills 📅 10 days ago
13
总安装量
3
周安装量
#24446
全站排名
安装命令
npx skills add https://github.com/heshamfs/materials-simulation-skills --skill nonlinear-solvers

Agent 安装分布

codex 3
gemini-cli 3
amp 2
kimi-cli 2
github-copilot 2

Skill 文档

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

  1. Characterize problem – Identify type, size, Jacobian availability
  2. Select solver – Run scripts/solver_selector.py
  3. Choose globalization – Run scripts/globalization_advisor.py
  4. Analyze Jacobian – If available, run scripts/jacobian_diagnostics.py
  5. Monitor residuals – During solve, use scripts/residual_monitor.py
  6. Analyze convergence – Run scripts/convergence_analyzer.py
  7. 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:

  1. 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
  2. Check globalization strategy: 
    python3 scripts/globalization_advisor.py --problem-type root-finding --jacobian-quality ill-conditioned --previous-failures 0 --json
  3. 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

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
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