CASCADEDecember 2024

Singularity-Skirting MOO: CASCADE + GlobalMOO

CASCADE enables access to "danger zones" near physics singularities where standard MOO fails. 61.9% win rate across 21 simulations, up to 93.4% closer to singularities.

VALIDATED

CASCADE Singularity Advantage

61.9%

Win Rate

The bigeometric derivative D*[1/r] = -1 (constant) enables evaluation near singularities where classical calculus diverges. This transforms "danger zones" into accessible search space.

13/21

Wins

93.4%

Best Gain

10

Domains

k=-1

Optimal

View full CASCADE singularity proof →
VAN LEER DOMINATED10/30 Strictly Dominate

The Proof: 10 of 30 MOO Solutions Strictly Dominate Van Leer

Pareto dominance proof showing Van Leer is dominated by Meta+MOO solutions
+0.64%
L2 Error Improvement
+21.6%
TV (Oscillation) Improvement
DOMINATED
Van Leer Status

The Key Insight

Traditional CFD uses fixed flux limiters (Van Leer, Superbee, etc.) with hand-tuned parameters. Meta-calculus proposes adaptive k(gradient)that varies based on local solution properties.

Question: Can MOO discover a k(gradient) function that beats fixed methods on multiple objectives (accuracy + smoothness + efficiency)?

Shock Tube: Dual MOO Results

How It Works

1. k(gradient) Function

Instead of fixed k, use a sigmoid that transitions smoothly based on local gradient magnitude. Parameters: k_base, k_max, threshold, width.

2. MOO Search

NSGA-II explores 4D parameter space, evaluating each candidate on: L2 error, Total Variation, and function evaluations.

3. Pareto Front

After 200 generations, MOO finds 30 non-dominated solutions. Van Leer is dominated - MOO finds better tradeoffs.

MOO-Discovered Formula

k(grad) = 0.493 + 0.958 * sigmoid((grad - 2.09) / 0.34)

This formula was discovered by optimization, not hand-tuned. The parameters have physical interpretation:

k_base
0.493
smooth regions
k_max
0.958
shock regions
threshold
2.09
gradient cutoff
width
0.34
transition sharpness

Robertson Stiff ODE: Honest Results

Application Boundary

The Meta+MOO synergy is real but specific. Testing revealed clear boundaries for when this approach adds value.

+ USE FOR

  • -Shock capturing - Spatial discontinuities where k(gradient) naturally adapts
  • -Power-law singularities - Problems with 1/r, fat-tail behavior
  • -Multi-scale physics - Where k correlates with energy density (R^2=0.82)

X DO NOT USE FOR

  • -Stiff ODEs - LSODA beats meta-solver by 2400x (Robertson benchmark)
  • -Implicit-method problems - k-blending breaks solver convergence
  • -Eigenvalue stiffness - Global timescale separation needs different tools

Bottom line: For CFD/shock problems, Meta+MOO adds real value (Van Leer dominated). For stiff ODEs, use LSODA instead. See docs/APPLICATION-GUIDE.md for details.

Summary

What Works

  • + MOO discovers non-obvious k(gradient) functions
  • + Pareto front dominates fixed limiters (Van Leer, Superbee)
  • + Automated parameter discovery vs hand-tuning
  • + Dual MOO (pymoo + GlobalMOO) provides robustness

What Does NOT Work

  • X Robertson stiff ODE: 2400x worse than LSODA
  • X k-blending breaks implicit solver convergence
  • X Eigenvalue stiffness is fundamentally different
  • X This is an honest negative result

Data Sources

  • results/shock_tube_dual_moo.json - Shock tube MOO results
  • results/robertson_stiff_ode.json - Robertson benchmark
  • simulations/shock_tube_moo.py - Dual MOO implementation
  • simulations/robertson_stiff_ode.py - Stiff ODE benchmark