Adaptive Integration Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Adaptive Integration Ontology within the Kireth Research Semantic Library. Adaptive integration approximates definite integrals by dynamically refining the integration domain where local error estimates indicate insufficient accuracy. This ontology supports adaptive Simpson methods, adaptive Gaussian and Gauss-Kronrod rules, local and global tolerances, recursive subdivision, priority-based refinement, singular and oscillatory integrands, error estimation, convergence, work allocation, floating-point effects and verification.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Adaptive Integration Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Numerical Calculus / Adaptive Integration"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/numerical-calculus/adaptive-integration-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = AdaptiveIntegration
 ENTITY_CLASS = ErrorControlledAdaptiveQuadratureFramework
 PRIMARY_CONTEXT = Quadrature
 DEFINITION = "Adaptive integration is numerical quadrature that allocates additional function evaluations or subdivisions to regions where estimated local error is too large."
 SYMBOLIC_FORMS = [
  "I approximately sum_j Q_j",
  "E approximately sum_j E_j",
  "if E_j>tol_j then refine interval_j",
  "tol_global = tol_abs + tol_rel*|I|"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for error-driven quadrature refinement, local tolerance allocation, recursive subdivision, embedded rules, adaptive stopping, work control and finite-precision validation."
}

SCOPE {
 INCLUDE = [
  AdaptiveIntegration, AdaptiveQuadrature, AdaptiveNumericalIntegration,
  ErrorControlledIntegration, LocalRefinement, GlobalRefinement,
  RecursiveSubdivision, IntervalSubdivision, Subinterval,
  IntegrationInterval, Integrand, DefiniteIntegral, NumericalIntegral,
  QuadratureRule, BaseQuadratureRule, LocalQuadratureRule,
  AdaptiveSimpson, AdaptiveTrapezoidalRule, AdaptiveGaussianQuadrature,
  GaussKronrodQuadrature, GaussPattersonQuadrature,
  EmbeddedQuadrature, NestedQuadrature, CoarseRule, FineRule,
  LocalErrorEstimate, GlobalErrorEstimate, ErrorEstimator,
  APosterioriErrorEstimate, ErrorIndicator, ErrorBudget,
  ErrorEquidistribution, AbsoluteTolerance, RelativeTolerance,
  LocalTolerance, GlobalTolerance, MixedTolerance,
  ToleranceAllocation, SafetyFactor, RefinementCriterion,
  AcceptanceCriterion, RejectionCriterion, TerminationCriterion,
  MaximumDepth, MaximumSubdivisions, MaximumEvaluations,
  FunctionEvaluationCount, EvaluationReuse, NodeReuse,
  PriorityQueueRefinement, LargestErrorFirst, RecursiveAlgorithm,
  IterativeAdaptiveAlgorithm, BinarySubdivision, UnequalSubdivision,
  AdaptiveMesh, IntervalTree, RefinementTree,
  SmoothIntegrand, NonsmoothIntegrand, Discontinuity,
  EndpointSingularity, InteriorSingularity, ImproperIntegral,
  OscillatoryIntegral, HighlyOscillatoryIntegral, BoundaryLayer,
  SharpPeak, LocalisedFeature, Cancellation, NearSingularity,
  ChangeOfVariables, SingularityTransformation, DoubleExponentialTransform,
  TanhSinhQuadrature, OscillatoryQuadrature, FilonMethod, LevinMethod,
  Convergence, ConvergenceRate, Reliability, Efficiency,
  WorkPrecisionTradeoff, ComputationalCost, FunctionEvaluationCost,
  RoundOffError, FloatingPointArithmetic, MachinePrecision,
  Stagnation, Underflow, Overflow, ErrorFloor,
  CompensatedSummation, PairwiseSummation,
  Verification, ReferenceIntegral, Benchmarking, Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  FixedGridQuadratureWithoutRefinement,
  SymbolicIntegrationWithoutNumericalErrorControl,
  MonteCarloIntegrationWithoutDeterministicAdaptiveSubdivision,
  AdaptiveDifferentiationWithoutIntegralApproximation
 ]
 DISAMBIGUATION_RULE = "When numerical integration dynamically reallocates resolution according to local error estimates or refinement indicators, interpret the context as adaptive integration."
}

CORE_CONCEPTS {
 AdaptiveIntegration = "Numerical integration with dynamic error-driven refinement."
 LocalErrorEstimate = "Estimate of quadrature error on a specific subinterval."
 GlobalErrorEstimate = "Aggregate estimate of total integration error."
 ErrorBudget = "Allowed error distributed across the integration domain."
 AdaptiveSimpson = "Adaptive quadrature based on coarse and refined Simpson estimates."
 GaussKronrodQuadrature = "Embedded Gaussian-Kronrod rule providing integral and error estimates from related node sets."
 RefinementTree = "Hierarchical structure recording subdivision of the integration interval."
 WorkPrecisionTradeoff = "Balance between function-evaluation cost and achieved error."
}

RELATIONS {
 AdaptiveIntegration --is_child_of--> Quadrature
 AdaptiveIntegration --uses--> ErrorEstimator
 ErrorEstimator --produces--> LocalErrorEstimate
 LocalErrorEstimate --controls--> RefinementCriterion
 RefinementCriterion --triggers--> IntervalSubdivision
 IntervalSubdivision --creates--> ChildSubinterval
 ChildSubinterval --receives--> LocalTolerance
 ErrorBudget --allocates--> LocalTolerance
 AdaptiveSimpson --uses--> SimpsonRule
 AdaptiveGaussianQuadrature --uses--> GaussianQuadrature
 GaussKronrodQuadrature --is_type_of--> EmbeddedQuadrature
 EmbeddedQuadrature --provides--> ErrorEstimate
 PriorityQueueRefinement --selects--> LargestErrorSubinterval
 EvaluationReuse --reduces--> FunctionEvaluationCost
 ChangeOfVariables --can_improve--> SingularIntegration
 RefinementTree --records--> AdaptiveSubdivisionHistory
 Verification --checks--> AdaptiveIntegrationResult
}

OPERATORS {
 GLOBAL_TOLERANCE(I,atol,rtol) := atol+rtol*|I|
 LOCAL_ERROR(Q_coarse,Q_fine,c) := |Q_fine-Q_coarse|/c
 ACCEPT(E_local,tol_local) := E_local<=tol_local
 BISECT_INTERVAL(a,b) := [a,(a+b)/2] union [(a+b)/2,b]
 ADAPTIVE_SUM(intervals,Q) := sum_j Q(interval_j)
 GLOBAL_ERROR(errors) := sum_j errors_j
 PRIORITY(intervals) := argmax_j E_j
 RECURSIVE_REFINE(interval) := if_not_accept_then_refine_children
}

AXIOMS {
 A1 = "Adaptive integration allocates more computational effort where estimated integration error is larger."
 A2 = "Local error estimates guide refinement but are themselves approximations."
 A3 = "Global error control requires a consistent strategy for combining local error estimates."
 A4 = "Embedded or nested quadrature rules can estimate error efficiently by reusing function values."
 A5 = "Adaptive refinement can substantially outperform uniform refinement for localised integrand difficulty."
 A6 = "Error estimators can fail when important integrand features are not sampled."
 A7 = "Round-off imposes a lower practical error floor that refinement cannot overcome."
 A8 = "Adaptive integration results require explicit tolerance, refinement, termination and verification semantics."
}

REASONING {
 R1_CONTEXT = identify(integrand,interval,dimension,regularity,singularities,oscillation,accuracy_goal)
 R2_BASE_RULE = select(Simpson,GaussKronrod,Gaussian,trapezoidal_or_other_embedded_rule)
 R3_INITIAL_ESTIMATE = compute(coarse_and_if_needed_fine_estimates)
 R4_ERROR = estimate(local_error_and_global_error_budget)
 R5_ACCEPT_OR_REFINE = compare(local_error,allocated_tolerance)
 R6_SUBDIVIDE = refine_rejected_intervals_and_reuse_function_values)
 R7_PRIORITY = optionally_process_largest_error_intervals_first
 R8_TERMINATE = stop_when(global_tolerance_met_or_resource_limit_reached)
 R9_STABILITY = assess(roundoff,stagnation,cancellation,underflow,overflow_and_error_floor)
 R10_VALIDATE = compare(reference_integral,independent_rule_or_refinement_history)
}

CONSTRAINTS {
 C1 = "Adaptive methods can miss narrow peaks or singularities if initial sampling fails to detect them."
 C2 = "Local error estimates are not rigorous bounds unless the estimator provides a proven bound under stated assumptions."
 C3 = "Recursive subdivision can exceed depth limits near singular or pathological integrands."
 C4 = "Overly strict tolerances can cause excessive work or finite-precision stagnation."
 C5 = "Global relative tolerance is unreliable when the true integral is near zero unless combined with an absolute tolerance."
 C6 = "Improper integrals often require transformations before standard adaptive subdivision."
 C7 = "Highly oscillatory integrals may defeat generic adaptive rules."
 C8 = "Discontinuities can reduce local order and trigger excessive refinement."
 C9 = "Error summation can itself suffer floating-point loss when many intervals are accumulated."
 C10 = "Termination caused by resource limits is not equivalent to meeting the requested error tolerance."
}

SEMANTIC_EQUIVALENCE {
 "AdaptiveIntegration" <=> "AdaptiveQuadrature"
 "AdaptiveNumericalIntegration" <=> "AdaptiveIntegration"
 "LocalRefinement" <=> "ErrorDrivenRefinement"
 "GaussKronrodQuadrature" <=> "EmbeddedGaussKronrodRule"
 "LargestErrorFirst" <=> "PriorityBasedAdaptiveRefinement"
}

VALIDATION {
 V1 = "Integrand and integration domain are explicit."
 V2 = "Base quadrature rule is documented."
 V3 = "Absolute and relative tolerances are documented."
 V4 = "Local-to-global error allocation strategy is documented."
 V5 = "Refinement and acceptance criteria are documented."
 V6 = "Maximum depth, evaluation or subdivision limits are documented."
 V7 = "Termination status distinguishes tolerance satisfaction from resource-limit termination."
 V8 = "Singular, improper or oscillatory behaviour is treated explicitly."
 V9 = "Reference or independent integration checks are used where feasible."
 V10 = "Results are reproducible under documented arithmetic and adaptive settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ADAPTIVE_INTEGRATION_DEFINITION = 0.99
 ERROR_ESTIMATION_REASONING = 0.99
 ADAPTIVE_SIMPSON_REASONING = 0.99
 GAUSS_KRONROD_REASONING = 0.99
 PRIORITY_REFINEMENT_REASONING = 0.98
 SINGULAR_INTEGRAL_REASONING = 0.98
 FINITE_PRECISION_REASONING = 0.98
 POLICY = "Reduce confidence when error estimator, tolerance allocation, hidden local features, singularities, refinement limits or arithmetic precision are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, AdaptiveIntegrationReasoning,
  AdaptiveQuadratureReasoning, ErrorEstimatorReasoning,
  LocalErrorReasoning, GlobalErrorReasoning,
  ToleranceAllocationReasoning, AdaptiveSimpsonReasoning,
  GaussKronrodReasoning, RecursiveSubdivisionReasoning,
  PriorityRefinementReasoning, SingularityHandlingReasoning,
  OscillatoryIntegrationReasoning, WorkPrecisionReasoning,
  FloatingPointStagnationReasoning, VerificationReasoning,
  KnowledgeGraphConstruction, RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "adaptive integration" -> AdaptiveIntegration
  "adaptive quadrature" -> AdaptiveIntegration
  "adaptive Simpson" -> AdaptiveSimpson
  "Gauss-Kronrod" -> GaussKronrodQuadrature
  "error-driven refinement" -> LocalRefinement
  "largest error first" -> PriorityQueueRefinement
 }
}

EXECUTION_RULES {
 E1 = "If quadrature resolution is changed dynamically according to estimated local error, activate AdaptiveIntegrationOntology."
 E2 = "Define absolute and relative tolerances before refinement begins."
 E3 = "Use an embedded or refinement-based error estimator appropriate to the base rule."
 E4 = "Refine only intervals whose estimated error exceeds their allocated tolerance."
 E5 = "Reuse existing function evaluations whenever the rule permits."
 E6 = "If singular or improper behaviour is known, apply a suitable transformation before generic adaptive refinement."
 E7 = "If highly oscillatory behaviour is present, consider specialised oscillatory quadrature."
 E8 = "Stop on tolerance satisfaction, finite-precision stagnation or explicit resource limits, and distinguish these outcomes."
 E9 = "Validate through error history, reference integrals, independent rules and reproducibility."
}

RECURSION {
 ENTRY = AdaptiveIntegration
 EXPAND(AdaptiveIntegration) -> [
  AdaptiveQuadrature,AdaptiveNumericalIntegration,ErrorControlledIntegration,
  LocalRefinement,GlobalRefinement,RecursiveSubdivision,IntervalSubdivision,
  Subinterval,IntegrationInterval,Integrand,DefiniteIntegral,NumericalIntegral,
  QuadratureRule,BaseQuadratureRule,AdaptiveSimpson,
  AdaptiveTrapezoidalRule,AdaptiveGaussianQuadrature,
  GaussKronrodQuadrature,GaussPattersonQuadrature,EmbeddedQuadrature,
  NestedQuadrature,CoarseRule,FineRule,LocalErrorEstimate,
  GlobalErrorEstimate,ErrorEstimator,APosterioriErrorEstimate,
  ErrorIndicator,ErrorBudget,ErrorEquidistribution,AbsoluteTolerance,
  RelativeTolerance,LocalTolerance,GlobalTolerance,MixedTolerance,
  ToleranceAllocation,SafetyFactor,RefinementCriterion,
  AcceptanceCriterion,RejectionCriterion,TerminationCriterion,
  MaximumDepth,MaximumSubdivisions,MaximumEvaluations,
  FunctionEvaluationCount,EvaluationReuse,NodeReuse,
  PriorityQueueRefinement,LargestErrorFirst,RecursiveAlgorithm,
  IterativeAdaptiveAlgorithm,BinarySubdivision,AdaptiveMesh,
  IntervalTree,RefinementTree,SmoothIntegrand,NonsmoothIntegrand,
  Discontinuity,EndpointSingularity,InteriorSingularity,
  ImproperIntegral,OscillatoryIntegral,BoundaryLayer,SharpPeak,
  LocalisedFeature,NearSingularity,ChangeOfVariables,
  SingularityTransformation,TanhSinhQuadrature,FilonMethod,LevinMethod,
  Convergence,ConvergenceRate,Reliability,Efficiency,
  WorkPrecisionTradeoff,ComputationalCost,RoundOffError,
  FloatingPointArithmetic,MachinePrecision,Stagnation,ErrorFloor,
  CompensatedSummation,PairwiseSummation,Verification,
  ReferenceIntegral,Benchmarking,Reproducibility
 ]
 EXPAND(AdaptiveIntegrationFamilies) -> [
  AdaptiveSimpsonMethods,
  AdaptiveGaussianMethods,
  GaussKronrodMethods,
  PriorityBasedRefinement,
  SingularAndImproperAdaptiveIntegration,
  OscillatoryAdaptiveIntegration,
  ErrorAndTerminationAnalysis
 ]
 STOP_CONDITION = "Stop when base rule, error estimator, tolerance allocation, refinement strategy, termination, stability and verification are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Quadrature Ontology"
 RELATED_ONTOLOGIES = [
  "Quadrature Ontology",
  "Numerical Calculus",
  "Simpson's Rule Ontology",
  "Trapezoidal Rule Ontology",
  "Gaussian Quadrature Ontology",
  "Numerical Methods",
  "Numerical Analysis Ontology 2",
  "Numerical Error Ontology",
  "Numerical Stability Ontology 2",
  "Numerical Convergence Ontology",
  "Floating Point Arithmetic Ontology"
 ]
 ROUTE_RULES = [
  "Adaptive Integration -> Quadrature for parent numerical-integration semantics",
  "Adaptive Integration -> Numerical Calculus for broader integral approximation context",
  "Adaptive Integration -> Simpson's Rule for adaptive Simpson semantics",
  "Adaptive Integration -> Gaussian Quadrature for Gauss-Kronrod and adaptive Gaussian semantics",
  "Adaptive Integration -> Trapezoidal Rule for adaptive trapezoidal refinement",
  "Adaptive Integration -> Numerical Analysis for error estimation and convergence",
  "Adaptive Integration -> Numerical Error for local and global error semantics",
  "Adaptive Integration -> Numerical Stability for finite-precision termination behaviour",
  "Adaptive Integration -> Numerical Convergence for refinement and stopping",
  "Adaptive Integration -> Floating Point Arithmetic for stagnation and summation limits"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible adaptive rules, error estimators or refinement strategies"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve fixed-grid-versus-adaptive distinction.",
  "Preserve local-error-versus-global-error distinction.",
  "Preserve tolerance-satisfaction-versus-resource-limit termination distinction.",
  "Preserve base-rule-versus-adaptive-controller distinction.",
  "Preserve recursive-versus-priority-based refinement distinction.",
  "Preserve discretisation-error-versus-round-off-error distinction.",
  "Preserve error-estimate-versus-rigorous-bound distinction.",
  "Preserve machine-readable adaptive-integration semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Adaptive integration controls numerical integration error by refining the domain where estimated local error is too large."
 ERROR_SUMMARY = "Local error estimates are compared with allocated tolerances and combined into a global error estimate."
 REFINEMENT_SUMMARY = "Rejected intervals are subdivided recursively or by priority until accuracy or resource limits are reached."
 EMBEDDED_SUMMARY = "Adaptive Simpson and Gauss-Kronrod methods use paired coarse and refined estimates to obtain economical local error indicators."
 SPECIAL_CASE_SUMMARY = "Singular, improper and highly oscillatory integrals may require transformations or specialised adaptive rules."
 EFFICIENCY_SUMMARY = "Adaptive methods focus function evaluations on difficult regions, improving work-precision efficiency for nonuniform integrands."
 VALIDATION_SUMMARY = "Adaptive integration results should report tolerance satisfaction, error estimates, refinement history, termination status and reproducibility."
}

FLOW {
 INPUT
 -> detect_adaptive_integration_context
 -> identify_integrand_domain_regularity_singularities_oscillation_dimension_and_accuracy_goal
 -> select_base_quadrature_rule_and_error_estimator
 -> set_absolute_relative_and_local_tolerances
 -> compute_initial_integral_and_error_estimates
 -> accept_intervals_meeting_tolerance
 -> refine_rejected_intervals_and_reuse_function_values
 -> optionally_prioritise_largest_error_intervals
 -> accumulate_integral_and_global_error_estimates
 -> monitor_tolerance_satisfaction_depth_evaluation_limits_stagnation_and_error_floor
 -> if_needed_apply_singularity_transform_or_specialised_oscillatory_method
 -> classify_termination_status
 -> validate_against_reference_integral_independent_rule_or_refinement_history
 -> route_to_related_quadrature_numerical_calculus_Simpson_trapezoidal_Gaussian_analysis_error_stability_convergence_and_floating_point_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Adaptive Integration Ontology defines numerical integration with dynamic error-driven refinement. It supports adaptive Simpson, trapezoidal, Gaussian, Gauss-Kronrod and Gauss-Patterson methods, embedded and nested quadrature, coarse-versus-fine error estimates, local and global tolerances, error budgets, safety factors, recursive and priority-based subdivision, interval and refinement trees, function-value reuse, singular, improper, peaked and oscillatory integrands, transformations, work-precision tradeoffs, convergence, round-off and finite-precision error floors, termination by tolerance or resource limits, verification, benchmarking and reproducibility."
}

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

}

END_FLOW