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Add comprehensive documentation for Numerical Analysis Workflow
- Introduced agent instruction template for task assignments in numerical analysis. - Defined CSV schema for tasks, including input, computed, and output columns. - Specified analysis dimensions across six phases of the workflow. - Established phase topology for the diamond deep tree structure of the workflow. - Outlined quality standards for assessing analysis reports, including criteria and quality gates.
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# Analysis Dimensions for Numerical Computation
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Defines the 18 analysis dimensions across 6 phases of the NADW workflow.
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## Purpose
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| Phase | Usage |
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|-------|-------|
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| Phase 1 (Decomposition) | Reference when assigning analysis_dimension to tasks |
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| Phase 2 (Execution) | Agents use to understand their analysis focus |
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| Phase 3 (Aggregation) | Organize findings by dimension |
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---
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## 1. Wave 1: Global Survey Dimensions
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### 1.1 Domain Modeling (`domain_modeling`)
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**Analyst Role**: Problem_Domain_Analyst
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**Focus Areas**:
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- Governing equations (PDEs, ODEs, integral equations)
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- Physical domain and boundary conditions
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- Conservation laws and constitutive relations
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- Problem classification (elliptic, parabolic, hyperbolic)
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- Dimensional analysis and non-dimensionalization
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**Key Outputs**:
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- Equation inventory with LaTeX notation
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- Boundary condition catalog
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- Problem classification matrix
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- Physical parameter ranges
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**Formula Types to Identify**:
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$$\frac{\partial u}{\partial t} + \mathcal{L}u = f \quad \text{(general PDE form)}$$
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$$u|_{\partial\Omega} = g \quad \text{(Dirichlet BC)}$$
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$$\frac{\partial u}{\partial n}|_{\partial\Omega} = h \quad \text{(Neumann BC)}$$
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### 1.2 Architecture Analysis (`architecture_analysis`)
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**Analyst Role**: Software_Architect
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**Focus Areas**:
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- Module decomposition and dependency graph
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- Data flow between computational stages
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- I/O patterns (mesh input, solution output, checkpointing)
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- Parallelism strategy (MPI, OpenMP, GPU)
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- Build system and dependency management
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**Key Outputs**:
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- High-level component diagram
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- Data flow diagram
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- Technology stack inventory
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- Parallelism strategy assessment
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### 1.3 Validation Design (`validation_design`)
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**Analyst Role**: Validation_Strategist
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**Focus Areas**:
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- Benchmark cases with known analytical solutions
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- Manufactured solution methodology
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- Grid convergence study design
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- Key Performance Indicators (KPIs)
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- Acceptance criteria definition
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**Key Outputs**:
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- Benchmark case catalog
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- Validation methodology matrix
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- KPI definitions with targets
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- Acceptance test specifications
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---
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## 2. Wave 2: Theoretical Foundation Dimensions
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### 2.1 Formula Derivation (`formula_derivation`)
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**Analyst Role**: Mathematician
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**Focus Areas**:
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- Strong-to-weak form transformation
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- Discretization schemes (FEM, FDM, FVM, spectral)
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- Variational formulations
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- Linearization techniques (Newton, Picard)
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- Stabilization methods (SUPG, GLS, VMS)
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**Key Formula Templates**:
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$$\text{Weak form: } a(u,v) = l(v) \quad \forall v \in V_h$$
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$$a(u,v) = \int_\Omega \nabla u \cdot \nabla v \, d\Omega$$
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$$l(v) = \int_\Omega f v \, d\Omega + \int_{\Gamma_N} g v \, dS$$
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### 2.2 Convergence Analysis (`convergence_analysis`)
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**Analyst Role**: Convergence_Analyst
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**Focus Areas**:
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- A priori error estimates
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- A posteriori error estimators
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- Convergence order verification
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- Lax equivalence theorem applicability
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- CFL conditions for time-dependent problems
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**Key Formula Templates**:
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$$\|u - u_h\|_{L^2} \leq C h^{k+1} |u|_{H^{k+1}} \quad \text{(optimal L2 rate)}$$
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$$\|u - u_h\|_{H^1} \leq C h^k |u|_{H^{k+1}} \quad \text{(optimal H1 rate)}$$
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$$\Delta t \leq \frac{C h}{\|v\|_\infty} \quad \text{(CFL condition)}$$
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### 2.3 Complexity Analysis (`complexity_analysis`)
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**Analyst Role**: Complexity_Analyst
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**Focus Areas**:
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- Assembly operation counts
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- Solver complexity (direct vs iterative)
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- Preconditioner cost analysis
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- Memory scaling with problem size
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- Communication overhead in parallel settings
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**Key Formula Templates**:
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$$T_{assembly} = O(N_{elem} \cdot p^{2d}) \quad \text{(FEM assembly)}$$
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$$T_{solve} = O(N^{3/2}) \quad \text{(2D direct)}, \quad O(N \log N) \quad \text{(multigrid)}$$
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$$M_{storage} = O(nnz) \quad \text{(sparse storage)}$$
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---
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## 3. Wave 3: Algorithm Design Dimensions
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### 3.1 Method Selection (`method_selection`)
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**Analyst Role**: Algorithm_Designer
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**Focus Areas**:
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- Spatial discretization method selection
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- Time integration scheme selection
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- Linear/nonlinear solver selection
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- Preconditioner selection
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- Mesh generation strategy
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**Decision Criteria**:
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| Criterion | Weight | Metrics |
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|-----------|--------|---------|
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| Accuracy order | High | Convergence rate, error bounds |
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| Stability | High | Unconditional vs conditional, CFL |
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| Efficiency | Medium | FLOPS per DOF, memory per DOF |
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| Parallelizability | Medium | Communication-to-computation ratio |
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| Implementation complexity | Low | Lines of code, library availability |
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### 3.2 Stability Analysis (`stability_analysis`)
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**Analyst Role**: Stability_Analyst
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**Focus Areas**:
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- Von Neumann stability analysis
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- Matrix condition numbers
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- Amplification factors
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- Inf-sup (LBB) stability for mixed methods
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- Mesh-dependent stability bounds
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**Key Formula Templates**:
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$$\kappa(A) = \|A\| \cdot \|A^{-1}\| \quad \text{(condition number)}$$
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$$|g(\xi)| \leq 1 \quad \forall \xi \quad \text{(von Neumann stability)}$$
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$$\inf_{q_h \in Q_h} \sup_{v_h \in V_h} \frac{b(v_h, q_h)}{\|v_h\| \|q_h\|} \geq \beta > 0 \quad \text{(inf-sup)}$$
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### 3.3 Performance Modeling (`performance_modeling`)
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**Analyst Role**: Performance_Modeler
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**Focus Areas**:
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- Arithmetic intensity (FLOPS/byte)
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- Roofline model analysis
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- Strong/weak scaling prediction
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- Memory bandwidth bottleneck identification
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- Cache utilization estimates
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**Key Formula Templates**:
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$$AI = \frac{\text{FLOPS}}{\text{Bytes transferred}} \quad \text{(arithmetic intensity)}$$
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$$P_{max} = \min(P_{peak}, AI \times BW_{mem}) \quad \text{(roofline bound)}$$
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$$E_{parallel}(p) = \frac{T_1}{p \cdot T_p} \quad \text{(parallel efficiency)}$$
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---
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## 4. Wave 4: Module Implementation Dimensions
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### 4.1 Implementation Analysis (`implementation_analysis`)
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**Focus**: Algorithm-to-code mapping, implementation correctness, coding patterns
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### 4.2 Data Structure Review (`data_structure_review`)
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**Focus**: Sparse matrix formats (CSR/CSC/COO), mesh data structures, memory layout optimization
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### 4.3 Interface Analysis (`interface_analysis`)
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**Focus**: Module APIs, data contracts between components, error handling patterns
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---
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## 5. Wave 5: Local Function-Level Dimensions
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### 5.1 Optimization (`optimization`)
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**Focus**: Hotspot identification, vectorization opportunities, cache optimization, loop restructuring
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### 5.2 Edge Case Analysis (`edge_case_analysis`)
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**Focus**: Division by zero, matrix singularity, degenerate mesh elements, boundary layer singularities
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### 5.3 Precision Audit (`precision_audit`)
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**Focus**: Catastrophic cancellation, accumulation errors, mixed-precision opportunities, compensated algorithms
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**Critical Patterns to Detect**:
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| Pattern | Risk | Mitigation |
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|---------|------|-----------|
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| `a - b` where `a ≈ b` | Catastrophic cancellation | Reformulate or use higher precision |
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| `sum += small_value` in loop | Accumulation error | Kahan summation |
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| `1.0/x` where `x → 0` | Overflow/loss of significance | Guard with threshold |
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| Mixed float32/float64 | Silent precision loss | Explicit type annotations |
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---
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## 6. Wave 6: Integration & QA Dimensions
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### 6.1 Integration Testing (`integration_testing`)
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**Focus**: End-to-end test design, regression suite, manufactured solutions verification
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### 6.2 Benchmarking (`benchmarking`)
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**Focus**: Actual vs predicted performance, scalability tests, profiling results
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### 6.3 Quality Assurance (`quality_assurance`)
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**Focus**: All-phase synthesis, risk matrix, improvement roadmap, final recommendations
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