Numerical Approximation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Numerical Approximation Ontology within the Kireth Research Semantic Library. Numerical approximation studies the construction and analysis of finite numerical representations used to approximate mathematical functions, quantities, operators and solutions. This ontology supports interpolation, polynomial and spline approximation, least-squares approximation, projection methods, spectral approximation, rational approximation, extrapolation, approximation error, convergence, best approximation, normed error, adaptive approximation and relationships with numerical methods, numerical analysis and approximation theory.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = NumericalApproximation
 ENTITY_CLASS = FiniteApproximationRepresentationFramework
 PRIMARY_CONTEXT = NumericalMethods
 DEFINITION = "Numerical approximation is the construction of finite computational representations that approximate mathematical functions, quantities, operators or solutions within a declared error measure."
 SYMBOLIC_FORMS = [
  "f approximately p_n",
  "||f-p_n|| -> 0",
  "u_h approximates u",
  "E_n(f)=inf_(v in V_n)||f-v||"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for interpolation, approximation spaces, best approximation, projection, splines, spectral methods, least squares, rational approximation, extrapolation, adaptive refinement and approximation error."
}

SCOPE {
 INCLUDE = [
  NumericalApproximation,
  Approximation,
  ApproximateSolution,
  ApproximationSpace,
  FiniteDimensionalSpace,
  BasisFunction,
  TrialSpace,
  TestSpace,
  PolynomialApproximation,
  PolynomialInterpolation,
  PiecewisePolynomialApproximation,
  SplineApproximation,
  CubicSpline,
  HermiteApproximation,
  TaylorApproximation,
  ChebyshevApproximation,
  FourierApproximation,
  SpectralApproximation,
  RationalApproximation,
  PadeApproximation,
  LeastSquaresApproximation,
  WeightedLeastSquaresApproximation,
  OrthogonalProjection,
  ProjectionMethod,
  GalerkinProjection,
  CollocationApproximation,
  Interpolation,
  LinearInterpolation,
  LagrangeInterpolation,
  NewtonInterpolation,
  HermiteInterpolation,
  SplineInterpolation,
  Node,
  InterpolationNode,
  ChebyshevNode,
  RungePhenomenon,
  Extrapolation,
  RichardsonExtrapolation,
  SequenceAcceleration,
  BestApproximation,
  MinimaxApproximation,
  UniformApproximation,
  MeanSquareApproximation,
  PointwiseApproximation,
  GlobalApproximation,
  LocalApproximation,
  ApproximationError,
  InterpolationError,
  ProjectionError,
  TruncationError,
  DiscretisationError,
  Residual,
  ErrorBound,
  APrioriErrorEstimate,
  APosterioriErrorEstimate,
  Norm,
  L1Norm,
  L2Norm,
  InfinityNorm,
  EnergyNorm,
  SobolevNorm,
  Convergence,
  ConvergenceRate,
  OrderOfAccuracy,
  SpectralConvergence,
  AlgebraicConvergence,
  Stability,
  Conditioning,
  ApproximationOrder,
  Consistency,
  Mesh,
  Grid,
  StepSize,
  PolynomialDegree,
  PRefinement,
  HRefinement,
  HPRefinement,
  AdaptiveApproximation,
  AdaptiveMesh,
  ErrorIndicator,
  ErrorEstimator,
  SparseApproximation,
  LowRankApproximation,
  MatrixApproximation,
  SingularValueDecomposition,
  TruncatedSVD,
  ProperOrthogonalDecomposition,
  ReducedBasisApproximation,
  SurrogateApproximation,
  ResponseSurface,
  FunctionApproximation,
  OperatorApproximation,
  NumericalVerification,
  ReferenceSolution,
  Benchmark,
  Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  ExactSymbolicRepresentationWithoutApproximation,
  PureInterpolationWithoutErrorOrNumericalApproximationContext,
  StatisticalEstimationWithoutApproximationSpaceSemantics,
  MachineLearningApproximationWithoutNumericalApproximationInterpretation
 ]
 DISAMBIGUATION_RULE = "When a finite numerical representation is constructed to approximate a mathematical object and approximation quality or convergence is central, interpret the context as numerical approximation."
}

CORE_CONCEPTS {
 NumericalApproximation = "Finite numerical representation of a mathematical object within an error measure."
 ApproximationSpace = "Finite or restricted space from which approximating functions or states are selected."
 BestApproximation = "Approximation minimising a declared error norm over an admissible approximation space."
 Interpolation = "Approximation constrained to match known values or derivatives at selected nodes."
 ProjectionMethod = "Approximation obtained by enforcing residual orthogonality or projection conditions."
 LeastSquaresApproximation = "Approximation minimising a sum or norm of squared discrepancies."
 SpectralApproximation = "Approximation using global basis functions, often with rapidly convergent representations for smooth problems."
 Approximation-Error = "Difference between the target object and its approximation under a declared norm or metric."
 AdaptiveApproximation = "Approximation strategy that changes resolution or model order according to error information."
}

RELATIONS {
 NumericalApproximation --is_child_of--> NumericalMethods
 NumericalApproximation --uses--> ApproximationSpace
 ApproximateSolution --belongs_to--> ApproximationSpace
 Interpolation --is_type_of--> NumericalApproximation
 PolynomialInterpolation --uses--> InterpolationNode
 SplineApproximation --uses--> PiecewisePolynomialApproximation
 ChebyshevApproximation --uses--> ChebyshevNode
 LeastSquaresApproximation --minimises--> SquaredResidual
 OrthogonalProjection --minimises_under_conditions--> NormedError
 GalerkinProjection --is_type_of--> ProjectionMethod
 CollocationApproximation --enforces--> ResidualAtPoints
 BestApproximation --minimises--> ApproximationError
 MinimaxApproximation --minimises--> MaximumError
 SpectralApproximation --uses--> GlobalBasisFunctions
 RationalApproximation --uses--> RationalFunction
 RichardsonExtrapolation --improves--> AsymptoticApproximation
 AdaptiveApproximation --uses--> ErrorEstimator
 PRefinement --increases--> ApproximationOrder
 HRefinement --decreases--> MeshSize
 HPRefinement --combines--> HAndPRefinement
 LowRankApproximation --reduces--> RepresentationDimension
 TruncatedSVD --produces--> LowRankApproximation
}

OPERATORS {
 APPROXIMATE(f,V_n) := choose(v_n in V_n)
 BEST_APPROXIMATION(f,V_n,norm) := argmin_(v in V_n)||f-v||
 APPROXIMATION_ERROR(f,v,norm) := ||f-v||
 INTERPOLATION_CONDITION(p,x_i,y_i) := p(x_i)=y_i
 LAGRANGE_FORM(x,x_i,y_i) := p(x)=sum_i y_i*L_i(x)
 LEAST_SQUARES(A,x,b) := x_hat=argmin_x ||A*x-b||_2
 ORTHOGONAL_PROJECTION(f,V) := find(v in V): <f-v,w>=0 for_all_w_in_V
 MINIMAX_ERROR(f,p) := max_x |f(x)-p(x)|
 RICHARDSON(A_h,A_h2,p) := A_star=A_h2+(A_h2-A_h)/(2^p-1)
 TRUNCATED_SVD(A,k) := A_k=sum_i=1^k sigma_i*u_i*v_i^T
}

AXIOMS {
 A1 = "Numerical approximation requires an explicit target object and approximation space."
 A2 = "Approximation quality requires an explicit norm, metric or error functional."
 A3 = "Interpolation and best approximation are distinct constructions."
 A4 = "Higher approximation order does not guarantee better numerical behaviour for arbitrary node choices."
 A5 = "Projection methods depend on the chosen trial, test and inner-product structures."
 A6 = "Convergence must be interpreted relative to refinement of space, mesh or model order."
 A7 = "Adaptive approximation requires meaningful local or global error information."
 A8 = "Approximation claims require error, convergence and reproducibility validation."
}

REASONING {
 R1_CONTEXT = identify(target_object,domain,regularity,desired_norm,accuracy_goal)
 R2_SPACE = choose(polynomial,spline,spectral,rational,low_rank_or_other_approximation_space)
 R3_CONSTRUCTION = select(interpolation,projection,least_squares,collocation,best_approximation_or_extrapolation)
 R4_NODES = if_interpolation choose_nodes_and_assess_node_quality)
 R5_ERROR = derive_or_estimate(approximation,interpolation,projection_or_truncation_error)
 R6_CONVERGENCE = determine(rate,order,spectral_or_algebraic_behaviour)
 R7_STABILITY = assess(conditioning,Runge_effect,basis_scaling,numerical_stability)
 R8_ADAPTIVITY = if_required refine(mesh,degree,basis_or_rank)
 R9_COMPRESSION = if_high_dimensional consider(low_rank,SVD,reduced_basis)
 R10_VALIDATION = compare(reference_solution,error_bounds,refinement_and_reproducibility)
}

CONSTRAINTS {
 C1 = "High polynomial degree can produce oscillatory interpolation on poorly chosen nodes."
 C2 = "Pointwise interpolation accuracy does not imply small global norm error."
 C3 = "Least-squares optimality depends on the chosen norm and weighting."
 C4 = "Spectral convergence requires sufficient regularity."
 C5 = "Rational approximation can introduce poles that require interpretation."
 C6 = "Low-rank approximation quality depends on singular-value decay."
 C7 = "Adaptive refinement can overfocus on an inappropriate error indicator."
 C8 = "Extrapolation outside the approximation domain is less reliable than interpolation."
 C9 = "Approximation error and floating-point error can interact."
 C10 = "A small residual does not always imply a small approximation error."
}

SEMANTIC_EQUIVALENCE {
 "NumericalApproximation" <=> "ComputationalApproximation"
 "LeastSquaresApproximation" <=> "LeastSquaresFitInApproximationContext"
 "MinimaxApproximation" <=> "UniformBestApproximation"
 "PRefinement" <=> "PolynomialOrderRefinement"
 "HRefinement" <=> "MeshRefinement"
 "HPRefinement" <=> "CombinedMeshAndOrderRefinement"
}

VALIDATION {
 V1 = "Target object and approximation domain are explicit."
 V2 = "Approximation space and basis are documented."
 V3 = "Error norm or metric is explicit."
 V4 = "Interpolation nodes or projection conditions are documented."
 V5 = "Convergence is assessed under refinement."
 V6 = "Conditioning and stability are considered."
 V7 = "Adaptive methods document their error indicators."
 V8 = "Low-rank approximations report retained rank or energy."
 V9 = "Reference or benchmark solutions are used where feasible."
 V10 = "Results are reproducible under documented approximation settings."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 NUMERICAL_APPROXIMATION_DEFINITION = 0.99
 INTERPOLATION_REASONING = 0.99
 PROJECTION_REASONING = 0.99
 LEAST_SQUARES_REASONING = 0.99
 SPECTRAL_APPROXIMATION_REASONING = 0.98
 ERROR_REASONING = 0.99
 ADAPTIVITY_REASONING = 0.98
 LOW_RANK_REASONING = 0.98
 POLICY = "Reduce confidence when approximation space, norm, regularity, node distribution, refinement regime, rank or error estimator are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  NumericalApproximationReasoning,
  ApproximationSpaceReasoning,
  InterpolationReasoning,
  PolynomialApproximationReasoning,
  SplineApproximationReasoning,
  SpectralApproximationReasoning,
  RationalApproximationReasoning,
  LeastSquaresReasoning,
  ProjectionReasoning,
  BestApproximationReasoning,
  ExtrapolationReasoning,
  ErrorEstimationReasoning,
  ConvergenceReasoning,
  AdaptiveApproximationReasoning,
  LowRankApproximationReasoning,
  ReducedBasisReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "numerical approximation" -> NumericalApproximation
  "computational approximation" -> NumericalApproximation
  "least squares approximation" -> LeastSquaresApproximation
  "minimax approximation" -> MinimaxApproximation
  "p refinement" -> PRefinement
  "h refinement" -> HRefinement
  "hp refinement" -> HPRefinement
 }
}

EXECUTION_RULES {
 E1 = "If a mathematical object is represented approximately in a finite computational space, activate NumericalApproximationOntology."
 E2 = "If exact matching at selected nodes is required, use interpolation reasoning."
 E3 = "If global squared error is minimised, use least-squares reasoning."
 E4 = "If residual orthogonality is central, use projection or Galerkin reasoning."
 E5 = "If high smoothness and global bases are available, consider spectral approximation."
 E6 = "If error is localised, consider adaptive h, p or hp refinement."
 E7 = "If dimensional compression is needed, consider low-rank or reduced-basis approximation."
 E8 = "If extrapolating beyond observed or discretised support, reduce confidence and document assumptions."
 E9 = "Any approximation should be validated for error, conditioning, convergence and reproducibility."
}

RECURSION {
 ENTRY = NumericalApproximation
 EXPAND(NumericalApproximation) -> [
  Approximation, ApproximateSolution, ApproximationSpace,
  FiniteDimensionalSpace, BasisFunction, TrialSpace, TestSpace,
  PolynomialApproximation, PolynomialInterpolation,
  PiecewisePolynomialApproximation, SplineApproximation, CubicSpline,
  HermiteApproximation, TaylorApproximation, ChebyshevApproximation,
  FourierApproximation, SpectralApproximation, RationalApproximation,
  PadeApproximation, LeastSquaresApproximation,
  WeightedLeastSquaresApproximation, OrthogonalProjection,
  ProjectionMethod, GalerkinProjection, CollocationApproximation,
  Interpolation, LinearInterpolation, LagrangeInterpolation,
  NewtonInterpolation, HermiteInterpolation, SplineInterpolation,
  Node, InterpolationNode, ChebyshevNode, RungePhenomenon,
  Extrapolation, RichardsonExtrapolation, SequenceAcceleration,
  BestApproximation, MinimaxApproximation, UniformApproximation,
  MeanSquareApproximation, PointwiseApproximation, GlobalApproximation,
  LocalApproximation, ApproximationError, InterpolationError,
  ProjectionError, TruncationError, DiscretisationError, Residual,
  ErrorBound, APrioriErrorEstimate, APosterioriErrorEstimate,
  Norm, L1Norm, L2Norm, InfinityNorm, EnergyNorm, SobolevNorm,
  Convergence, ConvergenceRate, OrderOfAccuracy, SpectralConvergence,
  AlgebraicConvergence, Stability, Conditioning, ApproximationOrder,
  Consistency, Mesh, Grid, StepSize, PolynomialDegree, PRefinement,
  HRefinement, HPRefinement, AdaptiveApproximation, AdaptiveMesh,
  ErrorIndicator, ErrorEstimator, SparseApproximation,
  LowRankApproximation, MatrixApproximation, SingularValueDecomposition,
  TruncatedSVD, ProperOrthogonalDecomposition, ReducedBasisApproximation,
  SurrogateApproximation, ResponseSurface, FunctionApproximation,
  OperatorApproximation, NumericalVerification, ReferenceSolution,
  Benchmark, Reproducibility
 ]
 EXPAND(ApproximationFamilies) -> [
  InterpolationMethods,
  PolynomialAndSplineApproximation,
  LeastSquaresApproximation,
  ProjectionMethods,
  SpectralAndRationalApproximation,
  AdaptiveApproximation,
  LowRankAndReducedApproximation
 ]
 STOP_CONDITION = "Stop when target, approximation space, construction, error norm, convergence, stability and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Numerical Methods"
 RELATED_ONTOLOGIES = [
  "Numerical Methods",
  "Numerical Methods Ontology 2",
  "Numerical Analysis Ontology 2",
  "Numerical Computation Ontology",
  "Approximation Theory Ontologies",
  "Linear Algebra Ontologies",
  "Functional Analysis Ontologies",
  "Finite Element Ontologies",
  "Spectral Method Ontologies",
  "Scientific Computing Ontology 2"
 ]
 ROUTE_RULES = [
  "Numerical Approximation -> Numerical Methods for parent semantics",
  "Numerical Approximation -> Numerical Methods Ontology 2 for core numerical method structure",
  "Numerical Approximation -> Numerical Analysis for error and convergence",
  "Numerical Approximation -> Numerical Computation for finite-precision semantics",
  "Numerical Approximation -> Approximation Theory for theoretical approximation structure",
  "Numerical Approximation -> Linear Algebra for least-squares and low-rank methods",
  "Numerical Approximation -> Functional Analysis for normed-space and projection semantics",
  "Numerical Approximation -> Finite Elements for piecewise approximation spaces",
  "Numerical Approximation -> Spectral Methods for global-basis approximation",
  "Numerical Approximation -> Scientific Computing for applied approximation models"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible approximation spaces, methods or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve target-versus-approximation distinction.",
  "Preserve interpolation-versus-best-approximation distinction.",
  "Preserve local-versus-global approximation distinction.",
  "Preserve approximation-space semantics.",
  "Preserve error-norm semantics.",
  "Preserve h-versus-p-versus-hp refinement distinction.",
  "Preserve full-rank-versus-low-rank distinction.",
  "Preserve machine-readable numerical-approximation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Numerical approximation constructs finite representations of mathematical objects and quantifies how closely they reproduce the target."
 SPACE_SUMMARY = "Approximation spaces and basis functions define the admissible finite representations."
 INTERPOLATION_SUMMARY = "Interpolation enforces exact matching of selected values or derivatives at nodes."
 PROJECTION_SUMMARY = "Projection and Galerkin methods select approximations using orthogonality or residual conditions."
 ERROR_SUMMARY = "Approximation quality is measured with declared norms, residuals or error functionals."
 ADAPTIVE_SUMMARY = "h, p and hp refinement improve approximation resolution based on error information."
 LOW_RANK_SUMMARY = "SVD, POD and reduced-basis methods compress high-dimensional representations while controlling approximation loss."
 VALIDATION_SUMMARY = "Numerical approximations should be checked for error, conditioning, convergence, refinement behaviour and reproducibility."
}

FLOW {
 INPUT
 -> detect_numerical_approximation_context
 -> identify_target_object_domain_regularities_accuracy_goal_and_error_norm
 -> choose_polynomial_spline_spectral_rational_low_rank_or_other_approximation_space
 -> select_interpolation_projection_least_squares_collocation_best_approximation_or_extrapolation_method
 -> choose_nodes_basis_trial_and_test_spaces_if_required
 -> derive_or_estimate_approximation_error
 -> assess_conditioning_stability_and_convergence
 -> if_adaptive_refine_mesh_degree_basis_or_rank
 -> if_high_dimensional_consider_SVD_POD_or_reduced_basis
 -> validate_against_error_bounds_reference_solutions_refinement_and_reproducibility
 -> route_to_numerical_methods_core_numerical_methods_numerical_analysis_numerical_computation_approximation_theory_linear_algebra_functional_analysis_finite_elements_spectral_methods_or_scientific_computing_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Numerical Approximation Ontology defines finite computational representation of mathematical objects. It supports approximation spaces, basis, trial and test spaces, polynomial, piecewise-polynomial, spline, Hermite, Taylor, Chebyshev, Fourier, spectral and rational approximation, least-squares and weighted least-squares methods, orthogonal projection, Galerkin and collocation methods, linear, Lagrange, Newton, Hermite and spline interpolation, Chebyshev nodes and the Runge phenomenon, Richardson extrapolation and sequence acceleration, best, minimax, uniform, mean-square, pointwise, global and local approximation, approximation, interpolation, projection, truncation and discretisation error, a-priori and a-posteriori error estimates, L1, L2, infinity, energy and Sobolev norms, convergence rates and approximation order, h, p and hp refinement, adaptive approximation, low-rank matrix approximation, truncated SVD, POD, reduced-basis methods, surrogate approximation, numerical verification, benchmarks and reproducibility."
}

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

}

END_FLOW