Claude-Code-Workflow/.codex/skill_lib/numerical-analysis-workflow/specs/analysis-dimensions.md

Analysis Dimensions for Numerical Computation

Defines the 18 analysis dimensions across 6 phases of the NADW workflow.

Purpose

Phase	Usage
Phase 1 (Decomposition)	Reference when assigning analysis_dimension to tasks
Phase 2 (Execution)	Agents use to understand their analysis focus
Phase 3 (Aggregation)	Organize findings by dimension

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1. Wave 1: Global Survey Dimensions

1.1 Domain Modeling (domain_modeling)

Analyst Role: Problem_Domain_Analyst

Focus Areas:

• Governing equations (PDEs, ODEs, integral equations)
• Physical domain and boundary conditions
• Conservation laws and constitutive relations
• Problem classification (elliptic, parabolic, hyperbolic)
• Dimensional analysis and non-dimensionalization

Key Outputs:

• Equation inventory with LaTeX notation
• Boundary condition catalog
• Problem classification matrix
• Physical parameter ranges

Formula Types to Identify:

\frac{\partial u}{\partial t} + \mathcal{L}u = f \quad \text{(general PDE form)} u|_{\partial\Omega} = g \quad \text{(Dirichlet BC)} \frac{\partial u}{\partial n}|_{\partial\Omega} = h \quad \text{(Neumann BC)}

1.2 Architecture Analysis (architecture_analysis)

Analyst Role: Software_Architect

Focus Areas:

• Module decomposition and dependency graph
• Data flow between computational stages
• I/O patterns (mesh input, solution output, checkpointing)
• Parallelism strategy (MPI, OpenMP, GPU)
• Build system and dependency management

Key Outputs:

• High-level component diagram
• Data flow diagram
• Technology stack inventory
• Parallelism strategy assessment

1.3 Validation Design (validation_design)

Analyst Role: Validation_Strategist

Focus Areas:

• Benchmark cases with known analytical solutions
• Manufactured solution methodology
• Grid convergence study design
• Key Performance Indicators (KPIs)
• Acceptance criteria definition

Key Outputs:

• Benchmark case catalog
• Validation methodology matrix
• KPI definitions with targets
• Acceptance test specifications

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2. Wave 2: Theoretical Foundation Dimensions

2.1 Formula Derivation (formula_derivation)

Analyst Role: Mathematician

Focus Areas:

• Strong-to-weak form transformation
• Discretization schemes (FEM, FDM, FVM, spectral)
• Variational formulations
• Linearization techniques (Newton, Picard)
• Stabilization methods (SUPG, GLS, VMS)

Key Formula Templates:

\text{Weak form: } a(u,v) = l(v) \quad \forall v \in V_h a(u,v) = \int_\Omega \nabla u \cdot \nabla v \, d\Omega l(v) = \int_\Omega f v \, d\Omega + \int_{\Gamma_N} g v \, dS

2.2 Convergence Analysis (convergence_analysis)

Analyst Role: Convergence_Analyst

Focus Areas:

• A priori error estimates
• A posteriori error estimators
• Convergence order verification
• Lax equivalence theorem applicability
• CFL conditions for time-dependent problems

Key Formula Templates:

\|u - u_h\|_{L^2} \leq C h^{k+1} |u|_{H^{k+1}} \quad \text{(optimal L2 rate)} \|u - u_h\|_{H^1} \leq C h^k |u|_{H^{k+1}} \quad \text{(optimal H1 rate)} \Delta t \leq \frac{C h}{\|v\|_\infty} \quad \text{(CFL condition)}

2.3 Complexity Analysis (complexity_analysis)

Analyst Role: Complexity_Analyst

Focus Areas:

• Assembly operation counts
• Solver complexity (direct vs iterative)
• Preconditioner cost analysis
• Memory scaling with problem size
• Communication overhead in parallel settings

Key Formula Templates:

T_{assembly} = O(N_{elem} \cdot p^{2d}) \quad \text{(FEM assembly)} T_{solve} = O(N^{3/2}) \quad \text{(2D direct)}, \quad O(N \log N) \quad \text{(multigrid)} M_{storage} = O(nnz) \quad \text{(sparse storage)}

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3. Wave 3: Algorithm Design Dimensions

3.1 Method Selection (method_selection)

Analyst Role: Algorithm_Designer

Focus Areas:

• Spatial discretization method selection
• Time integration scheme selection
• Linear/nonlinear solver selection
• Preconditioner selection
• Mesh generation strategy

Decision Criteria:

Criterion	Weight	Metrics
Accuracy order	High	Convergence rate, error bounds
Stability	High	Unconditional vs conditional, CFL
Efficiency	Medium	FLOPS per DOF, memory per DOF
Parallelizability	Medium	Communication-to-computation ratio
Implementation complexity	Low	Lines of code, library availability

3.2 Stability Analysis (stability_analysis)

Analyst Role: Stability_Analyst

Focus Areas:

• Von Neumann stability analysis
• Matrix condition numbers
• Amplification factors
• Inf-sup (LBB) stability for mixed methods
• Mesh-dependent stability bounds

Key Formula Templates:

\kappa(A) = \|A\| \cdot \|A^{-1}\| \quad \text{(condition number)} |g(\xi)| \leq 1 \quad \forall \xi \quad \text{(von Neumann stability)} \inf_{q_h \in Q_h} \sup_{v_h \in V_h} \frac{b(v_h, q_h)}{\|v_h\| \|q_h\|} \geq \beta > 0 \quad \text{(inf-sup)}

3.3 Performance Modeling (performance_modeling)

Analyst Role: Performance_Modeler

Focus Areas:

• Arithmetic intensity (FLOPS/byte)
• Roofline model analysis
• Strong/weak scaling prediction
• Memory bandwidth bottleneck identification
• Cache utilization estimates

Key Formula Templates:

AI = \frac{\text{FLOPS}}{\text{Bytes transferred}} \quad \text{(arithmetic intensity)} P_{max} = \min(P_{peak}, AI \times BW_{mem}) \quad \text{(roofline bound)} E_{parallel}(p) = \frac{T_1}{p \cdot T_p} \quad \text{(parallel efficiency)}

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4. Wave 4: Module Implementation Dimensions

4.1 Implementation Analysis (implementation_analysis)

Focus: Algorithm-to-code mapping, implementation correctness, coding patterns

4.2 Data Structure Review (data_structure_review)

Focus: Sparse matrix formats (CSR/CSC/COO), mesh data structures, memory layout optimization

4.3 Interface Analysis (interface_analysis)

Focus: Module APIs, data contracts between components, error handling patterns

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5. Wave 5: Local Function-Level Dimensions

5.1 Optimization (optimization)

Focus: Hotspot identification, vectorization opportunities, cache optimization, loop restructuring

5.2 Edge Case Analysis (edge_case_analysis)

Focus: Division by zero, matrix singularity, degenerate mesh elements, boundary layer singularities

5.3 Precision Audit (precision_audit)

Focus: Catastrophic cancellation, accumulation errors, mixed-precision opportunities, compensated algorithms

Critical Patterns to Detect:

Pattern	Risk	Mitigation
a - b where a ≈ b	Catastrophic cancellation	Reformulate or use higher precision
sum += small_value in loop	Accumulation error	Kahan summation
1.0/x where x → 0	Overflow/loss of significance	Guard with threshold
Mixed float32/float64	Silent precision loss	Explicit type annotations

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6. Wave 6: Integration & QA Dimensions

6.1 Integration Testing (integration_testing)

Focus: End-to-end test design, regression suite, manufactured solutions verification

6.2 Benchmarking (benchmarking)

Focus: Actual vs predicted performance, scalability tests, profiling results

6.3 Quality Assurance (quality_assurance)

Focus: All-phase synthesis, risk matrix, improvement roadmap, final recommendations