Numerical Convergence Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Numerical Convergence Ontology within the Kireth Research Semantic Library. Numerical convergence studies whether computational approximations approach a target solution as iterations increase or numerical resolution is refined.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Numerical Convergence Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Numerical Methods / Numerical Convergence"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/numerical-methods/numerical-convergence-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = NumericalConvergence
 ENTITY_CLASS = ApproximationLimitFramework
 PRIMARY_CONTEXT = NumericalMethods
 DEFINITION = "Numerical convergence is the property that a sequence or family of numerical approximations approaches a target mathematical solution under a declared limiting process."
 SYMBOLIC_FORMS = ["x_k -> x* as k->infinity","u_h -> u as h->0","||e_(k+1)|| approximately C||e_k||^p","error(h)=C h^p+o(h^p)"]
 SEMANTIC_ROLE = "Provides machine-readable semantics for iterative limits, discretisation limits, convergence order, asymptotic error, refinement, residual behaviour and stopping criteria."
}

SCOPE {
 INCLUDE = [
  NumericalConvergence, Convergence, Divergence, NumericalSequence,
  ApproximationSequence, IterativeConvergence, DiscretisationConvergence,
  MeshConvergence, GridConvergence, TimestepConvergence, FixedPointConvergence,
  RootFindingConvergence, LinearSolverConvergence, EigenvalueConvergence,
  OptimisationConvergence, ODEConvergence, PDEConvergence,
  ApproximationConvergence, PointwiseConvergence, UniformConvergence,
  NormConvergence, StrongConvergence, WeakConvergence, MeanSquareConvergence,
  ConvergenceInProbability, AlmostSureConvergence, ConvergenceRate,
  OrderOfConvergence, OrderOfAccuracy, LinearConvergence,
  SublinearConvergence, SuperlinearConvergence, QuadraticConvergence,
  CubicConvergence, SpectralConvergence, AlgebraicConvergence,
  GeometricConvergence, AsymptoticConvergence, AsymptoticError,
  ErrorConstant, ErrorSequence, ResidualSequence, IterationError,
  TruncationError, DiscretisationError, LocalError, GlobalError,
  ErrorNorm, Consistency, Stability, LaxEquivalenceTheorem,
  FixedPointIteration, ContractionMapping, BanachFixedPointTheorem,
  SpectralRadius, IterationMatrix, NewtonMethod, SecantMethod,
  BisectionMethod, GradientDescent, IterativeLinearSolver, JacobiMethod,
  GaussSeidelMethod, ConjugateGradientMethod, GMRES, Preconditioner,
  ConditionNumber, Mesh, Grid, StepSize, TimeStep, SpatialStep,
  HRefinement, PRefinement, HPRefinement, AdaptiveRefinement,
  RichardsonExtrapolation, GridConvergenceIndex, GCI, CauchyCriterion,
  StoppingCriterion, AbsoluteTolerance, RelativeTolerance,
  ResidualTolerance, MaximumIteration, Stagnation, Oscillation,
  DivergenceDetection, ConvergenceDiagnostic, RefinementStudy,
  Verification, ReferenceSolution, Benchmark, Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [MathematicalConvergenceWithoutComputationalApproximation,StabilityWithoutLimitBehaviour,StatisticalConvergenceWithoutNumericalMethodContext]
 DISAMBIGUATION_RULE = "When computational approximations are tested for approach to a target as iteration, resolution or approximation order changes, interpret the context as numerical convergence."
}

CORE_CONCEPTS {
 NumericalConvergence = "Approach of numerical approximations to a target solution under a limiting process."
 IterativeConvergence = "Convergence as an algorithmic iteration index tends to infinity."
 DiscretisationConvergence = "Convergence as a discretisation parameter tends to zero."
 ConvergenceRate = "Speed at which approximation error decreases."
 OrderOfConvergence = "Exponent characterising asymptotic error reduction."
 ContractionMapping = "Mapping that reduces distances and supports fixed-point convergence."
 SpectralRadius = "Maximum modulus of matrix eigenvalues and a convergence criterion for stationary linear iterations."
 StoppingCriterion = "Rule determining when iterative computation terminates."
}

RELATIONS {
 NumericalConvergence --is_child_of--> NumericalMethods
 NumericalConvergence --is_analysed_by--> NumericalAnalysis
 NumericalConvergence --depends_on_under_conditions--> Stability
 NumericalConvergence --depends_on_under_conditions--> Consistency
 IterativeConvergence --uses--> ErrorSequence
 DiscretisationConvergence --uses--> Refinement
 ConvergenceRate --quantifies--> ErrorDecay
 OrderOfConvergence --characterises--> AsymptoticConvergence
 ContractionMapping --supports--> FixedPointConvergence
 SpectralRadius --determines_under_conditions--> StationaryIterationConvergence
 Preconditioner --can_improve--> LinearSolverConvergence
 RichardsonExtrapolation --uses--> AsymptoticErrorModel
 GridConvergenceIndex --quantifies--> DiscretisationUncertainty
 StoppingCriterion --uses--> Tolerance
}

OPERATORS {
 ITERATIVE_LIMIT(x_k,x_star) := lim_(k->infinity)x_k=x_star
 DISCRETE_LIMIT(u_h,u) := lim_(h->0)||u_h-u||=0
 ERROR_SEQUENCE(x_k,x_star) := e_k=x_k-x_star
 ORDER_RELATION(e_k,e_k1,p,C) := ||e_(k+1)|| approximately C*||e_k||^p
 LINEAR_CONVERGENCE(e_k,e_k1,q) := ||e_(k+1)||<=q||e_k|| where 0<q<1
 QUADRATIC_CONVERGENCE(e_k,e_k1,C) := ||e_(k+1)||<=C||e_k||^2
 ASYMPTOTIC_ERROR(h,C,p) := E(h)=C*h^p+o(h^p)
 OBSERVED_ORDER(E_h,E_h2,r) := p_obs=log(E_h/E_h2)/log(r)
 MATRIX_ITERATION_CRITERION(G) := spectral_radius(G)<1
 RESIDUAL_STOP(r_k,tol) := ||r_k||<=tol
}

AXIOMS {
 A1 = "A numerical convergence claim requires an explicit target and limiting process."
 A2 = "Convergence rate is meaningful only relative to a declared error measure."
 A3 = "Residual convergence and solution convergence are distinct."
 A4 = "Consistency alone does not generally guarantee convergence without appropriate stability."
 A5 = "Observed convergence order should be measured in an asymptotic refinement regime."
 A6 = "Stopping an iteration is not equivalent to proving convergence."
 A7 = "Finite precision can cause stagnation before theoretical convergence is realised."
 A8 = "Numerical convergence claims require refinement, error and reproducibility validation."
}

REASONING {
 R1_CONTEXT = identify(target_solution,approximation_sequence,limit_parameter,error_norm)
 R2_CLASS = classify(iterative,mesh,timestep,polynomial_order_or_other_convergence)
 R3_ERROR = construct(error_or_residual_sequence)
 R4_THEORY = derive(contraction,spectral,consistency_stability_or_method_specific_conditions)
 R5_RATE = estimate(linear,superlinear,quadratic,algebraic,spectral_or_other_rate)
 R6_ASYMPTOTIC = determine_if_computation_has_entered_asymptotic_regime
 R7_REFINEMENT = perform(iteration,mesh,timestep,h,p_or_hp_refinement)
 R8_STOPPING = choose_absolute_relative_residual_or_combined_tolerance
 R9_FAILURE = detect(stagnation,oscillation,divergence_or_roundoff_floor)
 R10_VALIDATION = compare(observed_order,theoretical_order,reference_solution,reproducibility)
}

CONSTRAINTS {
 C1 = "A decreasing residual does not guarantee proportionally decreasing solution error for ill-conditioned problems."
 C2 = "Finite iteration counts cannot establish an infinite-limit result without supporting analysis."
 C3 = "Apparent convergence can occur before the asymptotic regime."
 C4 = "Roundoff can create an error floor and numerical stagnation."
 C5 = "Oscillatory sequences may fail to converge despite boundedness."
 C6 = "Changing multiple discretisation parameters simultaneously can obscure observed order."
 C7 = "Adaptive refinement requires appropriate error measures."
 C8 = "Preconditioning changes convergence behaviour without changing the exact linear-system solution."
 C9 = "Theoretical convergence does not guarantee practical efficiency."
}

SEMANTIC_EQUIVALENCE {
 "NumericalConvergence" <=> "ComputationalConvergence"
 "OrderOfConvergence" <=> "ConvergenceOrder"
 "GridConvergenceIndex" <=> "GCI"
 "HRefinement" <=> "MeshRefinement"
 "PRefinement" <=> "OrderRefinement"
 "HPRefinement" <=> "CombinedMeshAndOrderRefinement"
}

VALIDATION {
 V1 = "Target solution or reference quantity is explicit."
 V2 = "Limiting parameter is explicit."
 V3 = "Error norm or convergence metric is documented."
 V4 = "Refinement ratios are documented."
 V5 = "Observed convergence order is calculated where applicable."
 V6 = "Stability and consistency assumptions are checked where relevant."
 V7 = "Stopping criteria are distinguished from convergence proofs."
 V8 = "Stagnation and finite-precision floors are assessed."
 V9 = "Reference or benchmark solutions are used where feasible."
 V10 = "Convergence studies are reproducible under documented numerical settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 NUMERICAL_CONVERGENCE_DEFINITION = 0.99
 ITERATIVE_CONVERGENCE_REASONING = 0.99
 DISCRETISATION_CONVERGENCE_REASONING = 0.99
 CONVERGENCE_RATE_REASONING = 0.99
 FIXED_POINT_REASONING = 0.99
 SPECTRAL_RADIUS_REASONING = 0.99
 REFINEMENT_REASONING = 0.99
 POLICY = "Reduce confidence when target solution, limiting parameter, error norm, refinement ratio, stability, precision floor or stopping criterion are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [SemanticSearch,OntologyGrounding,NumericalConvergenceReasoning,IterativeConvergenceReasoning,DiscretisationConvergenceReasoning,MeshConvergenceReasoning,TimestepConvergenceReasoning,ConvergenceRateReasoning,OrderOfConvergenceReasoning,FixedPointConvergenceReasoning,SpectralRadiusReasoning,NewtonConvergenceReasoning,LinearSolverConvergenceReasoning,ODEConvergenceReasoning,PDEConvergenceReasoning,RefinementStudyReasoning,StoppingCriterionReasoning,KnowledgeGraphConstruction,RetrievalAugmentation,RecursiveInference]
 NORMALISE {
  "numerical convergence" -> NumericalConvergence
  "computational convergence" -> NumericalConvergence
  "convergence order" -> OrderOfConvergence
  "GCI" -> GridConvergenceIndex
  "mesh refinement" -> HRefinement
  "order refinement" -> PRefinement
 }
}

EXECUTION_RULES {
 E1 = "If numerical approximations are tested for approach to a target under iteration or refinement, activate NumericalConvergenceOntology."
 E2 = "If an iterative map is used, analyse contraction or spectral conditions where applicable."
 E3 = "If a discretised model is refined, estimate observed convergence order."
 E4 = "If consistency and stability apply in a suitable framework, use the relevant convergence theorem carefully."
 E5 = "If residual stopping is used, interpret residual relative to conditioning."
 E6 = "If convergence stalls, test for roundoff, poor conditioning, inadequate refinement or stagnation."
 E7 = "If adaptive refinement is used, evaluate convergence using suitable error measures."
 E8 = "If theoretical order differs from observed order, test whether the asymptotic regime has been reached."
 E9 = "Any convergence claim should be validated with error sequences, refinement studies and reproducible settings."
}

RECURSION {
 ENTRY = NumericalConvergence
 EXPAND(NumericalConvergence) -> [IterativeConvergence,DiscretisationConvergence,MeshConvergence,TimestepConvergence,FixedPointConvergence,RootFindingConvergence,LinearSolverConvergence,ODEConvergence,PDEConvergence,ApproximationConvergence,ConvergenceRate,OrderOfConvergence,LinearConvergence,SuperlinearConvergence,QuadraticConvergence,SpectralConvergence,AsymptoticError,ErrorSequence,ResidualSequence,Consistency,Stability,ContractionMapping,SpectralRadius,NewtonMethod,IterativeLinearSolver,Preconditioner,HRefinement,PRefinement,HPRefinement,AdaptiveRefinement,RichardsonExtrapolation,GridConvergenceIndex,StoppingCriterion,Stagnation,DivergenceDetection,RefinementStudy,Verification,Reproducibility]
 STOP_CONDITION = "Stop when target, limiting process, error measure, convergence conditions, rate, refinement and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Numerical Methods"
 RELATED_ONTOLOGIES = ["Numerical Methods","Numerical Methods Ontology 2","Numerical Analysis Ontology 2","Numerical Approximation Ontology","Numerical Stability Ontology 2","Numerical Computation Ontology","Numerical Linear Algebra Ontologies","Differential Equation Ontologies","Scientific Computing Ontology 2"]
 ROUTE_RULES = [
  "Numerical Convergence -> Numerical Methods for parent semantics",
  "Numerical Convergence -> Numerical Analysis for formal convergence theory",
  "Numerical Convergence -> Numerical Approximation for approximation-space convergence",
  "Numerical Convergence -> Numerical Stability for stability relationships",
  "Numerical Convergence -> Numerical Linear Algebra for iterative solver convergence",
  "Numerical Convergence -> Differential Equations for ODE and PDE refinement convergence"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible convergence concepts, criteria or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = ["Preserve convergence-versus-stability distinction.","Preserve residual-versus-solution-error distinction.","Preserve iterative-versus-discretisation convergence distinction.","Preserve theoretical-versus-observed order distinction.","Preserve stopping-versus-convergence-proof distinction.","Preserve asymptotic-regime semantics.","Preserve machine-readable numerical-convergence semantics."]
}

OUTPUT {
 CANONICAL_DEFINITION = "Numerical convergence describes whether computational approximations approach a target solution under iteration, refinement or increasing approximation order."
 ITERATIVE_SUMMARY = "Iterative convergence concerns the limit of repeated algorithmic updates."
 DISCRETISATION_SUMMARY = "Discretisation convergence concerns approach to a continuous target as numerical resolution is refined."
 RATE_SUMMARY = "Convergence rate and order quantify asymptotic error reduction."
 REFINEMENT_SUMMARY = "h, p and hp studies test approximation behaviour under increasing numerical resolution."
 VALIDATION_SUMMARY = "Convergence claims should be checked with error sequences, observed order, refinement studies, stability assumptions and reproducible settings."
}

FLOW {
 INPUT
 -> detect_numerical_convergence_context
 -> identify_target_solution_approximation_sequence_limit_parameter_and_error_norm
 -> classify_iterative_mesh_timestep_order_or_other_convergence
 -> construct_error_or_residual_sequence
 -> derive_convergence_conditions
 -> estimate_convergence_rate_and_order
 -> determine_asymptotic_regime
 -> perform_iteration_mesh_timestep_h_p_or_hp_refinement
 -> assess_stopping_tolerance_stagnation_oscillation_and_roundoff_floor
 -> compare_observed_and_theoretical_order
 -> validate_against_reference_solution
 -> route_to_related_numerical_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Numerical Convergence Ontology defines limit behaviour of computational approximations. It supports iterative, mesh, grid and timestep convergence, fixed-point, root-finding, linear-solver, eigenvalue, optimisation, ODE and PDE convergence, pointwise, uniform, norm, strong and weak convergence, convergence rates and orders, linear, superlinear, quadratic, cubic, spectral and algebraic convergence, asymptotic error, residual and error sequences, consistency and stability relationships, contraction mappings, spectral-radius criteria, Newton and iterative solver convergence, preconditioning, h, p and hp refinement, Richardson extrapolation, grid convergence indices, stopping tolerances, stagnation, oscillation, divergence detection, refinement studies, verification and reproducibility."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-COMPUTATIONAL-MATHEMATICS-NUMERICAL-CONVERGENCE-ONTOLOGY-V1"
}

}

END_FLOW