Spectral Method Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Spectral Method Ontology within the Kireth Research Semantic Library. Spectral methods approximate functions and solutions of differential equations using globally supported basis functions such as Fourier modes, Chebyshev polynomials or other orthogonal polynomial families. This ontology supports modal and nodal formulations, collocation, Galerkin projection, pseudospectral methods, spectral differentiation, transform methods, aliasing, dealiasing, exponential or spectral convergence for smooth problems, boundary treatment, eigenfunction expansions, tensor-product discretisations and relationships with numerical analysis, approximation theory, numerical differentiation, partial differential equations and scientific computing.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Spectral Method Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Numerical Analysis / Spectral Methods"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/numerical-analysis/spectral-method-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = SpectralMethod
 SYMBOLIC_FORMS = [
  "u_N(x)=sum_{k=0}^{N} a_k phi_k(x)",
  "D_N u",
  "F_k",
  "T_k(x)"
 ]
 ENTITY_CLASS = GlobalBasisDiscretisationFramework
 PRIMARY_CONTEXT = NumericalAnalysis
 DEFINITION = "A spectral method is a numerical technique that approximates functions or differential-equation solutions using global basis expansions, commonly Fourier or orthogonal polynomial bases."
 SEMANTIC_ROLE = "Provides the machine-readable framework for modal and collocation spectral discretisation, global basis functions, spectral differentiation, transforms, aliasing, convergence and boundary treatment."
}

SCOPE {
 INCLUDE = [
  SpectralMethod,
  SpectralApproximation,
  GlobalBasisFunction,
  ModalMethod,
  NodalMethod,
  SpectralGalerkinMethod,
  SpectralCollocationMethod,
  PseudospectralMethod,
  FourierSpectralMethod,
  ChebyshevSpectralMethod,
  LegendreSpectralMethod,
  HermiteSpectralMethod,
  LaguerreSpectralMethod,
  FourierSeries,
  ChebyshevPolynomial,
  LegendrePolynomial,
  OrthogonalPolynomial,
  SpectralCoefficient,
  CollocationPoint,
  GaussPoint,
  GaussLobattoPoint,
  SpectralDifferentiation,
  DifferentiationMatrix,
  TransformMethod,
  FastFourierTransform,
  DiscreteCosineTransform,
  Aliasing,
  Dealiasing,
  TwoThirdsRule,
  SpectralConvergence,
  ExponentialConvergence,
  GibbsPhenomenon,
  BoundaryCondition,
  PeriodicBoundaryCondition,
  DirichletCondition,
  NeumannCondition,
  TensorProductSpectralMethod,
  DomainMapping,
  SpectralFilter
 ]

 EXCLUDE_AS_PRIMARY = [
  FiniteDifferenceMethod,
  FiniteElementMethod,
  FiniteVolumeMethod,
  LocalPolynomialMethodWithoutGlobalBasisSemantics,
  FourierAnalysisWithoutNumericalApproximationContext
 ]

 DISAMBIGUATION_RULE =
  "When a numerical approximation uses global Fourier, Chebyshev, Legendre or related basis functions and determines global coefficients or nodal values, interpret the context as a spectral method."
}

CORE_CONCEPTS {

 SpectralMethod {
  definition = "A numerical method based on global basis expansion."
 }

 SpectralApproximation {
  definition = "An approximation represented by a truncated series of global basis functions."
 }

 ModalMethod {
  definition = "A spectral formulation whose unknowns are expansion coefficients."
 }

 NodalMethod {
  definition = "A spectral formulation whose unknowns are function values at selected collocation nodes."
 }

 SpectralGalerkinMethod {
  definition = "A method projecting the residual onto a global trial and test basis."
 }

 SpectralCollocationMethod {
  definition = "A method enforcing differential equations at selected spectral nodes."
 }

 PseudospectralMethod {
  definition = "A nodal spectral method using transforms between physical-space values and spectral coefficients."
 }

 SpectralDifferentiation {
  definition = "Derivative computation through analytic differentiation of the spectral basis or equivalent differentiation matrices."
 }

 Aliasing {
  definition = "Misrepresentation of unresolved high-frequency modes as lower-frequency modes in a discrete spectral representation."
 }

 SpectralConvergence {
  definition = "Rapid convergence associated with increasing spectral resolution, often faster than algebraic for sufficiently smooth functions."
 }

 GibbsPhenomenon {
  definition = "Persistent oscillatory overshoot near discontinuities in truncated global spectral expansions."
 }
}

RELATIONS {
 SpectralMethod --is_type_of--> NumericalMethod
 SpectralMethod --uses--> GlobalBasisFunction
 ModalMethod --represents_unknowns_as--> SpectralCoefficient
 NodalMethod --represents_unknowns_as--> CollocationValue
 SpectralGalerkinMethod --is_type_of--> SpectralMethod
 SpectralCollocationMethod --is_type_of--> SpectralMethod
 PseudospectralMethod --is_type_of--> SpectralCollocationMethod
 FourierSpectralMethod --uses--> FourierSeries
 ChebyshevSpectralMethod --uses--> ChebyshevPolynomial
 LegendreSpectralMethod --uses--> LegendrePolynomial
 SpectralDifferentiation --uses--> BasisDerivative
 SpectralDifferentiation --may_use--> DifferentiationMatrix
 TransformMethod --maps_between--> ModalAndNodalRepresentations
 FastFourierTransform --accelerates--> FourierTransform
 NonlinearProduct --may_generate--> Aliasing
 Dealiasing --reduces--> Aliasing
 Discontinuity --causes--> GibbsPhenomenon
 Smoothness --supports--> SpectralConvergence
}

OPERATORS {
 SPECTRAL_EXPANSION(phi,a,N,x) :=
  sum_{k=0}^{N} a_k*phi_k(x)

 FOURIER_EXPANSION(a_k,x) :=
  sum_k a_k*exp(i*k*x)

 CHEBYSHEV_EXPANSION(a_k,x) :=
  sum_{k=0}^{N} a_k*T_k(x)

 MODAL_DERIVATIVE({a_k},{phi_k'}) :=
  coefficients_or_function_obtained_by_differentiating_basis expansion

 NODAL_DERIVATIVE(D,u) :=
  D*u

 FORWARD_TRANSFORM(u_nodes) :=
  spectral_coefficients

 INVERSE_TRANSFORM(a_modes) :=
  nodal_values

 COLLOCATION_RESIDUAL(L,u_N,f,x_j) :=
  L(u_N)(x_j)-f(x_j)

 GALERKIN_CONDITION(residual,phi_j) :=
  inner_product(residual,phi_j)=0

 DEALIAS_TWO_THIRDS(a_modes) :=
  zero_modes_above_two_thirds_cutoff
  under standard Fourier pseudospectral convention

 APPLY_SPECTRAL_FILTER(a_k,sigma_k) :=
  sigma_k*a_k
}

DEFINITIONS {
 D1 = "Spectral methods use globally supported basis functions."
 D2 = "Modal formulations solve for spectral coefficients."
 D3 = "Nodal or collocation formulations solve for values at special points."
 D4 = "Fourier methods are natural for periodic problems."
 D5 = "Chebyshev and Legendre methods are common on finite nonperiodic intervals."
 D6 = "Spectral differentiation uses exact derivatives of the truncated approximation."
 D7 = "Smooth solutions can exhibit very rapid convergence with spectral resolution."
 D8 = "Discontinuities degrade spectral convergence and produce Gibbs oscillations."
 D9 = "Nonlinear products can cause aliasing unless resolution or dealiasing is used."
 D10 = "Transform methods can reduce computational cost dramatically."
}

AXIOMS {
 A1_GLOBAL_BASIS:
  SpectralMethod represents the approximate solution in a global basis

 A2_TRUNCATION:
  Practical spectral approximations use finitely many modes or collocation nodes

 A3_SMOOTHNESS:
  Spectral convergence depends strongly on solution regularity

 A4_PERIODICITY:
  Fourier bases naturally encode periodic boundary conditions

 A5_ORTHOGONALITY:
  Many spectral bases are orthogonal with respect to a weighting function

 A6_ALIASING:
  Discrete nonlinear operations can transfer unresolved spectral content into resolved modes

 A7_TRANSFORM_EQUIVALENCE:
  Modal and nodal representations can be related through suitable transforms

 A8_BOUNDARY_TREATMENT:
  Nonperiodic spectral methods require boundary-compatible basis or enforcement strategy
}

REASONING {

 R1_PROBLEM_CLASSIFICATION {
  INPUT = differential_equation_or_approximation_problem
  IDENTIFY = periodicity_domain_smoothness_dimension_boundary_conditions
  RETURN = spectral_problem_class
 }

 R2_BASIS_SELECTION {
  INPUT = problem_class
  IF periodic_domain
   SELECT = FourierBasis
  ELSE IF finite_interval_and_nonperiodic
   CONSIDER = Chebyshev_or_LegendreBasis
  ELSE IF unbounded_domain
   CONSIDER = Hermite_or_LaguerreBasis_or_domain_mapping
  RETURN = spectral_basis
 }

 R3_FORMULATION_SELECTION {
  INPUT = basis_and_equation
  IF coefficient_space_natural
   SELECT = ModalOrGalerkin
  ELSE
   SELECT = CollocationOrPseudospectral
  RETURN = spectral_formulation
 }

 R4_NODE_SELECTION {
  INPUT = polynomial_basis
  SELECT = Gauss_GaussLobatto_or_other orthogonal-polynomial nodes
  RETURN = collocation_nodes
 }

 R5_SPECTRAL_DIFFERENTIATION {
  INPUT = spectral_representation
  IF modal
   DIFFERENTIATE = basis_modes
  ELSE
   APPLY = differentiation_matrix
  RETURN = derivative approximation
 }

 R6_TRANSFORM_REASONING {
  INPUT = modal_or_nodal data
  APPLY = FFT_DCT_or_general transform
  RETURN = alternate_representation
 }

 R7_ALIASING_ANALYSIS {
  INPUT = nonlinear_product_and_resolution
  IDENTIFY = generated_high_modes
  IF unresolved
   APPLY = padding_or_two_thirds dealiasing
  RETURN = alias_controlled representation
 }

 R8_BOUNDARY_ENFORCEMENT {
  INPUT = spectral_system_and_boundary_conditions
  APPLY = basis_modification_tau_collocation_penalty_or_direct replacement
  RETURN = boundary_constrained system
 }

 R9_CONVERGENCE_ANALYSIS {
  INPUT = spectral_solutions_at_increasing_N
  MONITOR = coefficient_decay_and_solution error
  IDENTIFY = exponential_or_algebraic regime
  RETURN = convergence_profile
 }

 R10_GIBBS_ANALYSIS {
  INPUT = discontinuous_or_sharp solution
  DETECT = persistent_oscillatory overshoot
  CONSIDER = filtering_domain_decomposition_or_special basis
  RETURN = Gibbs_mitigation_strategy
 }
}

CONSTRAINTS {
 C1 = "Global basis choice must match domain and boundary structure."
 C2 = "Fourier periodicity assumptions must be respected."
 C3 = "Exponential convergence should not be claimed for nonsmooth solutions without qualification."
 C4 = "Differentiation matrices can become ill-conditioned at high order."
 C5 = "Aliasing must be considered in nonlinear pseudospectral calculations."
 C6 = "Spectral filtering alters high-frequency content and must be disclosed."
 C7 = "Boundary enforcement can affect stability and conditioning."
 C8 = "Mapped domains require correct Jacobian and derivative transformations."
 C9 = "Tensor-product spectral grids suffer dimensional growth."
 C10 = "Coefficient decay is a diagnostic, not an absolute proof of solution accuracy."
}

SEMANTIC_EQUIVALENCE {
 "SpectralMethod" <=> "GlobalBasisNumericalMethod"
 "PseudospectralMethod" <=> "NodalTransformSpectralMethod"
 "SpectralDifferentiation" <=> "GlobalBasisDerivativeApproximation"
 "SpectralCoefficient" <=> "GlobalBasisExpansionCoefficient"
 "Aliasing" <=> "UnresolvedModeFrequencyFolding"
 "SpectralConvergence" <=> "RapidGlobalBasisConvergence"
}

VALIDATION {
 V1 = "Basis matches domain and boundary conditions."
 V2 = "Collocation nodes are appropriate for basis."
 V3 = "Transforms use consistent normalisation."
 V4 = "Boundary conditions are enforced correctly."
 V5 = "Aliasing is controlled in nonlinear problems."
 V6 = "Coefficient decay is monitored."
 V7 = "Convergence is checked with increasing spectral resolution."
 V8 = "Mapped-domain derivatives use correct coordinate transformation."
 V9 = "Filtering effects are documented."
 V10 = "Approximate-versus-exact status is preserved."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SPECTRAL_METHOD_DEFINITION = 0.99
 BASIS_SELECTION_REASONING = 0.99
 FORMULATION_REASONING = 0.98
 SPECTRAL_DIFFERENTIATION_REASONING = 0.99
 TRANSFORM_REASONING = 0.99
 ALIASING_REASONING = 0.98
 BOUNDARY_REASONING = 0.97
 CONVERGENCE_REASONING = 0.98

 POLICY =
  "Reduce confidence when smoothness, periodicity, basis choice, collocation nodes, boundary enforcement, aliasing treatment, mapping or spectral-resolution evidence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SpectralMethodReasoning,
  SpectralApproximationReasoning,
  BasisSelection,
  FourierSpectralReasoning,
  ChebyshevSpectralReasoning,
  LegendreSpectralReasoning,
  SpectralGalerkinReasoning,
  PseudospectralReasoning,
  SpectralDifferentiation,
  TransformReasoning,
  AliasingDetection,
  DealiasingReasoning,
  SpectralConvergenceAnalysis,
  GibbsPhenomenonReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "spectral method" -> SpectralMethod
  "spectral methods" -> SpectralMethod
  "pseudospectral method" -> PseudospectralMethod
  "spectral collocation" -> SpectralCollocationMethod
  "spectral Galerkin" -> SpectralGalerkinMethod
  "Fourier spectral method" -> FourierSpectralMethod
  "Chebyshev spectral method" -> ChebyshevSpectralMethod
  "spectral differentiation" -> SpectralDifferentiation
  "aliasing" -> Aliasing
  "dealiasing" -> Dealiasing
 }

 INTERPRETATION_ORDER = [
  detect_spectral_method_context,
  identify domain_periodicity_smoothness_dimension_boundary_conditions_and_equation structure,
  select Fourier_Chebyshev_Legendre_or_other global basis,
  select modal_Galerkin_collocation_or_pseudospectral formulation,
  construct spectral coefficients_or_nodal representation,
  differentiate_or_apply operators spectrally,
  control aliasing_and_boundary conditions,
  solve spectral algebraic system,
  assess coefficient_decay_convergence_conditioning_and_Gibbs effects,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("spectral method")
  OR global_Fourier_Chebyshev_Legendre_discretisation_context
  THEN activate(SpectralMethodOntology)

 E2:
  IF periodic_problem
  THEN consider Fourier_basis

 E3:
  IF finite_nonperiodic_interval
  THEN consider Chebyshev_or_Legendre basis

 E4:
  IF collocation_formulation
  THEN select basis-compatible nodes

 E5:
  IF nonlinear_pseudospectral calculation
  THEN assess aliasing_and_apply dealiasing if needed

 E6:
  IF discontinuous_solution
  THEN expect degraded convergence_and_Gibbs phenomenon

 E7:
  IF spectral_coefficients_decay_slowly
  THEN increase_resolution_or_reconsider basis/domain decomposition

 E8:
  IF mapped_domain
  THEN transform derivatives_and_quadrature with correct Jacobian

 E9:
  IF convergence_claimed
  THEN verify solution_or_coefficient behaviour across increasing spectral order
}

RECURSION {
 ENTRY = SpectralMethod

 EXPAND(SpectralMethod) -> [
  SpectralApproximation,
  GlobalBasisFunction,
  ModalMethod,
  NodalMethod,
  SpectralGalerkinMethod,
  SpectralCollocationMethod,
  PseudospectralMethod,
  FourierSpectralMethod,
  ChebyshevSpectralMethod,
  LegendreSpectralMethod,
  OrthogonalPolynomial,
  SpectralCoefficient,
  CollocationPoint,
  SpectralDifferentiation,
  DifferentiationMatrix,
  TransformMethod,
  FastFourierTransform,
  Aliasing,
  Dealiasing,
  SpectralConvergence,
  GibbsPhenomenon,
  BoundaryCondition,
  TensorProductSpectralMethod,
  DomainMapping,
  SpectralFilter
 ]

 EXPAND(BasisFamilies) -> [
  FourierSeries,
  ChebyshevPolynomial,
  LegendrePolynomial,
  HermitePolynomial,
  LaguerrePolynomial
 ]

 EXPAND(CollocationStructure) -> [
  GaussPoint,
  GaussLobattoPoint,
  NodalValue,
  DifferentiationMatrix,
  InterpolationMatrix
 ]

 EXPAND(NonlinearSpectralStructure) -> [
  PhysicalSpaceProduct,
  ModalConvolution,
  Aliasing,
  ZeroPadding,
  TwoThirdsRule,
  SpectralFilter
 ]

 STOP_CONDITION =
  "Stop when the basis, spectral representation, collocation or modal formulation, differentiation, aliasing, boundary treatment, solution system and convergence behaviour are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Numerical Analysis"

 IMMEDIATE_CONTEXTS = [
  "Numerical Analysis Ontology",
  "Numerical Methods Ontology",
  "Numerical Differentiation Ontology",
  "Approximation Theory Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Numerical Analysis Ontology",
  "Numerical Methods Ontology",
  "Numerical Differentiation Ontology",
  "Numerical Integration Ontology",
  "Numerical Analysis",
  "Applied Mathematics Ontologies",
  "Approximation Theory Ontologies",
  "Fourier Analysis Ontologies",
  "Orthogonal Polynomial Ontologies",
  "Differential Equations Ontologies",
  "Partial Differential Equations Ontologies",
  "Linear Algebra Ontologies",
  "Scientific Computing Ontologies",
  "Computational Model Ontology",
  "Model Verification Ontology"
 ]

 ROUTE_RULES = [
  "Spectral Method -> Numerical Analysis Ontology for convergence and conditioning semantics",
  "Spectral Method -> Numerical Methods Ontology for algorithm family",
  "Spectral Method -> Numerical Differentiation for derivative approximation semantics",
  "Spectral Method -> Approximation Theory for global basis expansions",
  "Spectral Method -> Fourier Analysis for periodic modal structure",
  "Spectral Method -> Orthogonal Polynomials for Chebyshev and Legendre bases",
  "Spectral Method -> Differential Equations for governing-equation structure",
  "Spectral Method -> Linear Algebra for differentiation and operator matrices",
  "Spectral Method -> Scientific Computing for FFT-based large-scale execution",
  "Spectral Method -> Model Verification for resolution-convergence assessment"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible spectral bases, formulations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve global-basis approximation semantics.",
  "Preserve modal-versus-nodal distinction.",
  "Preserve spectral-differentiation semantics.",
  "Preserve smoothness-dependent convergence semantics.",
  "Preserve aliasing and dealiasing semantics.",
  "Preserve boundary-treatment semantics.",
  "Preserve Gibbs-phenomenon awareness.",
  "Preserve machine-readable spectral-method semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "A spectral method approximates functions or differential-equation solutions using global basis expansions such as Fourier or orthogonal polynomial series."

 BASIS_SUMMARY =
  "Fourier bases are natural for periodic domains, while Chebyshev and Legendre bases are common on finite nonperiodic intervals."

 MODAL_SUMMARY =
  "Modal spectral methods solve for expansion coefficients."

 NODAL_SUMMARY =
  "Collocation and pseudospectral methods solve for values at selected spectral nodes."

 DIFFERENTIATION_SUMMARY =
  "Spectral differentiation is obtained by differentiating the global basis analytically or using differentiation matrices."

 CONVERGENCE_SUMMARY =
  "Smooth problems can exhibit very rapid, often exponential-like, convergence with increasing spectral resolution."

 ALIASING_SUMMARY =
  "Nonlinear products can create unresolved modes that alias into the resolved spectrum unless controlled."

 GIBBS_SUMMARY =
  "Discontinuities produce Gibbs oscillations and reduce spectral convergence."

 COMPUTATION_SUMMARY =
  "FFT and related transforms allow efficient movement between physical-space and spectral representations."
}

FLOW {
 INPUT
 -> detect_spectral_method_context
 -> identify governing_problem_domain_periodicity_smoothness_dimension_boundary_conditions_and_accuracy requirements
 -> select Fourier_Chebyshev_Legendre_Hermite_Laguerre_or_other basis
 -> choose modal_Galerkin_collocation_or_pseudospectral formulation
 -> construct spectral_or_nodal representation
 -> apply spectral differentiation_and_operator evaluation
 -> enforce boundary_conditions
 -> if_nonlinear control aliasing_via padding_dealiasing_or_filtering
 -> solve resulting algebraic system
 -> assess spectral_coefficients_resolution_convergence_conditioning_and_Gibbs effects
 -> route to numerical_analysis_ontology numerical_methods numerical_differentiation approximation_theory Fourier_analysis orthogonal_polynomials differential_equations linear_algebra scientific_computing computational_model_or_model_verification ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Spectral Method Ontology defines global-basis numerical approximation. It supports Fourier, Chebyshev, Legendre and related bases, modal, Galerkin, collocation and pseudospectral formulations, spectral differentiation, transform methods, FFT acceleration, aliasing and dealiasing, spectral filters, boundary treatment, tensor-product discretisation, Gibbs phenomena and rapid convergence for smooth problems."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-SPECTRAL-METHOD-ONTOLOGY-V1"
}

}

END_FLOW