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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."

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npx skills add https://github.com/HeshamFS/materials-simulation-skills --skill nonlinear-solvers

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HeshamFS/materials-simulation-skills

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UpdatedMay 18, 2026 at 08:58

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convergence-study

HeshamFS/materials-simulation-skills

Perform spatial and temporal convergence analysis for solution verification — compute observed convergence orders from grid or timestep refinement studies, apply Richardson extrapolation to estimate discretization error, and calculate the Grid Convergence Index (GCI) per ASME V&V 20 standards. Use when verifying that a numerical solution converges at the expected rate, estimating the error on the finest mesh, checking whether grids are in the asymptotic range, or preparing formal verification reports, even if the user only asks "is my mesh fine enough" or "how accurate is my solution."

2026-05-1840

differentiation-schemes

HeshamFS/materials-simulation-skills

Select and apply numerical differentiation schemes for PDE and ODE discretization — generate finite-difference stencils at arbitrary order and accuracy, choose between central, upwind, compact (Pade), and spectral methods, handle boundary stencils, and estimate truncation error scaling. Use when discretizing spatial derivatives, picking a scheme for advection- or diffusion-dominated problems, building custom stencils for nonstandard operators, or comparing dispersion and dissipation properties of candidate schemes, even if the user just says "how do I approximate this derivative" or "my solution is too diffusive."

2026-05-1840

linear-solvers

HeshamFS/materials-simulation-skills

Select and configure linear solvers for Ax=b systems arising in numerical simulations — choose between direct (LU, Cholesky) and iterative (CG, GMRES, BiCGSTAB, MINRES) methods, analyze sparsity patterns and matrix conditioning, recommend preconditioners (AMG, ILU, IC), apply row/column scaling, and diagnose convergence stagnation from residual histories. Use when setting up a linear solve for FEM/FVM assembly, debugging slow or stalled Krylov iterations, choosing a preconditioner for SPD or nonsymmetric systems, or investigating ill-conditioning, even if the user only says "my solver is slow" or "GMRES won't converge."

2026-05-1840

mesh-generation

HeshamFS/materials-simulation-skills

Plan and evaluate mesh generation for numerical simulations — estimate grid resolution from physics scales (interface width, boundary layers, wavelengths), check aspect ratios and skewness against quality thresholds, choose between structured, unstructured, and adaptive mesh refinement strategies, and compute grid sizing for 1D/2D/3D domains. Use when setting up a new mesh, diagnosing poor solver convergence caused by mesh quality, deciding how many points to place across a phase-field interface or boundary layer, or preparing a mesh convergence study, even if the user only asks "what resolution do I need" or "why is my solver failing."

2026-05-1840

numerical-integration

HeshamFS/materials-simulation-skills

Select and configure time integration methods for ODE and PDE simulations — choose among explicit Runge-Kutta, BDF, Rosenbrock, and Adams families, set relative and absolute error tolerances, implement adaptive step-size control with I/PI/PID controllers, plan IMEX operator splitting for mixed stiff and non-stiff terms, and estimate splitting errors. Use when picking an integrator for a new simulation, diagnosing step rejections or tolerance failures, setting up operator splitting for phase-field or reaction-diffusion problems, or deciding between explicit and implicit time marching, even if the user only says "my solver keeps rejecting steps" or "which ODE method should I use."

2026-05-1840

numerical-stability

HeshamFS/materials-simulation-skills

Analyze numerical stability for time-dependent PDE simulations — check CFL and Fourier criteria, perform von Neumann stability analysis, detect stiffness, evaluate matrix conditioning, and recommend explicit vs implicit time-stepping schemes. Use when selecting time steps, diagnosing numerical blow-up or solver divergence, checking convergence criteria, or evaluating scheme stability for advection, diffusion, or reaction problems, even if the user doesn't explicitly mention "stability" or "CFL."

2026-05-1840

Source

HeshamFS

HeshamFS/materials-simulation-skills

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StatisticiansComputer and Mathematical Occupations15-2041L4

name	nonlinear-solvers
description	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."
allowed-tools	Read, Bash, Write, Grep, Glob
metadata	{"author":"HeshamFS","version":"1.1.0","security_tier":"high","security_reviewed":true,"tested_with":["claude-code","gemini-cli","vs-code-copilot"],"eval_cases":5,"last_reviewed":"2026-03-26"}

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

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