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

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

Script Outputs (JSON Fields)

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

# Select solver for a small nonlinear least-squares (data-fitting) problem
python3 scripts/solver_selector.py --problem-type least-squares --size 6 --jacobian-available --smooth --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

# Globalization for a distant initial guess (favors trust region)
python3 scripts/globalization_advisor.py --problem-type root-finding --jacobian-quality good --far-from-solution --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

Interpretation Guidance

Convergence Type

Jacobian Quality

Step Quality (Trust Region)

Verification checklist

Do not trust a "solved" claim until these concrete artifacts are recorded:

• Logged the full residual norm history and ran convergence_analyzer.py --residuals <history>; recorded convergence_type and estimated_rate, and confirmed converged: true against the actual solver tolerance (not the default 1e-10).
• Confirmed the residual sequence is monotone-decreasing or fed it to residual_monitor.py; recorded patterns_detected and verified it does NOT include diverging, oscillating, plateau, or slow_convergence while still above tolerance.
• If a Jacobian is available, ran jacobian_diagnostics.py --matrix J.txt and recorded condition_number and jacobian_quality; for an analytic Jacobian, passed --finite-diff-matrix and confirmed finite_diff_error is below ~1e-2 (no "Large discrepancy" note).
• Checked rank_deficient from jacobian_diagnostics.py is false (or documented why a rank-deficient/near-singular Jacobian is expected and that Levenberg-Marquardt regularization is in use).
• For a trust-region solve, evaluated accepted steps with step_quality.py and recorded the reduction ratio; confirmed accepted steps have ratio >= 0.25 (not very_poor/poor) and that the trust_radius_action matches the recorded ρ.
• Recorded the solver and globalization actually used and confirmed they match solver_selector.py and globalization_advisor.py recommendations for the stated problem type, size, and Jacobian quality (e.g., large/expensive-Jacobian → Newton-Krylov; least-squares → Levenberg-Marquardt trust region).
• Re-confirmed convergence after any change to tolerance, initial guess, or preconditioner — the convergence type can flip (e.g., quadratic → linear/stagnated) and must be re-classified, not assumed.

Common pitfalls & rationalizations

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

• v1.2.3 (2026-10-19): Use scale-relative rank diagnostics and correct infinity-norm handling.
• v1.2.2 (2026-06-24): Added a "Verification checklist" (evidence tied to each script's JSON outputs — convergence type/rate, residual patterns, Jacobian condition/finite-diff error, rank, trust-region step ratio, and solver/globalization agreement) and a "Common pitfalls & rationalizations" table covering constant-ratio-vs-superlinear, run-completion-vs-convergence, too-few-iterations, unverified analytic Jacobians, divergence handling, trust-region acceptance, and large/expensive-Jacobian routing
• v1.2.0 (2026-06-23): Added --problem-type to solver_selector.py with a nonlinear least-squares path (Levenberg-Marquardt / Gauss-Newton); reordered solver selection so problem size dominates high-accuracy and routes large/expensive-Jacobian problems to Newton-Krylov; added --far-from-solution to globalization_advisor.py and surfaced Levenberg-Marquardt as the trust-region type for least-squares; corrected convergence classification so constant-ratio sequences are linear (not superlinear); RFC-8259-safe JSON (no -Infinity); input-validation hardening
• v1.1.0 (2026-03-26): Optimized agent-discovery description, evaluation suite, security review docs, standardized metadata block, CHANGELOG
• v1.0.0: Initial release with 6 analysis scripts