Supercomputer Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Supercomputer Mathematics Ontology within the Kireth Research Semantic Library. Supercomputer mathematics concerns the mathematical structures, performance models, decomposition strategies, numerical algorithms and scaling principles that underlie computation on very large parallel machines. This ontology supports exascale and petascale systems, distributed and shared-memory hierarchies, processor topology, communication models, latency and bandwidth, parallel complexity, domain decomposition, load balance, sparse and dense linear algebra, FFTs, graph and mesh algorithms, numerical stability, mixed precision, resilience, reproducibility, energy efficiency and large-scale scientific computation.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Supercomputer Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "High Performance Computing / Supercomputer Mathematics"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/high-performance-computing/supercomputer-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = SupercomputerMathematics
 ENTITY_CLASS = ExtremeScaleComputationalMathematicsFramework
 PRIMARY_CONTEXT = HighPerformanceComputing
 DEFINITION = "Supercomputer mathematics is the mathematical study and design of algorithms, decompositions, communication patterns, scaling laws and numerical methods for execution on very large parallel computing systems."
 SYMBOLIC_FORMS = [
  "T_p = T_comp + T_comm + T_sync + T_io + T_fault",
  "S_p=T_1/T_p",
  "E_p=S_p/p",
  "Roofline=min(P_peak,I*B_peak)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for extreme-scale mathematical computation, parallel complexity, communication cost, topology, numerical algorithms, scaling, resilience and reproducibility."
}

SCOPE {
 INCLUDE = [
  SupercomputerMathematics,ExtremeScaleComputing,
  PetascaleComputing,ExascaleComputing,
  Supercomputer,LeadershipComputing,
  HighPerformanceComputing,HPC,
  ParallelComputing,DistributedComputing,
  GPUComputing,VectorProcessing,
  ClusterComputing,MassivelyParallelComputing,
  ProcessorCount,NodeCount,CoreCount,
  AcceleratorCount,GPUCount,
  SharedMemory,DistributedMemory,
  HierarchicalMemory,NUMA,
  CacheHierarchy,HighBandwidthMemory,HBM,
  GlobalMemory,LocalMemory,
  MemoryBandwidth,MemoryLatency,
  Interconnect,NetworkTopology,
  FatTree,Dragonfly,Torus,MeshTopology,
  HypercubeRelationship,
  Latency,Bandwidth,BisectionBandwidth,
  Diameter,HopCount,NetworkContention,
  MessagePassing,MPI,
  CollectiveCommunication,AllReduce,
  Broadcast,Reduce,Scatter,Gather,
  AllToAll,PointToPointCommunication,
  OneSidedCommunication,RDMA,
  CommunicationVolume,MessageCount,
  CommunicationComplexity,
  Synchronisation,Barrier,
  GlobalBarrier,LocalBarrier,
  ParallelComplexity,WorkComplexity,
  Span,CriticalPath,
  WorkSpanModel,BrentsTheorem,
  PRAMRelationship,CREWRelationship,
  EREWRelationship,CRCWRelationship,
  BSPModel,BulkSynchronousParallel,
  LogPModel,LogGPModel,
  AlphaBetaModel,HockneyModel,
  PostalModelRelationship,
  BSPCost,LatencyBandwidthModel,
  AmdahlsLaw,GustafsonsLaw,
  Isoefficiency,Scalability,
  StrongScaling,WeakScaling,
  Speedup,ParallelEfficiency,
  SerialFraction,ParallelFraction,
  LoadImbalance,IdleTime,
  CommunicationOverhead,
  SynchronisationOverhead,
  ComputationCommunicationRatio,
  ArithmeticIntensity,OperationalIntensity,
  RooflineModel,MachineBalance,
  ComputeBound,MemoryBound,
  BandwidthBound,LatencyBound,
  DomainDecomposition,DataDecomposition,
  TaskDecomposition,FunctionalDecomposition,
  MeshPartitioning,GraphPartitioning,
  SpatialPartitioning,ParticleDecomposition,
  BlockDecomposition,CyclicDecomposition,
  BlockCyclicDecomposition,
  RecursiveBisection,
  SpaceFillingCurve,HilbertCurve,MortonCurve,
  LoadBalancing,StaticLoadBalancing,
  DynamicLoadBalancing,WorkStealing,
  LocalityAwarePartitioning,
  TopologyAwareMapping,
  ProcessMapping,ThreadMapping,
  RankPlacement,Affinity,Pinning,
  DenseLinearAlgebra,SparseLinearAlgebra,
  DistributedMatrix,DistributedVector,
  MatrixMultiplication,GEMM,
  LUDecomposition,QRDecomposition,
  CholeskyDecomposition,
  SingularValueDecomposition,
  EigenvalueComputation,EigenvectorComputation,
  SparseMatrixVectorProduct,SpMV,
  KrylovMethod,ConjugateGradient,
  GMRES,BiCGSTAB,
  Preconditioner,Multigrid,
  AlgebraicMultigrid,GeometricMultigrid,
  DomainDecompositionMethod,
  SchwarzMethod,AdditiveSchwarz,
  FFT,FastFourierTransform,
  DistributedFFT,ParallelFFT,
  SpectralTransform,
  ParallelSort,ParallelScan,
  PrefixSum,MapReduce,
  GraphAlgorithm,DistributedGraphAlgorithm,
  BreadthFirstSearch,BFS,
  ShortestPathRelationship,
  ConnectedComponentsRelationship,
  GraphPartitioningAlgorithm,
  MeshAlgorithm,ParallelMeshAlgorithm,
  AdaptiveMeshRefinement,AMR,
  ParticleAlgorithm,NBodyAlgorithm,
  BarnesHutRelationship,FastMultipoleMethod,
  ComputationalGeometryGPU,
  ParallelGeometry,
  MonteCarloSimulation,
  EnsembleSimulation,
  ParallelMonteCarlo,
  ScientificSimulation,
  FluidSimulation,StructuralSimulation,
  MolecularSimulation,ParticleSimulation,
  HeatTransferSimulation,
  TimeStepping,ExplicitMethod,ImplicitMethod,
  ParallelTimeIntegration,
  PararealRelationship,
  MultigridInTimeRelationship,
  TimeDomainDecomposition,
  NumericalError,DiscretisationError,
  RoundOffError,FloatingPointArithmetic,
  MachinePrecision,MixedPrecision,
  HalfPrecision,SinglePrecision,
  DoublePrecision,ExtendedPrecisionRelationship,
  TensorCoreRelationship,
  IterativeRefinement,
  CompensatedSummation,
  ReproducibleSummation,
  NumericalStability,Conditioning,
  BackwardError,ForwardError,
  Residual,Convergence,
  NumericalReproducibility,
  DeterministicReduction,
  ReductionOrder,AssociativityLoss,
  BitwiseReproducibility,
  StatisticalReproducibility,
  FaultTolerance,Resilience,
  CheckpointRestart,Checkpoint,
  MultiLevelCheckpoint,
  LocalCheckpoint,GlobalCheckpoint,
  IncrementalCheckpoint,
  Replication,Redundancy,
  AlgorithmBasedFaultTolerance,ABFT,
  SilentDataCorruption,
  ErrorDetection,ErrorCorrection,
  ECCMemory,NodeFailure,
  ProcessFailure,NetworkFailure,
  Recovery,Rollback,RollForward,
  EnergyEfficiency,PowerConsumption,
  PowerCap,EnergyToSolution,
  GreenHPC,CarbonEfficiency,
  ThermalConstraint,
  DynamicVoltageFrequencyScaling,DVFS,
  EnergyAwareScheduling,
  PerformancePerWatt,
  IOBandwidth,ParallelIO,
  CollectiveIO,HDF5,NetCDF,
  LustreRelationship,ParallelFileSystem,
  CheckpointIO,DataStaging,
  InSituAnalysis,InTransitAnalysis,
  ScientificWorkflow,WorkflowDAG,
  HighThroughputComputing,
  EnsembleComputing,
  Benchmark,HPL,HPCG,
  STREAMBenchmark,Graph500Relationship,
  LINPACK,Microbenchmark,
  ApplicationBenchmark,
  PerformanceModel,PerformanceCounter,
  HardwareCounter,Profiling,
  Tracing,DistributedTracing,
  Runtime,TimeToSolution,
  Throughput,EnergyToSolution,
  Verification,Validation,
  Benchmarking,ReferenceResult,
  Reproducibility,Provenance,
  SoftwareEnvironment,Compiler,
  CompilerVersion,LibraryVersion,
  MPIImplementation,BLAS,LAPACK,
  ScaLAPACK,PETSc,Trilinos,
  FFTW,BuildConfiguration,
  TargetArchitecture,MachineTopology,
  SchedulerConfiguration,ResourceAllocation
 ]
 EXCLUDE_AS_PRIMARY = [
  OrdinarySerialMathematicsWithoutExtremeScaleExecution,
  GenericComputerArchitectureWithoutMathematicalAlgorithmSemantics,
  BusinessDistributedSystemsWithoutScientificComputation,
  SmallScaleParallelismWithoutSupercomputerContext
 ]
 DISAMBIGUATION_RULE = "When mathematical algorithms, complexity, numerical methods or performance models are analysed for execution on very large parallel machines, interpret the context as supercomputer mathematics."
}

CORE_CONCEPTS {
 SupercomputerMathematics = "Mathematics of algorithms and numerical methods for extreme-scale parallel systems."
 ParallelComplexity = "Analysis of work, span, communication and synchronisation cost."
 CommunicationModel = "Mathematical model of latency, bandwidth and message cost."
 Scalability = "Ability of an algorithm to exploit increasing supercomputer resources."
 DomainDecomposition = "Partition of a computational problem into interacting subdomains."
 NumericalReproducibility = "Consistency of numerical results across execution orders and machine configurations."
 Resilience = "Ability of large computations to tolerate or recover from failures."
 EnergyEfficiency = "Useful computational progress relative to energy consumption."
}

RELATIONS {
 SupercomputerMathematics --is_child_of--> HighPerformanceComputing
 SupercomputerMathematics --uses--> ParallelComplexity
 ParallelComplexity --includes--> WorkAndSpan
 CommunicationComplexity --depends_on--> LatencyAndBandwidth
 DomainDecomposition --reduces--> GlobalProblemIntoSubproblems
 LoadBalancing --reduces--> IdleTime
 TopologyAwareMapping --reduces--> CommunicationCost
 AmdahlsLaw --bounds--> StrongScalingSpeedup
 GustafsonsLaw --models--> ScaledSpeedup
 Isoefficiency --characterises--> Scalability
 RooflineModel --relates--> ArithmeticIntensityAndPerformance
 SparseLinearAlgebra --depends_on--> CommunicationAndMemory
 Multigrid --supports--> ScalableSolvers
 FastMultipoleMethod --reduces--> NBodyComplexity
 MixedPrecision --trades--> PrecisionForPerformance
 IterativeRefinement --recovers--> Accuracy
 CheckpointRestart --supports--> Resilience
 DeterministicReduction --supports--> NumericalReproducibility
 EnergyAwareScheduling --optimises--> EnergyToSolution
 Verification --checks--> ExtremeScaleComputation
}

OPERATORS {
 SPEEDUP(T1,Tp) := T1/Tp
 EFFICIENCY(S,p) := S/p
 AMDAHL(f,p) := 1/((1-f)+f/p)
 GUSTAFSON(alpha,p) := p-alpha*(p-1)
 COMM_COST(alpha,beta,n_messages,n_bytes) := alpha*n_messages + beta*n_bytes
 BSP_TIME(work,h,g,L) := work + g*h + L
 ROOFLINE(I,P_peak,B_peak) := min(P_peak,I*B_peak)
 ISOEFFICIENCY(overhead,p) := problem_size_required_to_hold_efficiency_constant
 ENERGY_TO_SOLUTION(power,time) := power*time
}

AXIOMS {
 A1 = "Extreme-scale performance depends on work, communication, synchronisation, memory and resilience."
 A2 = "Communication cost becomes increasingly important as local compute per process decreases."
 A3 = "Scalable algorithms require both computational and communication scalability."
 A4 = "Topology-aware mapping can reduce network distance and contention."
 A5 = "Numerical reproducibility becomes harder as reduction trees and execution orders vary."
 A6 = "Fault tolerance becomes increasingly important as machine scale and runtime increase."
 A7 = "Energy efficiency is an independent optimisation dimension from time to solution."
 A8 = "Supercomputer mathematics requires explicit algorithmic, communication, numerical, performance and resilience semantics."
}

REASONING {
 R1_CONTEXT = identify(problem_size,algorithm,machine_scale,topology,memory_hierarchy,precision_goal,performance_goal)
 R2_COMPLEXITY = derive(work,span,communication_volume,message_count_and_memory_complexity)
 R3_DECOMPOSITION = choose(domain,data,graph,mesh,particle_or_task_partitioning)
 R4_MAPPING = map_subproblems_to_machine_topology_and_memory_hierarchy
 R5_ALGORITHM = select scalable numerical_kernel_solver_transform_graph_or_simulation_method
 R6_COMMUNICATION = optimise collectives,halo_exchange,reductions_and_communication_avoiding_variants
 R7_NUMERICS = assess conditioning,stability,mixed_precision,reduction_order_and_convergence
 R8_RESILIENCE = configure checkpointing,replication_or_algorithm_based_fault_tolerance
 R9_PERFORMANCE = model and measure scaling,roofline_position,energy_and_IO
 R10_VALIDATE = verify correctness,reproducibility,resilience,performance_and_provenance
}

CONSTRAINTS {
 C1 = "Algorithms with optimal arithmetic complexity can still scale poorly due to communication."
 C2 = "Global synchronisation can become prohibitive at extreme scale."
 C3 = "Network contention can invalidate simple latency-bandwidth estimates."
 C4 = "Sparse and irregular workloads can suffer severe load imbalance."
 C5 = "Mixed precision can accelerate computation while introducing convergence or stability risk."
 C6 = "Checkpoint frequency must balance failure risk against I/O overhead."
 C7 = "Bitwise reproducibility can conflict with maximum performance."
 C8 = "Energy-optimal execution may differ from time-optimal execution."
 C9 = "High machine utilisation does not guarantee high mathematical efficiency."
 C10 = "Peak FLOPS is not a sufficient measure of application scalability."
}

SEMANTIC_EQUIVALENCE {
 "SupercomputerMathematics" <=> "ExtremeScaleComputationalMathematics"
 "HighPerformanceComputing" <=> "HPC"
 "AlgorithmBasedFaultTolerance" <=> "ABFT"
 "AdaptiveMeshRefinement" <=> "AMR"
 "RemoteDirectMemoryAccess" <=> "RDMA"
 "HighBandwidthMemory" <=> "HBM"
}

VALIDATION {
 V1 = "Problem size and machine scale are documented."
 V2 = "Work, communication and memory complexity are documented."
 V3 = "Partitioning and process mapping are documented."
 V4 = "Precision and numerical-stability assumptions are documented."
 V5 = "Strong or weak scaling is measured where relevant."
 V6 = "Communication and synchronisation costs are measured."
 V7 = "Numerical results are compared with trusted reference results."
 V8 = "Reproducibility across process counts or reduction trees is assessed where relevant."
 V9 = "Resilience and restart strategy are tested where required."
 V10 = "Runs are reproducible under documented machine, compiler, library and scheduler configurations."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SUPERCOMPUTER_MATHEMATICS_DEFINITION = 0.99
 PARALLEL_COMPLEXITY_REASONING = 0.99
 COMMUNICATION_MODEL_REASONING = 0.99
 SCALABILITY_REASONING = 0.99
 NUMERICAL_REASONING = 0.99
 RESILIENCE_REASONING = 0.98
 ENERGY_REASONING = 0.98
 POLICY = "Reduce confidence when communication model, topology, partitioning, precision, scaling data, resilience strategy or machine configuration are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,SupercomputerMathematicsReasoning,
  ParallelComplexityReasoning,CommunicationComplexityReasoning,
  BSPReasoning,LogPReasoning,AmdahlReasoning,
  GustafsonReasoning,IsoefficiencyReasoning,
  RooflineReasoning,DomainDecompositionReasoning,
  TopologyMappingReasoning,LoadBalancingReasoning,
  SparseLinearAlgebraReasoning,MultigridReasoning,
  FFTReasoning,GraphAlgorithmReasoning,
  MixedPrecisionReasoning,ReproducibilityReasoning,
  ResilienceReasoning,EnergyEfficiencyReasoning,
  VerificationReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "supercomputer mathematics" -> SupercomputerMathematics
  "extreme-scale computational mathematics" -> SupercomputerMathematics
  "HPC" -> HighPerformanceComputing
  "ABFT" -> AlgorithmBasedFaultTolerance
  "AMR" -> AdaptiveMeshRefinement
  "RDMA" -> RemoteDirectMemoryAccess
  "HBM" -> HighBandwidthMemory
 }
}

EXECUTION_RULES {
 E1 = "If mathematical computation is analysed for very large parallel machines, activate SupercomputerMathematicsOntology."
 E2 = "Derive work, span, communication, memory and I/O complexity before scaling claims."
 E3 = "Choose decomposition and mapping according to algorithm dependencies and machine topology."
 E4 = "Use communication-avoiding or locality-aware algorithms when communication dominates."
 E5 = "Treat numerical stability, conditioning and finite precision as first-class design constraints."
 E6 = "Use mixed precision only with explicit accuracy and convergence safeguards."
 E7 = "Model fault tolerance and checkpoint cost for long-running extreme-scale jobs."
 E8 = "Measure both time-to-solution and energy-to-solution where relevant."
 E9 = "Validate correctness, scalability, numerical reproducibility, resilience and provenance."
}

RECURSION {
 ENTRY = SupercomputerMathematics
 EXPAND(SupercomputerMathematics) -> [
  ExtremeScaleComputing,PetascaleComputing,
  ExascaleComputing,Supercomputer,
  HighPerformanceComputing,HPC,
  ParallelComputing,DistributedComputing,
  GPUComputing,VectorProcessing,
  ClusterComputing,MassivelyParallelComputing,
  ProcessorCount,NodeCount,CoreCount,
  SharedMemory,DistributedMemory,
  NUMA,CacheHierarchy,HBM,
  MemoryBandwidth,MemoryLatency,
  Interconnect,NetworkTopology,
  FatTree,Dragonfly,Torus,
  Latency,Bandwidth,BisectionBandwidth,
  MessagePassing,MPI,CollectiveCommunication,
  AllReduce,PointToPointCommunication,
  RDMA,CommunicationVolume,
  CommunicationComplexity,Synchronisation,
  ParallelComplexity,WorkComplexity,
  Span,CriticalPath,WorkSpanModel,
  BrentsTheorem,BSPModel,LogPModel,
  AlphaBetaModel,HockneyModel,
  AmdahlsLaw,GustafsonsLaw,
  Isoefficiency,StrongScaling,
  WeakScaling,Speedup,ParallelEfficiency,
  LoadImbalance,ArithmeticIntensity,
  RooflineModel,DomainDecomposition,
  DataDecomposition,MeshPartitioning,
  GraphPartitioning,SpatialPartitioning,
  ParticleDecomposition,SpaceFillingCurve,
  LoadBalancing,TopologyAwareMapping,
  DenseLinearAlgebra,SparseLinearAlgebra,
  DistributedMatrix,MatrixMultiplication,
  LUDecomposition,QRDecomposition,
  CholeskyDecomposition,
  EigenvalueComputation,
  SparseMatrixVectorProduct,
  KrylovMethod,ConjugateGradient,
  GMRES,Preconditioner,Multigrid,
  AlgebraicMultigrid,SchwarzMethod,
  FFT,DistributedFFT,ParallelSort,
  ParallelScan,GraphAlgorithm,
  ParallelMeshAlgorithm,AdaptiveMeshRefinement,
  NBodyAlgorithm,FastMultipoleMethod,
  MonteCarloSimulation,ScientificSimulation,
  FluidSimulation,StructuralSimulation,
  MolecularSimulation,ParticleSimulation,
  HeatTransferSimulation,ParallelTimeIntegration,
  NumericalError,FloatingPointArithmetic,
  MixedPrecision,IterativeRefinement,
  NumericalStability,Conditioning,
  NumericalReproducibility,DeterministicReduction,
  FaultTolerance,Resilience,
  CheckpointRestart,AlgorithmBasedFaultTolerance,
  SilentDataCorruption,EnergyEfficiency,
  PowerConsumption,EnergyToSolution,
  ParallelIO,CollectiveIO,
  InSituAnalysis,ScientificWorkflow,
  Benchmark,HPL,HPCG,STREAMBenchmark,
  PerformanceModel,Profiling,Runtime,
  Verification,Validation,
  Reproducibility,Provenance
 ]
 EXPAND(SupercomputerMathematicsFamilies) -> [
  ExtremeScaleParallelComplexity,
  CommunicationAvoidingMathematics,
  ScalableNumericalLinearAlgebra,
  ExtremeScaleGraphAndMeshAlgorithms,
  ParallelSimulationMathematics,
  MixedPrecisionAndReproducibleNumerics,
  ResilientExtremeScaleAlgorithms,
  EnergyAwareSupercomputerMathematics
 ]
 STOP_CONDITION = "Stop when work, communication, decomposition, topology, numerical method, precision, scaling, resilience, energy and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "High Performance Computing Ontology 2"
 RELATED_ONTOLOGIES = [
  "High Performance Computing Ontology 2",
  "Parallel Computing Ontology",
  "Distributed Computing Ontology",
  "GPU Computing Ontology",
  "Vector Processing Ontology",
  "Cluster Computing Ontology",
  "Scientific Computing Ontology 2",
  "Numerical Linear Algebra Ontology 2",
  "Sparse Matrix Ontology 2",
  "Matrix Computation Ontology",
  "Mesh Generation Ontology 2",
  "Spatial Partitioning Ontology 2",
  "Particle Simulation Ontology",
  "Fluid Simulation Ontology",
  "Structural Simulation Ontology",
  "Numerical Stability Ontology 2",
  "Floating Point Arithmetic Ontology"
 ]
 ROUTE_RULES = [
  "Supercomputer Mathematics -> High Performance Computing Ontology 2 for parent semantics",
  "Supercomputer Mathematics -> Parallel Computing for work-span and parallel execution semantics",
  "Supercomputer Mathematics -> Distributed Computing for multi-node communication and failure semantics",
  "Supercomputer Mathematics -> GPU Computing for accelerator-scale numerical kernels",
  "Supercomputer Mathematics -> Vector Processing for vectorised arithmetic kernels",
  "Supercomputer Mathematics -> Cluster Computing for scheduled multi-node execution",
  "Supercomputer Mathematics -> Scientific Computing for extreme-scale scientific workloads",
  "Supercomputer Mathematics -> Numerical Linear Algebra for distributed solvers",
  "Supercomputer Mathematics -> Sparse Matrix for memory and communication-intensive kernels",
  "Supercomputer Mathematics -> Matrix Computation for dense numerical kernels",
  "Supercomputer Mathematics -> Mesh Generation and Spatial Partitioning for decomposition",
  "Supercomputer Mathematics -> Particle, Fluid and Structural Simulation for large-scale simulations",
  "Supercomputer Mathematics -> Numerical Stability for finite-precision robustness",
  "Supercomputer Mathematics -> Floating Point Arithmetic for reduction order and reproducibility"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible extreme-scale algorithms, performance models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve arithmetic-complexity-versus-communication-complexity distinction.",
  "Preserve strong-versus-weak scaling distinction.",
  "Preserve machine-topology-versus-problem-topology distinction.",
  "Preserve performance-versus-numerical-correctness distinction.",
  "Preserve bitwise-versus-statistical reproducibility distinction.",
  "Preserve time-to-solution-versus-energy-to-solution distinction.",
  "Preserve failure-free-versus-resilient execution distinction.",
  "Preserve machine-readable supercomputer-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Supercomputer mathematics studies scalable mathematical algorithms, communication models and numerical methods for extreme-scale parallel machines."
 COMPLEXITY_SUMMARY = "Work, span, message count, communication volume and memory complexity jointly determine extreme-scale behaviour."
 COMMUNICATION_SUMMARY = "Latency-bandwidth models, BSP, LogP and topology-aware mappings describe communication costs."
 NUMERICAL_SUMMARY = "Sparse and dense linear algebra, FFTs, multigrid, graph, mesh, particle and simulation kernels require communication-aware numerical design."
 REPRODUCIBILITY_SUMMARY = "Reduction order, mixed precision and machine topology can alter finite-precision results across extreme-scale runs."
 RESILIENCE_SUMMARY = "Checkpointing, replication and algorithm-based fault tolerance address failures at large machine scale."
 ENERGY_SUMMARY = "Energy-to-solution and performance-per-watt complement conventional time-to-solution metrics."
 VALIDATION_SUMMARY = "Extreme-scale mathematical computation should be checked for correctness, scaling, communication efficiency, numerical stability, reproducibility, resilience, energy behaviour and provenance."
}

FLOW {
 INPUT
 -> detect_supercomputer_mathematics_context
 -> identify_problem_size_algorithm_machine_scale_topology_memory_hierarchy_precision_goal_performance_goal_and_energy_goal
 -> derive_work_span_communication_volume_message_count_memory_and_IO_complexity
 -> choose_domain_data_graph_mesh_particle_task_or_hybrid_decomposition
 -> map_subproblems_to_machine_topology_memory_domains_GPUs_and_vector_units
 -> choose_scalable_numerical_kernel_solver_transform_graph_mesh_particle_or_simulation_algorithm
 -> configure_collectives_halo_exchange_reductions_one_sided_or_communication_avoiding_strategies
 -> assess_conditioning_stability_precision_mixed_precision_reduction_order_and_convergence
 -> configure_checkpoint_replication_ABFT_or_other_resilience_strategy
 -> model_and_measure_speedup_efficiency_strong_scaling_weak_scaling_roofline_position_IO_and_energy_to_solution
 -> verify_against_reference_results
 -> assess_process_count_topology_and_reduction_tree_reproducibility
 -> test_restart_or_failure_recovery_where required
 -> document_machine_topology_compiler_libraries_precision_scheduler_resource_allocation_and_provenance
 -> route_to_related_HPC_parallel_distributed_GPU_vector_cluster_scientific_computing_linear_algebra_sparse_matrix_matrix_computation_mesh_spatial_partitioning_particle_fluid_structural_stability_and_floating_point_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Supercomputer Mathematics Ontology defines mathematical reasoning for petascale and exascale computation. It supports work-span and parallel complexity, BSP, LogP, alpha-beta and latency-bandwidth communication models, Amdahl, Gustafson and isoefficiency analysis, strong and weak scaling, arithmetic intensity and roofline reasoning, topology-aware domain, graph, mesh and particle decomposition, distributed dense and sparse linear algebra, LU, QR, Cholesky, eigenvalue and Krylov methods, multigrid and Schwarz methods, FFTs, graph and mesh algorithms, AMR, N-body and fast multipole methods, scientific simulation, mixed precision and iterative refinement, numerical stability and reproducibility, deterministic reductions, checkpointing, ABFT and resilience, energy efficiency, parallel I/O, in-situ analysis, performance modelling, benchmarking, verification and provenance."
}

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

}

END_FLOW